Research Article JPD-Coloring of the Monohedral Tiling for the Plane S. A. El-Shehawy 1 and M. Basher 2 1 Department of Mathematics, Faculty of Science, Menoufia University, Shebin El-Kom 32511, Egypt 2 Department of Mathematics and Computer Science, Faculty of Science, Suez University, Suez 43518, Egypt Correspondence should be addressed to S. A. El-Shehawy; [email protected] Received 24 November 2014; Revised 5 January 2015; Accepted 19 January 2015 Academic Editor: Gaston Mandata N’gu´ er´ ekata Copyright © 2015 S. A. El-Shehawy and M. Basher. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We introduce a definition of coloring by using joint probability distribution “JPD-coloring” for the plane which is equipped by tiling I. We investigate the JPD-coloring of the r-monohedral tiling for the plane by mutually congruent regular convex polygons which are equilateral triangles at r = 3 or squares at r = 4 or regular hexagons at r = 6. Moreover we present some computations for determining the corresponding probability values which are used to color in the three studied cases by MAPLE-Package. 1. Introduction A tiling of the plane is a family of sets—called tiles—that cover the plane without gaps or overlaps. Tilings are known as tessellations or pavings; they have appeared in human activities since prehistoric times. eir mathematical theory is mostly elementary, but nevertheless it contains a rich supply of interesting problems at various levels. e same is true for the special class of tiling called tiling by regular polygons [1]. e notions of tiling by regular polygons in the plane are introduced by Gr¨ unbaum and Shephard in [2]. For more details see [3–5]. Definition 1 (see [1, 6]). A tiling of the plane is a collection I = { : = 1,2,3,...} of closed topological discs (tiles) which covers the Euclidean plane 2 and is such that the interiors of its tiles are disjoint. More explicitly, the union of the sets 1 , 2 , 3 ,..., tiles, is to be the whole plane, and the interiors of the sets are pairwise disjoint. We will restrict our interest to the case where each tile is a topological disc; that is, it has a boundary that is a single simple closed curve. Two tiles are called adjacent if they have an edge in common, and then each is called an adjacent of the other. Two distinct edges are adjacent if they have a common endpoint. e word incident is used to denote the relation of a tile to each of its edges or vertices and also of an edge to each of its endpoints. Two tilings I 1 and I 2 are congruent if I 1 may be made to coincide with I 2 by a rigid motion of the plane, possibly including reflection [6]. Definition 2 (see [1, 6]). A tiling is called edge-to-edge if the relation of any two tiles is one of the following three possibilities: (a) they are disjoint, (b) they have precisely one common point which is a vertex of each of the polygons, (c) they share a segment that is an edge of each of the two polygons. Definition 3 (see [6]). A regular tiling I will be called -monohedral tiling if every tile in I is congruent to one fixed set . e set is called the prototile of I, where is the number of vertices for each tile. Hence a point of the plane that is a vertex of one of the polygons in an edge-to-edge tiling is also a vertex of every other polygon to which it belongs and it is called a vertex of the tiling. Similarly, each edge of one of the polygons, regular tiling, is an edge of precisely one other polygon and it is called an edge of the tiling. It should be noted that the only possible edge-to-edge tilings of the plane by mutually congruent regular convex polygons are the three regular Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2015, Article ID 258436, 8 pages http://dx.doi.org/10.1155/2015/258436
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Research ArticleJPD-Coloring of the Monohedral Tiling for the Plane
S A El-Shehawy1 and M Basher2
1Department of Mathematics Faculty of Science Menoufia University Shebin El-Kom 32511 Egypt2Department of Mathematics and Computer Science Faculty of Science Suez University Suez 43518 Egypt
Correspondence should be addressed to S A El-Shehawy shshehawy64yahoocom
Received 24 November 2014 Revised 5 January 2015 Accepted 19 January 2015
Academic Editor Gaston Mandata Nrsquoguerekata
Copyright copy 2015 S A El-Shehawy and M Basher This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited
We introduce a definition of coloring by using joint probability distribution ldquoJPD-coloringrdquo for the plane which is equipped bytilingI We investigate the JPD-coloring of the r-monohedral tiling for the plane by mutually congruent regular convex polygonswhich are equilateral triangles at r = 3 or squares at r = 4 or regular hexagons at r = 6 Moreover we present some computations fordetermining the corresponding probability values which are used to color in the three studied cases by MAPLE-Package
1 Introduction
A tiling of the plane is a family of setsmdashcalled tilesmdashthatcover the plane without gaps or overlaps Tilings are knownas tessellations or pavings they have appeared in humanactivities since prehistoric times Their mathematical theoryis mostly elementary but nevertheless it contains a richsupply of interesting problems at various levels The sameis true for the special class of tiling called tiling by regularpolygons [1] The notions of tiling by regular polygons in theplane are introduced by Grunbaum and Shephard in [2] Formore details see [3ndash5]
Definition 1 (see [1 6]) A tiling of the plane is a collectionI = 119879
119904 119904 = 1 2 3 of closed topological discs (tiles)
which covers the Euclidean plane 119877
2 and is such that theinteriors of its tiles are disjoint
More explicitly the union of the sets 1198791 1198792 1198793 tiles
is to be the whole plane and the interiors of the sets 119879119904are
pairwise disjoint We will restrict our interest to the casewhere each tile is a topological disc that is it has a boundarythat is a single simple closed curve Two tiles are calledadjacent if they have an edge in common and then each iscalled an adjacent of the other Twodistinct edges are adjacentif they have a common endpointTheword incident is used todenote the relation of a tile to each of its edges or vertices and
also of an edge to each of its endpoints Two tilings I1and
I2are congruent if I
1may be made to coincide with I
2by
a rigid motion of the plane possibly including reflection [6]
Definition 2 (see [1 6]) A tiling is called edge-to-edge ifthe relation of any two tiles is one of the following threepossibilities
(a) they are disjoint(b) they have precisely one common point which is a
vertex of each of the polygons(c) they share a segment that is an edge of each of the two
polygons
Definition 3 (see [6]) A regular tiling I will be called119903-monohedral tiling if every tile in I is congruent to onefixed set 119879 The set 119879 is called the prototile of I where 119903 isthe number of vertices for each tile
Hence a point of the plane that is a vertex of one of thepolygons in an edge-to-edge tiling is also a vertex of everyother polygon to which it belongs and it is called a vertexof the tiling Similarly each edge of one of the polygonsregular tiling is an edge of precisely one other polygon andit is called an edge of the tiling It should be noted that theonly possible edge-to-edge tilings of the plane by mutuallycongruent regular convex polygons are the three regular
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2015 Article ID 258436 8 pageshttpdxdoiorg1011552015258436
2 Abstract and Applied Analysis
tilings by equilateral triangles by squares or by regularhexagons
The notions of the coloring of the monohedral tiling forthe plane have been introduced by Grunbaum and Shephard[2] The 120590-coloring and the perfect 120590-coloring for the planeequipped by the r-monohedral tilingI have been introducedby Basher [7]
In this paper we redefine the coloring of the r-monohedral tiling for the plane by using joint probabilitydistribution (JPD) We aim to investigate the three regulartilings by equilateral triangles squares and regular hexagonsusing JPD These three tilings are shown graphically andcomputationally Some computations by MAPLE-Packageto determine the probability values (vertices) for the threestudied tilings are presented We introduce this alternativetechnique to expand and update the coloring technique toimplement tiling according to a probabilistic approach Theprobability values refer to percentages in the coloring processand this contributes to convert the coloring process into acomputational process in the future
Throughout this paper we consider two discrete randomvariables 119883 and 119884 with a joint probability mass function119891119883119884
(119909 119910) = 119875(119883 = 119909 119884 = 119910) which satisfies
(1) 119891119883119884
(119909119894 119910119895) ge 0 for all points (119909
119894 119910119895) in the range of
(119883 119884)(2) sum119909119894
sum119910119895
119891119883119884
(119909119894 119910119895) = 1
The value 119891119883119884
(119909119894 119910119895) is usually written as 119901
119894119895for each point
(119909119894 119910119895) in the range of (119883 119884) see [8 9] In this paper we
consider 119901119894119895having equal denominators (the large common
multiplication of the denominators of the probabilities) ldquo119899rdquo
2 JPD-Coloring of the Regular Tilings
In this section we will investigate the coloring of 119903-monohedral tiling
Let 1198772 be equipped by 119903-monohedral tiling I and let119881(I) be the set of all vertices of the tiling Here we considerthe probability values 119901
119894119895to represent the coloring of the set
119881(I) as in the following definition where
119901119894119895= 119875 (119883 = 119909
119894 119884 = 119910
119895) = 119891119883119884
(119909119894 119910119895)
=
nonzero valuewith equal denominators 119899
if (119909119894 119910119895) isin 119881 (I)
zero value if (119909119894 119910119895) notin 119881 (I)
(1)
For each 119899 a family of a corresponding JPD is denoted byldquo119865(JPD)rdquo
Definition 4 A coloring of the tilingI is a partition of 119881(I)
into 119896 color-classes such that
(i) each color 119896 represents a probability value 119901119894119895
(ii) the different colors appear on adjacent vertices(iii) for each prototile 119879
119904isin I there exists a corresponding
JPD isin 119865(JPD)
p31
p22
p11
Figure 1 The used equilateral triangle with the corresponding JPDvalues
Definition 5 The set of tiles colored by 119865(JPD) is called themesh of tiling
From the above definition the tiling I can be colored byhorizontal or vertical translation of the mesh
Definition 6 The order 119874(119865(JPD)) of 119865(JPD) is the numberof JPDs which construct the mesh
21 JPD-Coloring of the 3-Monohedral Tiling Here we con-sider the JPD 119891
119883119884(119909119894 119910119895) = 119901
119894119895 119894 = 1 2 3 119895 = 1 2 with
different nonzero values of 11990111 11990131 11990122
and zero values of119901211199011211990132 where119901
119894119895is with equal denominators 119899The used
equilateral triangle is illustrated in Figure 1
Theorem 7 If the plane is equipped by 3-monohedral tilingthen the number of colors ldquo119888rdquo equals 3 where 119899 ge 6 If 119899 lt 6then the tiling cannot be colored
Proof Let 1198772 be equipped by equilateral triangle tiling Wegive the proof in two cases
Case 1 If 119899 lt 6 then the tiling cannot be colored (ie thenumber of colors 119888 equals 0) because we cannot find threedifferent probability values (JPD) to color the three verticesof the mentioned equilateral triangle tiling (say at 119899 = 5 theprobability values are 15 25 25 at 119899 = 4 the probabilityvalues are 14 14 24 and at 119899 = 3 the probability valuesare 13 13 13)
Case 2 If 119899 ge 6 then for each 119899 the number of colors 119888 equals3 and we can find three different probability values (JPD)which satisfied the condition (ii) in Definition 4 to color thethree vertices of the mentioned equilateral triangle tiling Wecan find the following
(i) at 119899 = 6 the different probability values are only16 26 36 (see Figure 2(a))
(ii) at 119899 = 7 the different probability values are only17 27 47
(iii) at 119899 ge 8 there are more than one JPD with threedifferent probability values (Figures 2(b) and 2(c))
Abstract and Applied Analysis 3
16
26 36
(a)
58
1828
(b)
48
18
38
(c)
Figure 2 Some JPD values at 119899 = 6 and 119899 = 8 to color the three vertices of the equilateral triangle tiling
p12 p22
p11 p21
Figure 3 The used square with the corresponding JPD values
Remark 8 119874(119865(JPD)) of the 3-monohedral tiling equals 1
22 JPD-Coloring of the 4-Monohedral Tiling Here we con-sider the following JPD 119891
119883119884(119909119894 119910119895) = 119901
119894119895 119894 = 1 2 119895 = 1 2
with the assumptions 11990111
= 11990121 11990111
= 11990112 11990122
= 11990121
11990122
= 11990112
and where 119901119894119895is with equal denominators 119899 The
used square is illustrated in Figure 3
Theorem 9 If the plane is equipped by the 4-monohedraltiling then the greatest number of colors ldquo119888rdquo is given as follows
119888 =
119899 minus 4 119899 ge 6 119886119899119889 119899 119894119904 119890119907119890119899
3 119899 = 7
119899 minus 5 119899 ge 9 119886119899119889 119899 119894119904 119900119889119889
(2)
and if 119899 lt 6 then the tiling cannot be colored
Proof Let 1198772 be equipped by square tiling We give the proofin four cases
Case 1 If 119899 lt 6 then the tiling cannot be colored (ie thegreatest number of colors 119888 equals 0) because we cannot findfour probability values (JPD) to color the four vertices of thementioned square tiling which satisfied (ii) in Definition 4
17
1727
37
Figure 4 The JPD values at 119899 = 7 to color the four vertices of thesquare triangle tiling
for coloring (say at 119899 = 5 the probability values are15 15 25 15 and at the minimum value 119899 = 4 theprobability values are 14 14 14 14)
Case 2 For 119899 = 7 take the corresponding probabilityvalues of two adjacent vertices 17 and 27 Then the restcorresponding probability values must be 17 and 37 whichsatisfied the condition (ii) in Definition 4 So the greatestnumber of colors ldquo119888rdquo equals 3 and the different probabilityvalues are only 17 27 37 see Figure 4
Case 3 For 119899 ge 6 and 119899 is even take the correspondingprobability values of two adjacent vertices 1119899 and 2119899Then the rest corresponding probability value is (119899 minus 3)119899This probability value must be distributed on the other twovertices as follows (119899 minus 4)119899 1119899 (119899 minus 5)119899 2119899 (119899 minus
119894)119899 (119894minus3)119899 where 119894 is an integer number and 4 le 119894 le 119899minus1This implies that the available total probability values to
obtain themesh are 1119899 2119899 (119899minus4)119899 (119899minus5)119899 (119899minus119894)119899where 4 le 119894 le 119899 minus 1 To avoid the repetition the last twoprobability values (119899 minus 3)119899 (119899 minus 1)119899 are excluded In thiscase we obtain the following
(i) For 119899 = 6 the available total probability values16 26 (6 minus 4)6 (6 minus 5)6 are equivalent to16 26 and the greatest number of colors 119888 equals2 (ie 119899 minus 4)
4 Abstract and Applied Analysis
Table 1 The relation between 119896 119899 119874(119865(JPD)) and the 119865(JPD) of the square tiling at 119899 = 13
119896 119874(F(JPD)) Example of the corresponding 119865(JPD)3 1 513 213 113 513
4 3 213 113 113 913 113 213 913 113 213 813 113 213
5 3 213 113 113 913 113 213 913 113 213 313 113 713
6 5 213 113 113 913 113 213 913 113 213 813 113 213813 213 213 113 213 313 113 713
7 5 213 113 113 913 113 213 913 113 213 313 113 713313 213 713 113 213 613 113 413
8 7 213 113 113 913 113 213 913 113 213 813 113 213813 213 213 113 213 313 113 713 313 213 713 113 213 413 113 613
614
314
814
1014514414
714214 214 914
214
214
114
214
114114
114214
114114
(a)
213
213
213 613
413
813113
113 113913
213
113
213
113
713
313
(b)
Figure 5 Some JPD values at 119899 = 13 and 119899 = 14 to color the four vertices of the square tiling
(ii) For 119899 ge 8 and 119899 is even the available total probabilityvalues (without repetition) are 1119899 2119899 (119899minus4)119899 (119899minus5)119899 (119899 minus 119894)119899 where 4 le 119894 le 119899 minus 3 Thegreatest number of colors 119888 equals (119899minus4) for exampleat 119899 = 8 the available total probability valuesare 18 28 38 48 and so 119888 equals 4 (ie 119899 minus
4) at 119899 = 14 the available total probability val-ues are 114 214 314 414 514 614 714 814914 1014 and so the greatest number of colors 119888equals 10 (ie 119899 minus 4) see Figure 5(a)
Case 4 For 119899 ge 9 and 119899 is odd the proof is similar toCase 3 Since 119899 is odd then in this case the rest correspondingprobability value (119899 minus 3)119899 has even value of its numeratorThen this probability value can be distributed on the othertwo vertices by two equal probability values ldquo((119899 minus 3)2)119899rdquo
As in Case 3 the last two probability values (119899minus3)119899 (119899minus1)119899 and ((119899 minus 3)2)119899 are excluded In this case we obtainthat for 119899 ge 9 and 119899 is odd the available total probabilityvalues are 1119899 2119899 (119899minus4)119899 (119899minus5)119899 (((119899minus3)2)+1)119899(((119899 minus 3)2) minus 1)119899 (119899 minus 119894)119899 where 4 le 119894 le 119899 minus 3 Thegreatest number of colors 119888 equals (119899 minus 5) for example
(i) at 119899 = 9 the available total distinct probability valuesare 19 29 49 59 and the greatest number ofcolors 119888 equals 4 (ie 119899 minus 5)
(ii) at 119899 = 13 the total distinct probability values are113 213 313 413 613 713 813 913 andthe greatest number of colors 119888 equals 8 (ie 119899 minus 5)see Figure 5(b)
p23
p12
p21 p31
p42
p33
Figure 6 The used regular hexagon with the corresponding JPDvalues
There are a relation between 119896 119899 119874(119865(JPD)) and thecorresponding 119865(JPD) Tables 1 and 2 show this relation
(i) for 119899 = 13 see Table 1
(ii) for 119899 = 14 see Table 2
Corollary 10 The smallest number of colors for square tilingis 2 if 119899 is even and 3 if 119899 is odd
23 JPD-Coloring of the 6-Monohedral Tiling Here we con-sider the JPD 119891
119883119884(119909119894 119910119895) = 119901119894119895 119894 = 1 2 3 4 119895 = 1 2 3 with
the assumptions 11990121
= 11990131 11990131
= 11990142 11990142
= 11990133 11990133
= 11990123
11990123
= 11990112 11990112
= 11990121(where 119901
119894119895is with equal denominators 119899)
and zero values of 11990111 11990141 11990122
11990132 11990113 11990143 The used regular
hexagon is illustrated in Figure 6
Abstract and Applied Analysis 5
Table 2 The relation between 119896 119899 119874(119865(JPD)) and the 119865(JPD) of the square tiling at 119899 = 14
119896 119874(119865(JPD)) Example of the corresponding 119865(JPD)2 414 314 314 414
3 1 814 114 114 414
4 214 714 114 414
5 3 314 114 114 914 114 314 914 114 314 414 114 614
6 4 614 114 114 614 114 514 614 214 514 114 214 614114 414 614 314
7 5214 714 114 414 714 214 414 114 214 614 114 514 614 214 514 114214 114 114 1014
8 214 714 114 414 714 214 414 114 214 614 114 514 614 214 514 114214 814 114 314
9 7 214 714 114 414 714 214 414 114 214 614 114 514 614 214 514 114214 114 114 1014 114 214 114 1014 214 814 114 314
10 9214 714 114 414 714 214 414 114 214 614 114 514 614 214 514 114214 114 114 1014 114 214 1014 114 214 914 114 214 914 214 214 114214 814 114 314
118
118
118
118
118118
118818
118
118
118 118 118
118
118
118118118
118118418
418 418
418
418
418
418518
518518718 718
718
718
618518
118 518
618
618618
618
618
418
118 118
118
118
818
118
318318
318
318
318318318
318
318 318 318
318 918
318
118
618
418218
218
218 218
218
218
218 218
218
218
218
218
218
218
218 218
218 218
218
218
218
218
218
218818
218
218
1018
1118
318
318
Figure 7 The JPD values at 119899 = 18 to color the six vertices of the regular hexagon tiling
Theorem 11 If the plane is equipped by 6-monohedral tilingthen the greatest number of colors ldquo119888rdquo is given as 119888 = 119899minus7 where119899 ge 9 If 119899 lt 9 then the tiling cannot be colored
Proof Let 1198772 be equipped by hexagon tiling The proof canbe given as follows
Case 1 If 119899 lt 9 then the tiling cannot be colored (ie thegreatest number of colors 119888 equals 0) because we cannot findsix probability values (JPD) which satisfied the condition (ii)in Definition 4 to color the six vertices of the mentioned
hexagon tiling (say at 119899 = 8 the available probabilityvalues are 18 28 18 28 18 18 at 119899 = 7 the availableprobability values are 17 27 17 17 17 17 and at thesmallest value 119899 = 6 the available probability values are16 16 16 16 16 16)
Case 2 For 119899 ge 9 take the corresponding probability valueof a vertex (119899 minus 119894)119899 where 119894 ge 7 because it is impossible totake the value of 119894 less than 7 Then the rest correspondingprobability value is 119894119899 This probability value must bedistributed on the other five vertices under consideration of
6 Abstract and Applied Analysis
125
125
425
325
325
225
625
225
425
425
324 325 225
225225
225
225
725 325 325
225625225
125 825
525
525525
725225
325425
425625
625
925
525 325 825
725425
825
325225 625 425
325
225
325 325
125
125125
125125 125
325
325
325
325
325
525525
525625
325125
125
125
125
125
125
125125
125125
125
125125
125
125
125325
325
625
725
125
125 125
125125
125
125
125
125125125
125125
225
225 225
225 225
225
225
225
225
225225
225
225225525
425
225
225225325
325
425
425
425
125
125
125
1025
1325
1625
1224
1425
1025
10251025 1725
1225
1325
1225
1125
1225
1525
1825
1325 525
Figure 8 The JPD values at 119899 = 25 to color the six vertices of the regular hexagon tiling
the conditions in Definition 4 for an integer number 119894 ge
7 This implies that the available total probability values toobtain the mesh under consideration of the conditions inDefinition 4 are 1119899 2119899 3119899 (119899 minus 119894 minus 1)119899 where 7 le
119894 le 119899 minus 2 Avoiding the repetition we obtain the following
(i) for 119899 = 9 the total probability values are 19 29 and119888 equals 2
(ii) for 119899 = 18 the total probability values are 118 218318 418 518 618 718 818 918 1018 1118 and119888 equals 11 (see Figure 7)
(iii) for 119899 = 25 the total probability values are 125 225325 425 525 625 725 825 925 1025 11251225 1325 1425 1525 1625 1725 1825 and 119888
equals 18 (see Figure 8)
In this case the greatest number of colors 119888 equals 119899 minus 7
Corollary 12 The relation between 119896 and 119899 can be shown inTable 3
Table 3The relation between 119896 and 119899 of the regular hexagon tiling
119896 119899
2 6119903 + 3 119903 ge 1
3 6119903 + 3 6119903 + 4 6119903 + 5 119903 ge 1
6119903 6119903 + 1 6119903 + 2 119903 ge 2
4 6119903 6119903 + 1 6119903 + 2 6119903 + 3 6119903 + 4 6119903 + 5 119903 ge 2
5 6119903 + 5 119903 ge 2
6119903 6119903 + 1 6119903 + 2 6119903 + 3 6119903 + 4 119903 ge 3
6 6119903 + 3 6119903 + 4 6119903 + 5 119903 ge 3
6119903 6119903 + 1 6119903 + 2 119903 ge 4
Appendix
As a MAPLE programming guide see [10]
Determination of the Probability Values by Using the MAPLEProgram
Case A1 (JPD-coloring of the 3-monohedral tiling) SeeBox 1
Case A2 (JPD-coloring of the 4-monohedral tiling) SeeBox 2
Abstract and Applied Analysis 7
gt restart with(linalg)
Probability Values=proc(nposintiposintjposint)
local AbTrvC global PV
A[T]= matrix([[111]]) b= vector([1]) C=sum(V[w]w=12)
linsolve(A[T]blsquor1015840) r linsolve(A[T]blsquor1015840v)PV[T[nij]]=subs(v[1]=inv[2]=jnv[3]=1-C)
end proc
n=3i=1j=1 PV[T[nij]]=Probability Values(nij)
n=4i=1j=2 PV[T[nij]]=Probability Values(nij)
n=5i=1j=2 PV[T[nij]]=Probability Values(nij)
n=6i=1j=2 PV[T[nij]]=Probability Values(nij)
n=7i=1j=2 PV[T[nij]]=Probability Values(nij)
n=8i=1j=2 PV[T[nij]]=Probability Values(nij)
n=8i=1j=3 PV[T[nij]]=Probability Values(nij)
PV119879311
= [
1
3
1
3
1
3
] PV119879412
= [
1
2
1
4
1
4
] PV119879512
= [
2
5
1
5
2
5
] PV119879612
= [
1
2
1
6
1
3
]
PV119879712
= [
2
7
4
7
1
7
] PV119879812
= [
5
8
1
8
1
4
] PV119879813
= [
1
2
1
8
3
8
]
Box 1
gt restart with(linalg)
Probability Values=proc(nposintiposintjposintkposint)
local AbFrvC global PV
A[F]=matrix([[1111]]) b=vector([1]) C=sum(V[w]w=13)
linsolve(A[F]blsquor1015840) r linsolve(A[F]blsquor1015840v)PV[S[nijk]]=subs(v[1]=inv[2]=jnv[3]=knv[4]=1-C)
end proc
n=4 i=1j=1k=1 PV[F[nijk]]=Probability Values(nijk)
n=5 i=1j=2k=1 PV[F[nijk]]=Probability Values(nijk)
n=6 i=1j=2k=1 PV[F[nijk]]=Probability Values(nijk)
n=7 i=1j=2k=1 PV[F[nijk]]=Probability Values(nijk)
n=8 i=1j=2k=1 PV[F[nijk]]=Probability Values(nijk)
n=8 i=1j=2k=3 PV[F[nijk]]=Probability Values(nijk)
n=8 i=1j=3k=1 PV[F[nijk]]=Probability Values(nijk)
n=9 i=1j=2k=1 PV[F[nijk]]=Probability Values(nijk)
n=9 i=1j=4k=3 PV[F[nijk]]=Probability Values(nijk)
n=13i=1j=2k=1 PV[F[nijk]]=Probability Values(nijk)
n=13i=1j=3k=4 PV[F[nijk]]=Probability Values(nijk)
n=13i=1j=1k=6 PV[F[nijk]]=Probability Values(nijk)
n=13i=1j=1k=7 PV[F[nijk]]=Probability Values(nijk)
n=13i=1j=1k=8 PV[F[nijk]]=Probability Values(nijk)
n=14i=1j=1k=2 PV[F[nijk]]=Probability Values(nijk)
n=14i=2j=3k=4 PV[F[nijk]]=Probability Values(nijk)
n=14i=1j=5k=6 PV[F[nijk]]=Probability Values(nijk)
n=14i=1j=2k=7 PV[F[nijk]]=Probability Values(nijk)
n=14i=1j=2k=8 PV[F[nijk]]=Probability Values(nijk)
n=14i=2j=2k=9 PV[F[nijk]]=Probability Values(nijk)
PV1198654111
= [
1
4
1
4
1
4
1
4
] PV1198655121
= [
1
5
1
5
2
5
1
5
] PV1198656121
= [
1
6
1
3
1
6
1
3
] PV1198657121
= [
3
7
1
7
2
7
1
7
]
PV1198658121
= [
1
8
1
4
1
2
1
8
] PV1198658123
= [
1
4
1
8
1
4
3
8
] PV1198658131
= [
3
8
1
8
3
8
1
8
] PV1198659121
= [
5
9
1
9
2
9
1
9
]
PV1198659143
= [
1
3
4
9
1
9
1
9
] PV11986513121
= [
9
13
1
13
1
13
2
13
] PV11986513134
= [
5
13
4
13
3
13
1
13
] PV11986513116
= [
1
13
6
13
5
13
1
13
]
PV11986513117
= [
1
13
7
13
4
13
1
13
] PV11986513118
= [
8
13
1
13
1
13
3
13
] PV11986514112
= [
1
7
5
7
1
14
1
14
] PV11986514234
= [
1
7
3
14
2
7
5
14
]
PV11986514156
= [
1
14
5
14
3
7
1
7
] PV11986514127
= [
2
7
1
14
1
7
1
4
] PV11986514128
= [
1
14
1
7
4
7
3
14
] PV11986514229
= [
1
14
1
7
1
7
9
14
]
Box 2
8 Abstract and Applied Analysis
gt restart with(linalg)
Probability Values=proc(nposintiposintjposintkposintlposintmposint)
local AbSrvC global PV
A[S]=matrix([[111111]]) b=vector([1]) C=sum(v[w]w=15)
linsolve(A[S]blsquor1015840) r linsolve(A[S]blsquor1015840v)PV[S[nijklm]]=subs(v[1]=inv[2]=jnv[3]=knv[4]=lnv[5]=mnv[6]=1-C)
end proc
n=6 i=1j=1k=1l=1m=1 PV[S[nijklm]]=Probability Values(nijklm)
n=7 i=1j=2k=1l=1m=1 PV[S[nijklm]]=Probability Values(nijklm)
n=8 i=1j=2k=1l=2m=1 PV[S[nijklm]]=Probability Values(nijklm)
n=9 i=1j=2k=1l=2m=1 PV[S[nijklm]]=Probability Values(nijklm)
n=18i=1j=2k=3l=4m=6 PV[S[nijklm]]=Probability Values(nijklm)
n=25i=1j=2k=3l=4m=5 PV[S[nijklm]]=Probability Values(nijklm)
n=25i=1j=2k=3l=4m=5 PV[S[nijklm]]=Probability Values(nijklm)
PV119878611111
= [
1
6
1
6
1
6
1
6
1
6
1
6
] PV119878712111
= [
1
7
1
7
2
7
1
7
1
7
1
7
] PV119878812121
= [
1
4
1
8
1
4
1
8
1
8
1
8
]
PV119878912121
= [
2
9
1
9
2
9
1
9
2
9
1
9
] PV1198781812346
= [
1
9
1
18
1
9
1
6
2
9
1
3
] PV1198782512345
= [
1
5
4
25
3
25
2
25
1
25
2
25
]
PV1198782512345
= [
1
5
4
25
2
5
3
25
2
25
1
25
]
Box 3
Case A3 (JPD-coloring of the 6-monohedral tiling) SeeBox 3
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] B Grunbaum and G C Shephard ldquoTiling by regular polygonsrdquoMathematics Magazine vol 50 no 5 pp 227ndash247 1977
[2] B Grunbaum and G C Shephard ldquoPerfect colorings of transi-tive tilings and patterns in the planerdquoDiscrete Mathematics vol20 no 3 pp 235ndash247 1977
[3] S Eigen J Navarro andV S Prasad ldquoAn aperiodic tiling using adynamical system and Beatty sequencesrdquo in Dynamics ErgodicTheory andGeometry vol 54 ofMathematical Sciences ResearchInstitute pp 223ndash241 Cambridge University Press 2007
[4] C Mann L Asaro J Hyde M Jensen and T Schroeder ldquoUni-form edge-119888-colorings of the Archimedean tilingsrdquo DiscreteMathematics vol 338 no 1 pp 10ndash22 2015
[5] R Santos andR Felix ldquoPerfect precise colorings of plane regulartilingsrdquo Zeitschrift fur Kristallographie vol 226 no 9 pp 726ndash730 2011
[6] B Grunbaum P Mani-Levitska and G C Shephard ldquoTilingthree-dimensional space with polyhedral tiles of a given iso-morphism typerdquo Journal of the London Mathematical Societyvol 29 no 1 pp 181ndash191 1984
[7] M E Basher ldquo120590-Coloring of the monohedral tilingrdquo Interna-tional Journal of Mathematical Combinatorics vol 2 pp 46ndash522009
[8] J E Freund IMiller andMMillerMathematical Statistics withApplications Prentice Hall PTR 7th edition 2003
[9] D C Montgomery and G C Runger Applied Statistics andProbability for Engineers John Wiley amp Sons New York NYUSA 2nd edition 1999
[10] L Bernardin P Chin P DeMarco et al Maple ProgrammingGuide Maplesoft 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Abstract and Applied Analysis
tilings by equilateral triangles by squares or by regularhexagons
The notions of the coloring of the monohedral tiling forthe plane have been introduced by Grunbaum and Shephard[2] The 120590-coloring and the perfect 120590-coloring for the planeequipped by the r-monohedral tilingI have been introducedby Basher [7]
In this paper we redefine the coloring of the r-monohedral tiling for the plane by using joint probabilitydistribution (JPD) We aim to investigate the three regulartilings by equilateral triangles squares and regular hexagonsusing JPD These three tilings are shown graphically andcomputationally Some computations by MAPLE-Packageto determine the probability values (vertices) for the threestudied tilings are presented We introduce this alternativetechnique to expand and update the coloring technique toimplement tiling according to a probabilistic approach Theprobability values refer to percentages in the coloring processand this contributes to convert the coloring process into acomputational process in the future
Throughout this paper we consider two discrete randomvariables 119883 and 119884 with a joint probability mass function119891119883119884
(119909 119910) = 119875(119883 = 119909 119884 = 119910) which satisfies
(1) 119891119883119884
(119909119894 119910119895) ge 0 for all points (119909
119894 119910119895) in the range of
(119883 119884)(2) sum119909119894
sum119910119895
119891119883119884
(119909119894 119910119895) = 1
The value 119891119883119884
(119909119894 119910119895) is usually written as 119901
119894119895for each point
(119909119894 119910119895) in the range of (119883 119884) see [8 9] In this paper we
consider 119901119894119895having equal denominators (the large common
multiplication of the denominators of the probabilities) ldquo119899rdquo
2 JPD-Coloring of the Regular Tilings
In this section we will investigate the coloring of 119903-monohedral tiling
Let 1198772 be equipped by 119903-monohedral tiling I and let119881(I) be the set of all vertices of the tiling Here we considerthe probability values 119901
119894119895to represent the coloring of the set
119881(I) as in the following definition where
119901119894119895= 119875 (119883 = 119909
119894 119884 = 119910
119895) = 119891119883119884
(119909119894 119910119895)
=
nonzero valuewith equal denominators 119899
if (119909119894 119910119895) isin 119881 (I)
zero value if (119909119894 119910119895) notin 119881 (I)
(1)
For each 119899 a family of a corresponding JPD is denoted byldquo119865(JPD)rdquo
Definition 4 A coloring of the tilingI is a partition of 119881(I)
into 119896 color-classes such that
(i) each color 119896 represents a probability value 119901119894119895
(ii) the different colors appear on adjacent vertices(iii) for each prototile 119879
119904isin I there exists a corresponding
JPD isin 119865(JPD)
p31
p22
p11
Figure 1 The used equilateral triangle with the corresponding JPDvalues
Definition 5 The set of tiles colored by 119865(JPD) is called themesh of tiling
From the above definition the tiling I can be colored byhorizontal or vertical translation of the mesh
Definition 6 The order 119874(119865(JPD)) of 119865(JPD) is the numberof JPDs which construct the mesh
21 JPD-Coloring of the 3-Monohedral Tiling Here we con-sider the JPD 119891
119883119884(119909119894 119910119895) = 119901
119894119895 119894 = 1 2 3 119895 = 1 2 with
different nonzero values of 11990111 11990131 11990122
and zero values of119901211199011211990132 where119901
119894119895is with equal denominators 119899The used
equilateral triangle is illustrated in Figure 1
Theorem 7 If the plane is equipped by 3-monohedral tilingthen the number of colors ldquo119888rdquo equals 3 where 119899 ge 6 If 119899 lt 6then the tiling cannot be colored
Proof Let 1198772 be equipped by equilateral triangle tiling Wegive the proof in two cases
Case 1 If 119899 lt 6 then the tiling cannot be colored (ie thenumber of colors 119888 equals 0) because we cannot find threedifferent probability values (JPD) to color the three verticesof the mentioned equilateral triangle tiling (say at 119899 = 5 theprobability values are 15 25 25 at 119899 = 4 the probabilityvalues are 14 14 24 and at 119899 = 3 the probability valuesare 13 13 13)
Case 2 If 119899 ge 6 then for each 119899 the number of colors 119888 equals3 and we can find three different probability values (JPD)which satisfied the condition (ii) in Definition 4 to color thethree vertices of the mentioned equilateral triangle tiling Wecan find the following
(i) at 119899 = 6 the different probability values are only16 26 36 (see Figure 2(a))
(ii) at 119899 = 7 the different probability values are only17 27 47
(iii) at 119899 ge 8 there are more than one JPD with threedifferent probability values (Figures 2(b) and 2(c))
Abstract and Applied Analysis 3
16
26 36
(a)
58
1828
(b)
48
18
38
(c)
Figure 2 Some JPD values at 119899 = 6 and 119899 = 8 to color the three vertices of the equilateral triangle tiling
p12 p22
p11 p21
Figure 3 The used square with the corresponding JPD values
Remark 8 119874(119865(JPD)) of the 3-monohedral tiling equals 1
22 JPD-Coloring of the 4-Monohedral Tiling Here we con-sider the following JPD 119891
119883119884(119909119894 119910119895) = 119901
119894119895 119894 = 1 2 119895 = 1 2
with the assumptions 11990111
= 11990121 11990111
= 11990112 11990122
= 11990121
11990122
= 11990112
and where 119901119894119895is with equal denominators 119899 The
used square is illustrated in Figure 3
Theorem 9 If the plane is equipped by the 4-monohedraltiling then the greatest number of colors ldquo119888rdquo is given as follows
119888 =
119899 minus 4 119899 ge 6 119886119899119889 119899 119894119904 119890119907119890119899
3 119899 = 7
119899 minus 5 119899 ge 9 119886119899119889 119899 119894119904 119900119889119889
(2)
and if 119899 lt 6 then the tiling cannot be colored
Proof Let 1198772 be equipped by square tiling We give the proofin four cases
Case 1 If 119899 lt 6 then the tiling cannot be colored (ie thegreatest number of colors 119888 equals 0) because we cannot findfour probability values (JPD) to color the four vertices of thementioned square tiling which satisfied (ii) in Definition 4
17
1727
37
Figure 4 The JPD values at 119899 = 7 to color the four vertices of thesquare triangle tiling
for coloring (say at 119899 = 5 the probability values are15 15 25 15 and at the minimum value 119899 = 4 theprobability values are 14 14 14 14)
Case 2 For 119899 = 7 take the corresponding probabilityvalues of two adjacent vertices 17 and 27 Then the restcorresponding probability values must be 17 and 37 whichsatisfied the condition (ii) in Definition 4 So the greatestnumber of colors ldquo119888rdquo equals 3 and the different probabilityvalues are only 17 27 37 see Figure 4
Case 3 For 119899 ge 6 and 119899 is even take the correspondingprobability values of two adjacent vertices 1119899 and 2119899Then the rest corresponding probability value is (119899 minus 3)119899This probability value must be distributed on the other twovertices as follows (119899 minus 4)119899 1119899 (119899 minus 5)119899 2119899 (119899 minus
119894)119899 (119894minus3)119899 where 119894 is an integer number and 4 le 119894 le 119899minus1This implies that the available total probability values to
obtain themesh are 1119899 2119899 (119899minus4)119899 (119899minus5)119899 (119899minus119894)119899where 4 le 119894 le 119899 minus 1 To avoid the repetition the last twoprobability values (119899 minus 3)119899 (119899 minus 1)119899 are excluded In thiscase we obtain the following
(i) For 119899 = 6 the available total probability values16 26 (6 minus 4)6 (6 minus 5)6 are equivalent to16 26 and the greatest number of colors 119888 equals2 (ie 119899 minus 4)
4 Abstract and Applied Analysis
Table 1 The relation between 119896 119899 119874(119865(JPD)) and the 119865(JPD) of the square tiling at 119899 = 13
119896 119874(F(JPD)) Example of the corresponding 119865(JPD)3 1 513 213 113 513
4 3 213 113 113 913 113 213 913 113 213 813 113 213
5 3 213 113 113 913 113 213 913 113 213 313 113 713
6 5 213 113 113 913 113 213 913 113 213 813 113 213813 213 213 113 213 313 113 713
7 5 213 113 113 913 113 213 913 113 213 313 113 713313 213 713 113 213 613 113 413
8 7 213 113 113 913 113 213 913 113 213 813 113 213813 213 213 113 213 313 113 713 313 213 713 113 213 413 113 613
614
314
814
1014514414
714214 214 914
214
214
114
214
114114
114214
114114
(a)
213
213
213 613
413
813113
113 113913
213
113
213
113
713
313
(b)
Figure 5 Some JPD values at 119899 = 13 and 119899 = 14 to color the four vertices of the square tiling
(ii) For 119899 ge 8 and 119899 is even the available total probabilityvalues (without repetition) are 1119899 2119899 (119899minus4)119899 (119899minus5)119899 (119899 minus 119894)119899 where 4 le 119894 le 119899 minus 3 Thegreatest number of colors 119888 equals (119899minus4) for exampleat 119899 = 8 the available total probability valuesare 18 28 38 48 and so 119888 equals 4 (ie 119899 minus
4) at 119899 = 14 the available total probability val-ues are 114 214 314 414 514 614 714 814914 1014 and so the greatest number of colors 119888equals 10 (ie 119899 minus 4) see Figure 5(a)
Case 4 For 119899 ge 9 and 119899 is odd the proof is similar toCase 3 Since 119899 is odd then in this case the rest correspondingprobability value (119899 minus 3)119899 has even value of its numeratorThen this probability value can be distributed on the othertwo vertices by two equal probability values ldquo((119899 minus 3)2)119899rdquo
As in Case 3 the last two probability values (119899minus3)119899 (119899minus1)119899 and ((119899 minus 3)2)119899 are excluded In this case we obtainthat for 119899 ge 9 and 119899 is odd the available total probabilityvalues are 1119899 2119899 (119899minus4)119899 (119899minus5)119899 (((119899minus3)2)+1)119899(((119899 minus 3)2) minus 1)119899 (119899 minus 119894)119899 where 4 le 119894 le 119899 minus 3 Thegreatest number of colors 119888 equals (119899 minus 5) for example
(i) at 119899 = 9 the available total distinct probability valuesare 19 29 49 59 and the greatest number ofcolors 119888 equals 4 (ie 119899 minus 5)
(ii) at 119899 = 13 the total distinct probability values are113 213 313 413 613 713 813 913 andthe greatest number of colors 119888 equals 8 (ie 119899 minus 5)see Figure 5(b)
p23
p12
p21 p31
p42
p33
Figure 6 The used regular hexagon with the corresponding JPDvalues
There are a relation between 119896 119899 119874(119865(JPD)) and thecorresponding 119865(JPD) Tables 1 and 2 show this relation
(i) for 119899 = 13 see Table 1
(ii) for 119899 = 14 see Table 2
Corollary 10 The smallest number of colors for square tilingis 2 if 119899 is even and 3 if 119899 is odd
23 JPD-Coloring of the 6-Monohedral Tiling Here we con-sider the JPD 119891
119883119884(119909119894 119910119895) = 119901119894119895 119894 = 1 2 3 4 119895 = 1 2 3 with
the assumptions 11990121
= 11990131 11990131
= 11990142 11990142
= 11990133 11990133
= 11990123
11990123
= 11990112 11990112
= 11990121(where 119901
119894119895is with equal denominators 119899)
and zero values of 11990111 11990141 11990122
11990132 11990113 11990143 The used regular
hexagon is illustrated in Figure 6
Abstract and Applied Analysis 5
Table 2 The relation between 119896 119899 119874(119865(JPD)) and the 119865(JPD) of the square tiling at 119899 = 14
119896 119874(119865(JPD)) Example of the corresponding 119865(JPD)2 414 314 314 414
3 1 814 114 114 414
4 214 714 114 414
5 3 314 114 114 914 114 314 914 114 314 414 114 614
6 4 614 114 114 614 114 514 614 214 514 114 214 614114 414 614 314
7 5214 714 114 414 714 214 414 114 214 614 114 514 614 214 514 114214 114 114 1014
8 214 714 114 414 714 214 414 114 214 614 114 514 614 214 514 114214 814 114 314
9 7 214 714 114 414 714 214 414 114 214 614 114 514 614 214 514 114214 114 114 1014 114 214 114 1014 214 814 114 314
10 9214 714 114 414 714 214 414 114 214 614 114 514 614 214 514 114214 114 114 1014 114 214 1014 114 214 914 114 214 914 214 214 114214 814 114 314
118
118
118
118
118118
118818
118
118
118 118 118
118
118
118118118
118118418
418 418
418
418
418
418518
518518718 718
718
718
618518
118 518
618
618618
618
618
418
118 118
118
118
818
118
318318
318
318
318318318
318
318 318 318
318 918
318
118
618
418218
218
218 218
218
218
218 218
218
218
218
218
218
218
218 218
218 218
218
218
218
218
218
218818
218
218
1018
1118
318
318
Figure 7 The JPD values at 119899 = 18 to color the six vertices of the regular hexagon tiling
Theorem 11 If the plane is equipped by 6-monohedral tilingthen the greatest number of colors ldquo119888rdquo is given as 119888 = 119899minus7 where119899 ge 9 If 119899 lt 9 then the tiling cannot be colored
Proof Let 1198772 be equipped by hexagon tiling The proof canbe given as follows
Case 1 If 119899 lt 9 then the tiling cannot be colored (ie thegreatest number of colors 119888 equals 0) because we cannot findsix probability values (JPD) which satisfied the condition (ii)in Definition 4 to color the six vertices of the mentioned
hexagon tiling (say at 119899 = 8 the available probabilityvalues are 18 28 18 28 18 18 at 119899 = 7 the availableprobability values are 17 27 17 17 17 17 and at thesmallest value 119899 = 6 the available probability values are16 16 16 16 16 16)
Case 2 For 119899 ge 9 take the corresponding probability valueof a vertex (119899 minus 119894)119899 where 119894 ge 7 because it is impossible totake the value of 119894 less than 7 Then the rest correspondingprobability value is 119894119899 This probability value must bedistributed on the other five vertices under consideration of
6 Abstract and Applied Analysis
125
125
425
325
325
225
625
225
425
425
324 325 225
225225
225
225
725 325 325
225625225
125 825
525
525525
725225
325425
425625
625
925
525 325 825
725425
825
325225 625 425
325
225
325 325
125
125125
125125 125
325
325
325
325
325
525525
525625
325125
125
125
125
125
125
125125
125125
125
125125
125
125
125325
325
625
725
125
125 125
125125
125
125
125
125125125
125125
225
225 225
225 225
225
225
225
225
225225
225
225225525
425
225
225225325
325
425
425
425
125
125
125
1025
1325
1625
1224
1425
1025
10251025 1725
1225
1325
1225
1125
1225
1525
1825
1325 525
Figure 8 The JPD values at 119899 = 25 to color the six vertices of the regular hexagon tiling
the conditions in Definition 4 for an integer number 119894 ge
7 This implies that the available total probability values toobtain the mesh under consideration of the conditions inDefinition 4 are 1119899 2119899 3119899 (119899 minus 119894 minus 1)119899 where 7 le
119894 le 119899 minus 2 Avoiding the repetition we obtain the following
(i) for 119899 = 9 the total probability values are 19 29 and119888 equals 2
(ii) for 119899 = 18 the total probability values are 118 218318 418 518 618 718 818 918 1018 1118 and119888 equals 11 (see Figure 7)
(iii) for 119899 = 25 the total probability values are 125 225325 425 525 625 725 825 925 1025 11251225 1325 1425 1525 1625 1725 1825 and 119888
equals 18 (see Figure 8)
In this case the greatest number of colors 119888 equals 119899 minus 7
Corollary 12 The relation between 119896 and 119899 can be shown inTable 3
Table 3The relation between 119896 and 119899 of the regular hexagon tiling
119896 119899
2 6119903 + 3 119903 ge 1
3 6119903 + 3 6119903 + 4 6119903 + 5 119903 ge 1
6119903 6119903 + 1 6119903 + 2 119903 ge 2
4 6119903 6119903 + 1 6119903 + 2 6119903 + 3 6119903 + 4 6119903 + 5 119903 ge 2
5 6119903 + 5 119903 ge 2
6119903 6119903 + 1 6119903 + 2 6119903 + 3 6119903 + 4 119903 ge 3
6 6119903 + 3 6119903 + 4 6119903 + 5 119903 ge 3
6119903 6119903 + 1 6119903 + 2 119903 ge 4
Appendix
As a MAPLE programming guide see [10]
Determination of the Probability Values by Using the MAPLEProgram
Case A1 (JPD-coloring of the 3-monohedral tiling) SeeBox 1
Case A2 (JPD-coloring of the 4-monohedral tiling) SeeBox 2
Abstract and Applied Analysis 7
gt restart with(linalg)
Probability Values=proc(nposintiposintjposint)
local AbTrvC global PV
A[T]= matrix([[111]]) b= vector([1]) C=sum(V[w]w=12)
linsolve(A[T]blsquor1015840) r linsolve(A[T]blsquor1015840v)PV[T[nij]]=subs(v[1]=inv[2]=jnv[3]=1-C)
end proc
n=3i=1j=1 PV[T[nij]]=Probability Values(nij)
n=4i=1j=2 PV[T[nij]]=Probability Values(nij)
n=5i=1j=2 PV[T[nij]]=Probability Values(nij)
n=6i=1j=2 PV[T[nij]]=Probability Values(nij)
n=7i=1j=2 PV[T[nij]]=Probability Values(nij)
n=8i=1j=2 PV[T[nij]]=Probability Values(nij)
n=8i=1j=3 PV[T[nij]]=Probability Values(nij)
PV119879311
= [
1
3
1
3
1
3
] PV119879412
= [
1
2
1
4
1
4
] PV119879512
= [
2
5
1
5
2
5
] PV119879612
= [
1
2
1
6
1
3
]
PV119879712
= [
2
7
4
7
1
7
] PV119879812
= [
5
8
1
8
1
4
] PV119879813
= [
1
2
1
8
3
8
]
Box 1
gt restart with(linalg)
Probability Values=proc(nposintiposintjposintkposint)
local AbFrvC global PV
A[F]=matrix([[1111]]) b=vector([1]) C=sum(V[w]w=13)
linsolve(A[F]blsquor1015840) r linsolve(A[F]blsquor1015840v)PV[S[nijk]]=subs(v[1]=inv[2]=jnv[3]=knv[4]=1-C)
end proc
n=4 i=1j=1k=1 PV[F[nijk]]=Probability Values(nijk)
n=5 i=1j=2k=1 PV[F[nijk]]=Probability Values(nijk)
n=6 i=1j=2k=1 PV[F[nijk]]=Probability Values(nijk)
n=7 i=1j=2k=1 PV[F[nijk]]=Probability Values(nijk)
n=8 i=1j=2k=1 PV[F[nijk]]=Probability Values(nijk)
n=8 i=1j=2k=3 PV[F[nijk]]=Probability Values(nijk)
n=8 i=1j=3k=1 PV[F[nijk]]=Probability Values(nijk)
n=9 i=1j=2k=1 PV[F[nijk]]=Probability Values(nijk)
n=9 i=1j=4k=3 PV[F[nijk]]=Probability Values(nijk)
n=13i=1j=2k=1 PV[F[nijk]]=Probability Values(nijk)
n=13i=1j=3k=4 PV[F[nijk]]=Probability Values(nijk)
n=13i=1j=1k=6 PV[F[nijk]]=Probability Values(nijk)
n=13i=1j=1k=7 PV[F[nijk]]=Probability Values(nijk)
n=13i=1j=1k=8 PV[F[nijk]]=Probability Values(nijk)
n=14i=1j=1k=2 PV[F[nijk]]=Probability Values(nijk)
n=14i=2j=3k=4 PV[F[nijk]]=Probability Values(nijk)
n=14i=1j=5k=6 PV[F[nijk]]=Probability Values(nijk)
n=14i=1j=2k=7 PV[F[nijk]]=Probability Values(nijk)
n=14i=1j=2k=8 PV[F[nijk]]=Probability Values(nijk)
n=14i=2j=2k=9 PV[F[nijk]]=Probability Values(nijk)
PV1198654111
= [
1
4
1
4
1
4
1
4
] PV1198655121
= [
1
5
1
5
2
5
1
5
] PV1198656121
= [
1
6
1
3
1
6
1
3
] PV1198657121
= [
3
7
1
7
2
7
1
7
]
PV1198658121
= [
1
8
1
4
1
2
1
8
] PV1198658123
= [
1
4
1
8
1
4
3
8
] PV1198658131
= [
3
8
1
8
3
8
1
8
] PV1198659121
= [
5
9
1
9
2
9
1
9
]
PV1198659143
= [
1
3
4
9
1
9
1
9
] PV11986513121
= [
9
13
1
13
1
13
2
13
] PV11986513134
= [
5
13
4
13
3
13
1
13
] PV11986513116
= [
1
13
6
13
5
13
1
13
]
PV11986513117
= [
1
13
7
13
4
13
1
13
] PV11986513118
= [
8
13
1
13
1
13
3
13
] PV11986514112
= [
1
7
5
7
1
14
1
14
] PV11986514234
= [
1
7
3
14
2
7
5
14
]
PV11986514156
= [
1
14
5
14
3
7
1
7
] PV11986514127
= [
2
7
1
14
1
7
1
4
] PV11986514128
= [
1
14
1
7
4
7
3
14
] PV11986514229
= [
1
14
1
7
1
7
9
14
]
Box 2
8 Abstract and Applied Analysis
gt restart with(linalg)
Probability Values=proc(nposintiposintjposintkposintlposintmposint)
local AbSrvC global PV
A[S]=matrix([[111111]]) b=vector([1]) C=sum(v[w]w=15)
linsolve(A[S]blsquor1015840) r linsolve(A[S]blsquor1015840v)PV[S[nijklm]]=subs(v[1]=inv[2]=jnv[3]=knv[4]=lnv[5]=mnv[6]=1-C)
end proc
n=6 i=1j=1k=1l=1m=1 PV[S[nijklm]]=Probability Values(nijklm)
n=7 i=1j=2k=1l=1m=1 PV[S[nijklm]]=Probability Values(nijklm)
n=8 i=1j=2k=1l=2m=1 PV[S[nijklm]]=Probability Values(nijklm)
n=9 i=1j=2k=1l=2m=1 PV[S[nijklm]]=Probability Values(nijklm)
n=18i=1j=2k=3l=4m=6 PV[S[nijklm]]=Probability Values(nijklm)
n=25i=1j=2k=3l=4m=5 PV[S[nijklm]]=Probability Values(nijklm)
n=25i=1j=2k=3l=4m=5 PV[S[nijklm]]=Probability Values(nijklm)
PV119878611111
= [
1
6
1
6
1
6
1
6
1
6
1
6
] PV119878712111
= [
1
7
1
7
2
7
1
7
1
7
1
7
] PV119878812121
= [
1
4
1
8
1
4
1
8
1
8
1
8
]
PV119878912121
= [
2
9
1
9
2
9
1
9
2
9
1
9
] PV1198781812346
= [
1
9
1
18
1
9
1
6
2
9
1
3
] PV1198782512345
= [
1
5
4
25
3
25
2
25
1
25
2
25
]
PV1198782512345
= [
1
5
4
25
2
5
3
25
2
25
1
25
]
Box 3
Case A3 (JPD-coloring of the 6-monohedral tiling) SeeBox 3
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] B Grunbaum and G C Shephard ldquoTiling by regular polygonsrdquoMathematics Magazine vol 50 no 5 pp 227ndash247 1977
[2] B Grunbaum and G C Shephard ldquoPerfect colorings of transi-tive tilings and patterns in the planerdquoDiscrete Mathematics vol20 no 3 pp 235ndash247 1977
[3] S Eigen J Navarro andV S Prasad ldquoAn aperiodic tiling using adynamical system and Beatty sequencesrdquo in Dynamics ErgodicTheory andGeometry vol 54 ofMathematical Sciences ResearchInstitute pp 223ndash241 Cambridge University Press 2007
[4] C Mann L Asaro J Hyde M Jensen and T Schroeder ldquoUni-form edge-119888-colorings of the Archimedean tilingsrdquo DiscreteMathematics vol 338 no 1 pp 10ndash22 2015
[5] R Santos andR Felix ldquoPerfect precise colorings of plane regulartilingsrdquo Zeitschrift fur Kristallographie vol 226 no 9 pp 726ndash730 2011
[6] B Grunbaum P Mani-Levitska and G C Shephard ldquoTilingthree-dimensional space with polyhedral tiles of a given iso-morphism typerdquo Journal of the London Mathematical Societyvol 29 no 1 pp 181ndash191 1984
[7] M E Basher ldquo120590-Coloring of the monohedral tilingrdquo Interna-tional Journal of Mathematical Combinatorics vol 2 pp 46ndash522009
[8] J E Freund IMiller andMMillerMathematical Statistics withApplications Prentice Hall PTR 7th edition 2003
[9] D C Montgomery and G C Runger Applied Statistics andProbability for Engineers John Wiley amp Sons New York NYUSA 2nd edition 1999
[10] L Bernardin P Chin P DeMarco et al Maple ProgrammingGuide Maplesoft 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 3
16
26 36
(a)
58
1828
(b)
48
18
38
(c)
Figure 2 Some JPD values at 119899 = 6 and 119899 = 8 to color the three vertices of the equilateral triangle tiling
p12 p22
p11 p21
Figure 3 The used square with the corresponding JPD values
Remark 8 119874(119865(JPD)) of the 3-monohedral tiling equals 1
22 JPD-Coloring of the 4-Monohedral Tiling Here we con-sider the following JPD 119891
119883119884(119909119894 119910119895) = 119901
119894119895 119894 = 1 2 119895 = 1 2
with the assumptions 11990111
= 11990121 11990111
= 11990112 11990122
= 11990121
11990122
= 11990112
and where 119901119894119895is with equal denominators 119899 The
used square is illustrated in Figure 3
Theorem 9 If the plane is equipped by the 4-monohedraltiling then the greatest number of colors ldquo119888rdquo is given as follows
119888 =
119899 minus 4 119899 ge 6 119886119899119889 119899 119894119904 119890119907119890119899
3 119899 = 7
119899 minus 5 119899 ge 9 119886119899119889 119899 119894119904 119900119889119889
(2)
and if 119899 lt 6 then the tiling cannot be colored
Proof Let 1198772 be equipped by square tiling We give the proofin four cases
Case 1 If 119899 lt 6 then the tiling cannot be colored (ie thegreatest number of colors 119888 equals 0) because we cannot findfour probability values (JPD) to color the four vertices of thementioned square tiling which satisfied (ii) in Definition 4
17
1727
37
Figure 4 The JPD values at 119899 = 7 to color the four vertices of thesquare triangle tiling
for coloring (say at 119899 = 5 the probability values are15 15 25 15 and at the minimum value 119899 = 4 theprobability values are 14 14 14 14)
Case 2 For 119899 = 7 take the corresponding probabilityvalues of two adjacent vertices 17 and 27 Then the restcorresponding probability values must be 17 and 37 whichsatisfied the condition (ii) in Definition 4 So the greatestnumber of colors ldquo119888rdquo equals 3 and the different probabilityvalues are only 17 27 37 see Figure 4
Case 3 For 119899 ge 6 and 119899 is even take the correspondingprobability values of two adjacent vertices 1119899 and 2119899Then the rest corresponding probability value is (119899 minus 3)119899This probability value must be distributed on the other twovertices as follows (119899 minus 4)119899 1119899 (119899 minus 5)119899 2119899 (119899 minus
119894)119899 (119894minus3)119899 where 119894 is an integer number and 4 le 119894 le 119899minus1This implies that the available total probability values to
obtain themesh are 1119899 2119899 (119899minus4)119899 (119899minus5)119899 (119899minus119894)119899where 4 le 119894 le 119899 minus 1 To avoid the repetition the last twoprobability values (119899 minus 3)119899 (119899 minus 1)119899 are excluded In thiscase we obtain the following
(i) For 119899 = 6 the available total probability values16 26 (6 minus 4)6 (6 minus 5)6 are equivalent to16 26 and the greatest number of colors 119888 equals2 (ie 119899 minus 4)
4 Abstract and Applied Analysis
Table 1 The relation between 119896 119899 119874(119865(JPD)) and the 119865(JPD) of the square tiling at 119899 = 13
119896 119874(F(JPD)) Example of the corresponding 119865(JPD)3 1 513 213 113 513
4 3 213 113 113 913 113 213 913 113 213 813 113 213
5 3 213 113 113 913 113 213 913 113 213 313 113 713
6 5 213 113 113 913 113 213 913 113 213 813 113 213813 213 213 113 213 313 113 713
7 5 213 113 113 913 113 213 913 113 213 313 113 713313 213 713 113 213 613 113 413
8 7 213 113 113 913 113 213 913 113 213 813 113 213813 213 213 113 213 313 113 713 313 213 713 113 213 413 113 613
614
314
814
1014514414
714214 214 914
214
214
114
214
114114
114214
114114
(a)
213
213
213 613
413
813113
113 113913
213
113
213
113
713
313
(b)
Figure 5 Some JPD values at 119899 = 13 and 119899 = 14 to color the four vertices of the square tiling
(ii) For 119899 ge 8 and 119899 is even the available total probabilityvalues (without repetition) are 1119899 2119899 (119899minus4)119899 (119899minus5)119899 (119899 minus 119894)119899 where 4 le 119894 le 119899 minus 3 Thegreatest number of colors 119888 equals (119899minus4) for exampleat 119899 = 8 the available total probability valuesare 18 28 38 48 and so 119888 equals 4 (ie 119899 minus
4) at 119899 = 14 the available total probability val-ues are 114 214 314 414 514 614 714 814914 1014 and so the greatest number of colors 119888equals 10 (ie 119899 minus 4) see Figure 5(a)
Case 4 For 119899 ge 9 and 119899 is odd the proof is similar toCase 3 Since 119899 is odd then in this case the rest correspondingprobability value (119899 minus 3)119899 has even value of its numeratorThen this probability value can be distributed on the othertwo vertices by two equal probability values ldquo((119899 minus 3)2)119899rdquo
As in Case 3 the last two probability values (119899minus3)119899 (119899minus1)119899 and ((119899 minus 3)2)119899 are excluded In this case we obtainthat for 119899 ge 9 and 119899 is odd the available total probabilityvalues are 1119899 2119899 (119899minus4)119899 (119899minus5)119899 (((119899minus3)2)+1)119899(((119899 minus 3)2) minus 1)119899 (119899 minus 119894)119899 where 4 le 119894 le 119899 minus 3 Thegreatest number of colors 119888 equals (119899 minus 5) for example
(i) at 119899 = 9 the available total distinct probability valuesare 19 29 49 59 and the greatest number ofcolors 119888 equals 4 (ie 119899 minus 5)
(ii) at 119899 = 13 the total distinct probability values are113 213 313 413 613 713 813 913 andthe greatest number of colors 119888 equals 8 (ie 119899 minus 5)see Figure 5(b)
p23
p12
p21 p31
p42
p33
Figure 6 The used regular hexagon with the corresponding JPDvalues
There are a relation between 119896 119899 119874(119865(JPD)) and thecorresponding 119865(JPD) Tables 1 and 2 show this relation
(i) for 119899 = 13 see Table 1
(ii) for 119899 = 14 see Table 2
Corollary 10 The smallest number of colors for square tilingis 2 if 119899 is even and 3 if 119899 is odd
23 JPD-Coloring of the 6-Monohedral Tiling Here we con-sider the JPD 119891
119883119884(119909119894 119910119895) = 119901119894119895 119894 = 1 2 3 4 119895 = 1 2 3 with
the assumptions 11990121
= 11990131 11990131
= 11990142 11990142
= 11990133 11990133
= 11990123
11990123
= 11990112 11990112
= 11990121(where 119901
119894119895is with equal denominators 119899)
and zero values of 11990111 11990141 11990122
11990132 11990113 11990143 The used regular
hexagon is illustrated in Figure 6
Abstract and Applied Analysis 5
Table 2 The relation between 119896 119899 119874(119865(JPD)) and the 119865(JPD) of the square tiling at 119899 = 14
119896 119874(119865(JPD)) Example of the corresponding 119865(JPD)2 414 314 314 414
3 1 814 114 114 414
4 214 714 114 414
5 3 314 114 114 914 114 314 914 114 314 414 114 614
6 4 614 114 114 614 114 514 614 214 514 114 214 614114 414 614 314
7 5214 714 114 414 714 214 414 114 214 614 114 514 614 214 514 114214 114 114 1014
8 214 714 114 414 714 214 414 114 214 614 114 514 614 214 514 114214 814 114 314
9 7 214 714 114 414 714 214 414 114 214 614 114 514 614 214 514 114214 114 114 1014 114 214 114 1014 214 814 114 314
10 9214 714 114 414 714 214 414 114 214 614 114 514 614 214 514 114214 114 114 1014 114 214 1014 114 214 914 114 214 914 214 214 114214 814 114 314
118
118
118
118
118118
118818
118
118
118 118 118
118
118
118118118
118118418
418 418
418
418
418
418518
518518718 718
718
718
618518
118 518
618
618618
618
618
418
118 118
118
118
818
118
318318
318
318
318318318
318
318 318 318
318 918
318
118
618
418218
218
218 218
218
218
218 218
218
218
218
218
218
218
218 218
218 218
218
218
218
218
218
218818
218
218
1018
1118
318
318
Figure 7 The JPD values at 119899 = 18 to color the six vertices of the regular hexagon tiling
Theorem 11 If the plane is equipped by 6-monohedral tilingthen the greatest number of colors ldquo119888rdquo is given as 119888 = 119899minus7 where119899 ge 9 If 119899 lt 9 then the tiling cannot be colored
Proof Let 1198772 be equipped by hexagon tiling The proof canbe given as follows
Case 1 If 119899 lt 9 then the tiling cannot be colored (ie thegreatest number of colors 119888 equals 0) because we cannot findsix probability values (JPD) which satisfied the condition (ii)in Definition 4 to color the six vertices of the mentioned
hexagon tiling (say at 119899 = 8 the available probabilityvalues are 18 28 18 28 18 18 at 119899 = 7 the availableprobability values are 17 27 17 17 17 17 and at thesmallest value 119899 = 6 the available probability values are16 16 16 16 16 16)
Case 2 For 119899 ge 9 take the corresponding probability valueof a vertex (119899 minus 119894)119899 where 119894 ge 7 because it is impossible totake the value of 119894 less than 7 Then the rest correspondingprobability value is 119894119899 This probability value must bedistributed on the other five vertices under consideration of
6 Abstract and Applied Analysis
125
125
425
325
325
225
625
225
425
425
324 325 225
225225
225
225
725 325 325
225625225
125 825
525
525525
725225
325425
425625
625
925
525 325 825
725425
825
325225 625 425
325
225
325 325
125
125125
125125 125
325
325
325
325
325
525525
525625
325125
125
125
125
125
125
125125
125125
125
125125
125
125
125325
325
625
725
125
125 125
125125
125
125
125
125125125
125125
225
225 225
225 225
225
225
225
225
225225
225
225225525
425
225
225225325
325
425
425
425
125
125
125
1025
1325
1625
1224
1425
1025
10251025 1725
1225
1325
1225
1125
1225
1525
1825
1325 525
Figure 8 The JPD values at 119899 = 25 to color the six vertices of the regular hexagon tiling
the conditions in Definition 4 for an integer number 119894 ge
7 This implies that the available total probability values toobtain the mesh under consideration of the conditions inDefinition 4 are 1119899 2119899 3119899 (119899 minus 119894 minus 1)119899 where 7 le
119894 le 119899 minus 2 Avoiding the repetition we obtain the following
(i) for 119899 = 9 the total probability values are 19 29 and119888 equals 2
(ii) for 119899 = 18 the total probability values are 118 218318 418 518 618 718 818 918 1018 1118 and119888 equals 11 (see Figure 7)
(iii) for 119899 = 25 the total probability values are 125 225325 425 525 625 725 825 925 1025 11251225 1325 1425 1525 1625 1725 1825 and 119888
equals 18 (see Figure 8)
In this case the greatest number of colors 119888 equals 119899 minus 7
Corollary 12 The relation between 119896 and 119899 can be shown inTable 3
Table 3The relation between 119896 and 119899 of the regular hexagon tiling
119896 119899
2 6119903 + 3 119903 ge 1
3 6119903 + 3 6119903 + 4 6119903 + 5 119903 ge 1
6119903 6119903 + 1 6119903 + 2 119903 ge 2
4 6119903 6119903 + 1 6119903 + 2 6119903 + 3 6119903 + 4 6119903 + 5 119903 ge 2
5 6119903 + 5 119903 ge 2
6119903 6119903 + 1 6119903 + 2 6119903 + 3 6119903 + 4 119903 ge 3
6 6119903 + 3 6119903 + 4 6119903 + 5 119903 ge 3
6119903 6119903 + 1 6119903 + 2 119903 ge 4
Appendix
As a MAPLE programming guide see [10]
Determination of the Probability Values by Using the MAPLEProgram
Case A1 (JPD-coloring of the 3-monohedral tiling) SeeBox 1
Case A2 (JPD-coloring of the 4-monohedral tiling) SeeBox 2
Abstract and Applied Analysis 7
gt restart with(linalg)
Probability Values=proc(nposintiposintjposint)
local AbTrvC global PV
A[T]= matrix([[111]]) b= vector([1]) C=sum(V[w]w=12)
linsolve(A[T]blsquor1015840) r linsolve(A[T]blsquor1015840v)PV[T[nij]]=subs(v[1]=inv[2]=jnv[3]=1-C)
end proc
n=3i=1j=1 PV[T[nij]]=Probability Values(nij)
n=4i=1j=2 PV[T[nij]]=Probability Values(nij)
n=5i=1j=2 PV[T[nij]]=Probability Values(nij)
n=6i=1j=2 PV[T[nij]]=Probability Values(nij)
n=7i=1j=2 PV[T[nij]]=Probability Values(nij)
n=8i=1j=2 PV[T[nij]]=Probability Values(nij)
n=8i=1j=3 PV[T[nij]]=Probability Values(nij)
PV119879311
= [
1
3
1
3
1
3
] PV119879412
= [
1
2
1
4
1
4
] PV119879512
= [
2
5
1
5
2
5
] PV119879612
= [
1
2
1
6
1
3
]
PV119879712
= [
2
7
4
7
1
7
] PV119879812
= [
5
8
1
8
1
4
] PV119879813
= [
1
2
1
8
3
8
]
Box 1
gt restart with(linalg)
Probability Values=proc(nposintiposintjposintkposint)
local AbFrvC global PV
A[F]=matrix([[1111]]) b=vector([1]) C=sum(V[w]w=13)
linsolve(A[F]blsquor1015840) r linsolve(A[F]blsquor1015840v)PV[S[nijk]]=subs(v[1]=inv[2]=jnv[3]=knv[4]=1-C)
end proc
n=4 i=1j=1k=1 PV[F[nijk]]=Probability Values(nijk)
n=5 i=1j=2k=1 PV[F[nijk]]=Probability Values(nijk)
n=6 i=1j=2k=1 PV[F[nijk]]=Probability Values(nijk)
n=7 i=1j=2k=1 PV[F[nijk]]=Probability Values(nijk)
n=8 i=1j=2k=1 PV[F[nijk]]=Probability Values(nijk)
n=8 i=1j=2k=3 PV[F[nijk]]=Probability Values(nijk)
n=8 i=1j=3k=1 PV[F[nijk]]=Probability Values(nijk)
n=9 i=1j=2k=1 PV[F[nijk]]=Probability Values(nijk)
n=9 i=1j=4k=3 PV[F[nijk]]=Probability Values(nijk)
n=13i=1j=2k=1 PV[F[nijk]]=Probability Values(nijk)
n=13i=1j=3k=4 PV[F[nijk]]=Probability Values(nijk)
n=13i=1j=1k=6 PV[F[nijk]]=Probability Values(nijk)
n=13i=1j=1k=7 PV[F[nijk]]=Probability Values(nijk)
n=13i=1j=1k=8 PV[F[nijk]]=Probability Values(nijk)
n=14i=1j=1k=2 PV[F[nijk]]=Probability Values(nijk)
n=14i=2j=3k=4 PV[F[nijk]]=Probability Values(nijk)
n=14i=1j=5k=6 PV[F[nijk]]=Probability Values(nijk)
n=14i=1j=2k=7 PV[F[nijk]]=Probability Values(nijk)
n=14i=1j=2k=8 PV[F[nijk]]=Probability Values(nijk)
n=14i=2j=2k=9 PV[F[nijk]]=Probability Values(nijk)
PV1198654111
= [
1
4
1
4
1
4
1
4
] PV1198655121
= [
1
5
1
5
2
5
1
5
] PV1198656121
= [
1
6
1
3
1
6
1
3
] PV1198657121
= [
3
7
1
7
2
7
1
7
]
PV1198658121
= [
1
8
1
4
1
2
1
8
] PV1198658123
= [
1
4
1
8
1
4
3
8
] PV1198658131
= [
3
8
1
8
3
8
1
8
] PV1198659121
= [
5
9
1
9
2
9
1
9
]
PV1198659143
= [
1
3
4
9
1
9
1
9
] PV11986513121
= [
9
13
1
13
1
13
2
13
] PV11986513134
= [
5
13
4
13
3
13
1
13
] PV11986513116
= [
1
13
6
13
5
13
1
13
]
PV11986513117
= [
1
13
7
13
4
13
1
13
] PV11986513118
= [
8
13
1
13
1
13
3
13
] PV11986514112
= [
1
7
5
7
1
14
1
14
] PV11986514234
= [
1
7
3
14
2
7
5
14
]
PV11986514156
= [
1
14
5
14
3
7
1
7
] PV11986514127
= [
2
7
1
14
1
7
1
4
] PV11986514128
= [
1
14
1
7
4
7
3
14
] PV11986514229
= [
1
14
1
7
1
7
9
14
]
Box 2
8 Abstract and Applied Analysis
gt restart with(linalg)
Probability Values=proc(nposintiposintjposintkposintlposintmposint)
local AbSrvC global PV
A[S]=matrix([[111111]]) b=vector([1]) C=sum(v[w]w=15)
linsolve(A[S]blsquor1015840) r linsolve(A[S]blsquor1015840v)PV[S[nijklm]]=subs(v[1]=inv[2]=jnv[3]=knv[4]=lnv[5]=mnv[6]=1-C)
end proc
n=6 i=1j=1k=1l=1m=1 PV[S[nijklm]]=Probability Values(nijklm)
n=7 i=1j=2k=1l=1m=1 PV[S[nijklm]]=Probability Values(nijklm)
n=8 i=1j=2k=1l=2m=1 PV[S[nijklm]]=Probability Values(nijklm)
n=9 i=1j=2k=1l=2m=1 PV[S[nijklm]]=Probability Values(nijklm)
n=18i=1j=2k=3l=4m=6 PV[S[nijklm]]=Probability Values(nijklm)
n=25i=1j=2k=3l=4m=5 PV[S[nijklm]]=Probability Values(nijklm)
n=25i=1j=2k=3l=4m=5 PV[S[nijklm]]=Probability Values(nijklm)
PV119878611111
= [
1
6
1
6
1
6
1
6
1
6
1
6
] PV119878712111
= [
1
7
1
7
2
7
1
7
1
7
1
7
] PV119878812121
= [
1
4
1
8
1
4
1
8
1
8
1
8
]
PV119878912121
= [
2
9
1
9
2
9
1
9
2
9
1
9
] PV1198781812346
= [
1
9
1
18
1
9
1
6
2
9
1
3
] PV1198782512345
= [
1
5
4
25
3
25
2
25
1
25
2
25
]
PV1198782512345
= [
1
5
4
25
2
5
3
25
2
25
1
25
]
Box 3
Case A3 (JPD-coloring of the 6-monohedral tiling) SeeBox 3
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] B Grunbaum and G C Shephard ldquoTiling by regular polygonsrdquoMathematics Magazine vol 50 no 5 pp 227ndash247 1977
[2] B Grunbaum and G C Shephard ldquoPerfect colorings of transi-tive tilings and patterns in the planerdquoDiscrete Mathematics vol20 no 3 pp 235ndash247 1977
[3] S Eigen J Navarro andV S Prasad ldquoAn aperiodic tiling using adynamical system and Beatty sequencesrdquo in Dynamics ErgodicTheory andGeometry vol 54 ofMathematical Sciences ResearchInstitute pp 223ndash241 Cambridge University Press 2007
[4] C Mann L Asaro J Hyde M Jensen and T Schroeder ldquoUni-form edge-119888-colorings of the Archimedean tilingsrdquo DiscreteMathematics vol 338 no 1 pp 10ndash22 2015
[5] R Santos andR Felix ldquoPerfect precise colorings of plane regulartilingsrdquo Zeitschrift fur Kristallographie vol 226 no 9 pp 726ndash730 2011
[6] B Grunbaum P Mani-Levitska and G C Shephard ldquoTilingthree-dimensional space with polyhedral tiles of a given iso-morphism typerdquo Journal of the London Mathematical Societyvol 29 no 1 pp 181ndash191 1984
[7] M E Basher ldquo120590-Coloring of the monohedral tilingrdquo Interna-tional Journal of Mathematical Combinatorics vol 2 pp 46ndash522009
[8] J E Freund IMiller andMMillerMathematical Statistics withApplications Prentice Hall PTR 7th edition 2003
[9] D C Montgomery and G C Runger Applied Statistics andProbability for Engineers John Wiley amp Sons New York NYUSA 2nd edition 1999
[10] L Bernardin P Chin P DeMarco et al Maple ProgrammingGuide Maplesoft 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Abstract and Applied Analysis
Table 1 The relation between 119896 119899 119874(119865(JPD)) and the 119865(JPD) of the square tiling at 119899 = 13
119896 119874(F(JPD)) Example of the corresponding 119865(JPD)3 1 513 213 113 513
4 3 213 113 113 913 113 213 913 113 213 813 113 213
5 3 213 113 113 913 113 213 913 113 213 313 113 713
6 5 213 113 113 913 113 213 913 113 213 813 113 213813 213 213 113 213 313 113 713
7 5 213 113 113 913 113 213 913 113 213 313 113 713313 213 713 113 213 613 113 413
8 7 213 113 113 913 113 213 913 113 213 813 113 213813 213 213 113 213 313 113 713 313 213 713 113 213 413 113 613
614
314
814
1014514414
714214 214 914
214
214
114
214
114114
114214
114114
(a)
213
213
213 613
413
813113
113 113913
213
113
213
113
713
313
(b)
Figure 5 Some JPD values at 119899 = 13 and 119899 = 14 to color the four vertices of the square tiling
(ii) For 119899 ge 8 and 119899 is even the available total probabilityvalues (without repetition) are 1119899 2119899 (119899minus4)119899 (119899minus5)119899 (119899 minus 119894)119899 where 4 le 119894 le 119899 minus 3 Thegreatest number of colors 119888 equals (119899minus4) for exampleat 119899 = 8 the available total probability valuesare 18 28 38 48 and so 119888 equals 4 (ie 119899 minus
4) at 119899 = 14 the available total probability val-ues are 114 214 314 414 514 614 714 814914 1014 and so the greatest number of colors 119888equals 10 (ie 119899 minus 4) see Figure 5(a)
Case 4 For 119899 ge 9 and 119899 is odd the proof is similar toCase 3 Since 119899 is odd then in this case the rest correspondingprobability value (119899 minus 3)119899 has even value of its numeratorThen this probability value can be distributed on the othertwo vertices by two equal probability values ldquo((119899 minus 3)2)119899rdquo
As in Case 3 the last two probability values (119899minus3)119899 (119899minus1)119899 and ((119899 minus 3)2)119899 are excluded In this case we obtainthat for 119899 ge 9 and 119899 is odd the available total probabilityvalues are 1119899 2119899 (119899minus4)119899 (119899minus5)119899 (((119899minus3)2)+1)119899(((119899 minus 3)2) minus 1)119899 (119899 minus 119894)119899 where 4 le 119894 le 119899 minus 3 Thegreatest number of colors 119888 equals (119899 minus 5) for example
(i) at 119899 = 9 the available total distinct probability valuesare 19 29 49 59 and the greatest number ofcolors 119888 equals 4 (ie 119899 minus 5)
(ii) at 119899 = 13 the total distinct probability values are113 213 313 413 613 713 813 913 andthe greatest number of colors 119888 equals 8 (ie 119899 minus 5)see Figure 5(b)
p23
p12
p21 p31
p42
p33
Figure 6 The used regular hexagon with the corresponding JPDvalues
There are a relation between 119896 119899 119874(119865(JPD)) and thecorresponding 119865(JPD) Tables 1 and 2 show this relation
(i) for 119899 = 13 see Table 1
(ii) for 119899 = 14 see Table 2
Corollary 10 The smallest number of colors for square tilingis 2 if 119899 is even and 3 if 119899 is odd
23 JPD-Coloring of the 6-Monohedral Tiling Here we con-sider the JPD 119891
119883119884(119909119894 119910119895) = 119901119894119895 119894 = 1 2 3 4 119895 = 1 2 3 with
the assumptions 11990121
= 11990131 11990131
= 11990142 11990142
= 11990133 11990133
= 11990123
11990123
= 11990112 11990112
= 11990121(where 119901
119894119895is with equal denominators 119899)
and zero values of 11990111 11990141 11990122
11990132 11990113 11990143 The used regular
hexagon is illustrated in Figure 6
Abstract and Applied Analysis 5
Table 2 The relation between 119896 119899 119874(119865(JPD)) and the 119865(JPD) of the square tiling at 119899 = 14
119896 119874(119865(JPD)) Example of the corresponding 119865(JPD)2 414 314 314 414
3 1 814 114 114 414
4 214 714 114 414
5 3 314 114 114 914 114 314 914 114 314 414 114 614
6 4 614 114 114 614 114 514 614 214 514 114 214 614114 414 614 314
7 5214 714 114 414 714 214 414 114 214 614 114 514 614 214 514 114214 114 114 1014
8 214 714 114 414 714 214 414 114 214 614 114 514 614 214 514 114214 814 114 314
9 7 214 714 114 414 714 214 414 114 214 614 114 514 614 214 514 114214 114 114 1014 114 214 114 1014 214 814 114 314
10 9214 714 114 414 714 214 414 114 214 614 114 514 614 214 514 114214 114 114 1014 114 214 1014 114 214 914 114 214 914 214 214 114214 814 114 314
118
118
118
118
118118
118818
118
118
118 118 118
118
118
118118118
118118418
418 418
418
418
418
418518
518518718 718
718
718
618518
118 518
618
618618
618
618
418
118 118
118
118
818
118
318318
318
318
318318318
318
318 318 318
318 918
318
118
618
418218
218
218 218
218
218
218 218
218
218
218
218
218
218
218 218
218 218
218
218
218
218
218
218818
218
218
1018
1118
318
318
Figure 7 The JPD values at 119899 = 18 to color the six vertices of the regular hexagon tiling
Theorem 11 If the plane is equipped by 6-monohedral tilingthen the greatest number of colors ldquo119888rdquo is given as 119888 = 119899minus7 where119899 ge 9 If 119899 lt 9 then the tiling cannot be colored
Proof Let 1198772 be equipped by hexagon tiling The proof canbe given as follows
Case 1 If 119899 lt 9 then the tiling cannot be colored (ie thegreatest number of colors 119888 equals 0) because we cannot findsix probability values (JPD) which satisfied the condition (ii)in Definition 4 to color the six vertices of the mentioned
hexagon tiling (say at 119899 = 8 the available probabilityvalues are 18 28 18 28 18 18 at 119899 = 7 the availableprobability values are 17 27 17 17 17 17 and at thesmallest value 119899 = 6 the available probability values are16 16 16 16 16 16)
Case 2 For 119899 ge 9 take the corresponding probability valueof a vertex (119899 minus 119894)119899 where 119894 ge 7 because it is impossible totake the value of 119894 less than 7 Then the rest correspondingprobability value is 119894119899 This probability value must bedistributed on the other five vertices under consideration of
6 Abstract and Applied Analysis
125
125
425
325
325
225
625
225
425
425
324 325 225
225225
225
225
725 325 325
225625225
125 825
525
525525
725225
325425
425625
625
925
525 325 825
725425
825
325225 625 425
325
225
325 325
125
125125
125125 125
325
325
325
325
325
525525
525625
325125
125
125
125
125
125
125125
125125
125
125125
125
125
125325
325
625
725
125
125 125
125125
125
125
125
125125125
125125
225
225 225
225 225
225
225
225
225
225225
225
225225525
425
225
225225325
325
425
425
425
125
125
125
1025
1325
1625
1224
1425
1025
10251025 1725
1225
1325
1225
1125
1225
1525
1825
1325 525
Figure 8 The JPD values at 119899 = 25 to color the six vertices of the regular hexagon tiling
the conditions in Definition 4 for an integer number 119894 ge
7 This implies that the available total probability values toobtain the mesh under consideration of the conditions inDefinition 4 are 1119899 2119899 3119899 (119899 minus 119894 minus 1)119899 where 7 le
119894 le 119899 minus 2 Avoiding the repetition we obtain the following
(i) for 119899 = 9 the total probability values are 19 29 and119888 equals 2
(ii) for 119899 = 18 the total probability values are 118 218318 418 518 618 718 818 918 1018 1118 and119888 equals 11 (see Figure 7)
(iii) for 119899 = 25 the total probability values are 125 225325 425 525 625 725 825 925 1025 11251225 1325 1425 1525 1625 1725 1825 and 119888
equals 18 (see Figure 8)
In this case the greatest number of colors 119888 equals 119899 minus 7
Corollary 12 The relation between 119896 and 119899 can be shown inTable 3
Table 3The relation between 119896 and 119899 of the regular hexagon tiling
119896 119899
2 6119903 + 3 119903 ge 1
3 6119903 + 3 6119903 + 4 6119903 + 5 119903 ge 1
6119903 6119903 + 1 6119903 + 2 119903 ge 2
4 6119903 6119903 + 1 6119903 + 2 6119903 + 3 6119903 + 4 6119903 + 5 119903 ge 2
5 6119903 + 5 119903 ge 2
6119903 6119903 + 1 6119903 + 2 6119903 + 3 6119903 + 4 119903 ge 3
6 6119903 + 3 6119903 + 4 6119903 + 5 119903 ge 3
6119903 6119903 + 1 6119903 + 2 119903 ge 4
Appendix
As a MAPLE programming guide see [10]
Determination of the Probability Values by Using the MAPLEProgram
Case A1 (JPD-coloring of the 3-monohedral tiling) SeeBox 1
Case A2 (JPD-coloring of the 4-monohedral tiling) SeeBox 2
Abstract and Applied Analysis 7
gt restart with(linalg)
Probability Values=proc(nposintiposintjposint)
local AbTrvC global PV
A[T]= matrix([[111]]) b= vector([1]) C=sum(V[w]w=12)
linsolve(A[T]blsquor1015840) r linsolve(A[T]blsquor1015840v)PV[T[nij]]=subs(v[1]=inv[2]=jnv[3]=1-C)
end proc
n=3i=1j=1 PV[T[nij]]=Probability Values(nij)
n=4i=1j=2 PV[T[nij]]=Probability Values(nij)
n=5i=1j=2 PV[T[nij]]=Probability Values(nij)
n=6i=1j=2 PV[T[nij]]=Probability Values(nij)
n=7i=1j=2 PV[T[nij]]=Probability Values(nij)
n=8i=1j=2 PV[T[nij]]=Probability Values(nij)
n=8i=1j=3 PV[T[nij]]=Probability Values(nij)
PV119879311
= [
1
3
1
3
1
3
] PV119879412
= [
1
2
1
4
1
4
] PV119879512
= [
2
5
1
5
2
5
] PV119879612
= [
1
2
1
6
1
3
]
PV119879712
= [
2
7
4
7
1
7
] PV119879812
= [
5
8
1
8
1
4
] PV119879813
= [
1
2
1
8
3
8
]
Box 1
gt restart with(linalg)
Probability Values=proc(nposintiposintjposintkposint)
local AbFrvC global PV
A[F]=matrix([[1111]]) b=vector([1]) C=sum(V[w]w=13)
linsolve(A[F]blsquor1015840) r linsolve(A[F]blsquor1015840v)PV[S[nijk]]=subs(v[1]=inv[2]=jnv[3]=knv[4]=1-C)
end proc
n=4 i=1j=1k=1 PV[F[nijk]]=Probability Values(nijk)
n=5 i=1j=2k=1 PV[F[nijk]]=Probability Values(nijk)
n=6 i=1j=2k=1 PV[F[nijk]]=Probability Values(nijk)
n=7 i=1j=2k=1 PV[F[nijk]]=Probability Values(nijk)
n=8 i=1j=2k=1 PV[F[nijk]]=Probability Values(nijk)
n=8 i=1j=2k=3 PV[F[nijk]]=Probability Values(nijk)
n=8 i=1j=3k=1 PV[F[nijk]]=Probability Values(nijk)
n=9 i=1j=2k=1 PV[F[nijk]]=Probability Values(nijk)
n=9 i=1j=4k=3 PV[F[nijk]]=Probability Values(nijk)
n=13i=1j=2k=1 PV[F[nijk]]=Probability Values(nijk)
n=13i=1j=3k=4 PV[F[nijk]]=Probability Values(nijk)
n=13i=1j=1k=6 PV[F[nijk]]=Probability Values(nijk)
n=13i=1j=1k=7 PV[F[nijk]]=Probability Values(nijk)
n=13i=1j=1k=8 PV[F[nijk]]=Probability Values(nijk)
n=14i=1j=1k=2 PV[F[nijk]]=Probability Values(nijk)
n=14i=2j=3k=4 PV[F[nijk]]=Probability Values(nijk)
n=14i=1j=5k=6 PV[F[nijk]]=Probability Values(nijk)
n=14i=1j=2k=7 PV[F[nijk]]=Probability Values(nijk)
n=14i=1j=2k=8 PV[F[nijk]]=Probability Values(nijk)
n=14i=2j=2k=9 PV[F[nijk]]=Probability Values(nijk)
PV1198654111
= [
1
4
1
4
1
4
1
4
] PV1198655121
= [
1
5
1
5
2
5
1
5
] PV1198656121
= [
1
6
1
3
1
6
1
3
] PV1198657121
= [
3
7
1
7
2
7
1
7
]
PV1198658121
= [
1
8
1
4
1
2
1
8
] PV1198658123
= [
1
4
1
8
1
4
3
8
] PV1198658131
= [
3
8
1
8
3
8
1
8
] PV1198659121
= [
5
9
1
9
2
9
1
9
]
PV1198659143
= [
1
3
4
9
1
9
1
9
] PV11986513121
= [
9
13
1
13
1
13
2
13
] PV11986513134
= [
5
13
4
13
3
13
1
13
] PV11986513116
= [
1
13
6
13
5
13
1
13
]
PV11986513117
= [
1
13
7
13
4
13
1
13
] PV11986513118
= [
8
13
1
13
1
13
3
13
] PV11986514112
= [
1
7
5
7
1
14
1
14
] PV11986514234
= [
1
7
3
14
2
7
5
14
]
PV11986514156
= [
1
14
5
14
3
7
1
7
] PV11986514127
= [
2
7
1
14
1
7
1
4
] PV11986514128
= [
1
14
1
7
4
7
3
14
] PV11986514229
= [
1
14
1
7
1
7
9
14
]
Box 2
8 Abstract and Applied Analysis
gt restart with(linalg)
Probability Values=proc(nposintiposintjposintkposintlposintmposint)
local AbSrvC global PV
A[S]=matrix([[111111]]) b=vector([1]) C=sum(v[w]w=15)
linsolve(A[S]blsquor1015840) r linsolve(A[S]blsquor1015840v)PV[S[nijklm]]=subs(v[1]=inv[2]=jnv[3]=knv[4]=lnv[5]=mnv[6]=1-C)
end proc
n=6 i=1j=1k=1l=1m=1 PV[S[nijklm]]=Probability Values(nijklm)
n=7 i=1j=2k=1l=1m=1 PV[S[nijklm]]=Probability Values(nijklm)
n=8 i=1j=2k=1l=2m=1 PV[S[nijklm]]=Probability Values(nijklm)
n=9 i=1j=2k=1l=2m=1 PV[S[nijklm]]=Probability Values(nijklm)
n=18i=1j=2k=3l=4m=6 PV[S[nijklm]]=Probability Values(nijklm)
n=25i=1j=2k=3l=4m=5 PV[S[nijklm]]=Probability Values(nijklm)
n=25i=1j=2k=3l=4m=5 PV[S[nijklm]]=Probability Values(nijklm)
PV119878611111
= [
1
6
1
6
1
6
1
6
1
6
1
6
] PV119878712111
= [
1
7
1
7
2
7
1
7
1
7
1
7
] PV119878812121
= [
1
4
1
8
1
4
1
8
1
8
1
8
]
PV119878912121
= [
2
9
1
9
2
9
1
9
2
9
1
9
] PV1198781812346
= [
1
9
1
18
1
9
1
6
2
9
1
3
] PV1198782512345
= [
1
5
4
25
3
25
2
25
1
25
2
25
]
PV1198782512345
= [
1
5
4
25
2
5
3
25
2
25
1
25
]
Box 3
Case A3 (JPD-coloring of the 6-monohedral tiling) SeeBox 3
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] B Grunbaum and G C Shephard ldquoTiling by regular polygonsrdquoMathematics Magazine vol 50 no 5 pp 227ndash247 1977
[2] B Grunbaum and G C Shephard ldquoPerfect colorings of transi-tive tilings and patterns in the planerdquoDiscrete Mathematics vol20 no 3 pp 235ndash247 1977
[3] S Eigen J Navarro andV S Prasad ldquoAn aperiodic tiling using adynamical system and Beatty sequencesrdquo in Dynamics ErgodicTheory andGeometry vol 54 ofMathematical Sciences ResearchInstitute pp 223ndash241 Cambridge University Press 2007
[4] C Mann L Asaro J Hyde M Jensen and T Schroeder ldquoUni-form edge-119888-colorings of the Archimedean tilingsrdquo DiscreteMathematics vol 338 no 1 pp 10ndash22 2015
[5] R Santos andR Felix ldquoPerfect precise colorings of plane regulartilingsrdquo Zeitschrift fur Kristallographie vol 226 no 9 pp 726ndash730 2011
[6] B Grunbaum P Mani-Levitska and G C Shephard ldquoTilingthree-dimensional space with polyhedral tiles of a given iso-morphism typerdquo Journal of the London Mathematical Societyvol 29 no 1 pp 181ndash191 1984
[7] M E Basher ldquo120590-Coloring of the monohedral tilingrdquo Interna-tional Journal of Mathematical Combinatorics vol 2 pp 46ndash522009
[8] J E Freund IMiller andMMillerMathematical Statistics withApplications Prentice Hall PTR 7th edition 2003
[9] D C Montgomery and G C Runger Applied Statistics andProbability for Engineers John Wiley amp Sons New York NYUSA 2nd edition 1999
[10] L Bernardin P Chin P DeMarco et al Maple ProgrammingGuide Maplesoft 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 5
Table 2 The relation between 119896 119899 119874(119865(JPD)) and the 119865(JPD) of the square tiling at 119899 = 14
119896 119874(119865(JPD)) Example of the corresponding 119865(JPD)2 414 314 314 414
3 1 814 114 114 414
4 214 714 114 414
5 3 314 114 114 914 114 314 914 114 314 414 114 614
6 4 614 114 114 614 114 514 614 214 514 114 214 614114 414 614 314
7 5214 714 114 414 714 214 414 114 214 614 114 514 614 214 514 114214 114 114 1014
8 214 714 114 414 714 214 414 114 214 614 114 514 614 214 514 114214 814 114 314
9 7 214 714 114 414 714 214 414 114 214 614 114 514 614 214 514 114214 114 114 1014 114 214 114 1014 214 814 114 314
10 9214 714 114 414 714 214 414 114 214 614 114 514 614 214 514 114214 114 114 1014 114 214 1014 114 214 914 114 214 914 214 214 114214 814 114 314
118
118
118
118
118118
118818
118
118
118 118 118
118
118
118118118
118118418
418 418
418
418
418
418518
518518718 718
718
718
618518
118 518
618
618618
618
618
418
118 118
118
118
818
118
318318
318
318
318318318
318
318 318 318
318 918
318
118
618
418218
218
218 218
218
218
218 218
218
218
218
218
218
218
218 218
218 218
218
218
218
218
218
218818
218
218
1018
1118
318
318
Figure 7 The JPD values at 119899 = 18 to color the six vertices of the regular hexagon tiling
Theorem 11 If the plane is equipped by 6-monohedral tilingthen the greatest number of colors ldquo119888rdquo is given as 119888 = 119899minus7 where119899 ge 9 If 119899 lt 9 then the tiling cannot be colored
Proof Let 1198772 be equipped by hexagon tiling The proof canbe given as follows
Case 1 If 119899 lt 9 then the tiling cannot be colored (ie thegreatest number of colors 119888 equals 0) because we cannot findsix probability values (JPD) which satisfied the condition (ii)in Definition 4 to color the six vertices of the mentioned
hexagon tiling (say at 119899 = 8 the available probabilityvalues are 18 28 18 28 18 18 at 119899 = 7 the availableprobability values are 17 27 17 17 17 17 and at thesmallest value 119899 = 6 the available probability values are16 16 16 16 16 16)
Case 2 For 119899 ge 9 take the corresponding probability valueof a vertex (119899 minus 119894)119899 where 119894 ge 7 because it is impossible totake the value of 119894 less than 7 Then the rest correspondingprobability value is 119894119899 This probability value must bedistributed on the other five vertices under consideration of
6 Abstract and Applied Analysis
125
125
425
325
325
225
625
225
425
425
324 325 225
225225
225
225
725 325 325
225625225
125 825
525
525525
725225
325425
425625
625
925
525 325 825
725425
825
325225 625 425
325
225
325 325
125
125125
125125 125
325
325
325
325
325
525525
525625
325125
125
125
125
125
125
125125
125125
125
125125
125
125
125325
325
625
725
125
125 125
125125
125
125
125
125125125
125125
225
225 225
225 225
225
225
225
225
225225
225
225225525
425
225
225225325
325
425
425
425
125
125
125
1025
1325
1625
1224
1425
1025
10251025 1725
1225
1325
1225
1125
1225
1525
1825
1325 525
Figure 8 The JPD values at 119899 = 25 to color the six vertices of the regular hexagon tiling
the conditions in Definition 4 for an integer number 119894 ge
7 This implies that the available total probability values toobtain the mesh under consideration of the conditions inDefinition 4 are 1119899 2119899 3119899 (119899 minus 119894 minus 1)119899 where 7 le
119894 le 119899 minus 2 Avoiding the repetition we obtain the following
(i) for 119899 = 9 the total probability values are 19 29 and119888 equals 2
(ii) for 119899 = 18 the total probability values are 118 218318 418 518 618 718 818 918 1018 1118 and119888 equals 11 (see Figure 7)
(iii) for 119899 = 25 the total probability values are 125 225325 425 525 625 725 825 925 1025 11251225 1325 1425 1525 1625 1725 1825 and 119888
equals 18 (see Figure 8)
In this case the greatest number of colors 119888 equals 119899 minus 7
Corollary 12 The relation between 119896 and 119899 can be shown inTable 3
Table 3The relation between 119896 and 119899 of the regular hexagon tiling
119896 119899
2 6119903 + 3 119903 ge 1
3 6119903 + 3 6119903 + 4 6119903 + 5 119903 ge 1
6119903 6119903 + 1 6119903 + 2 119903 ge 2
4 6119903 6119903 + 1 6119903 + 2 6119903 + 3 6119903 + 4 6119903 + 5 119903 ge 2
5 6119903 + 5 119903 ge 2
6119903 6119903 + 1 6119903 + 2 6119903 + 3 6119903 + 4 119903 ge 3
6 6119903 + 3 6119903 + 4 6119903 + 5 119903 ge 3
6119903 6119903 + 1 6119903 + 2 119903 ge 4
Appendix
As a MAPLE programming guide see [10]
Determination of the Probability Values by Using the MAPLEProgram
Case A1 (JPD-coloring of the 3-monohedral tiling) SeeBox 1
Case A2 (JPD-coloring of the 4-monohedral tiling) SeeBox 2
Abstract and Applied Analysis 7
gt restart with(linalg)
Probability Values=proc(nposintiposintjposint)
local AbTrvC global PV
A[T]= matrix([[111]]) b= vector([1]) C=sum(V[w]w=12)
linsolve(A[T]blsquor1015840) r linsolve(A[T]blsquor1015840v)PV[T[nij]]=subs(v[1]=inv[2]=jnv[3]=1-C)
end proc
n=3i=1j=1 PV[T[nij]]=Probability Values(nij)
n=4i=1j=2 PV[T[nij]]=Probability Values(nij)
n=5i=1j=2 PV[T[nij]]=Probability Values(nij)
n=6i=1j=2 PV[T[nij]]=Probability Values(nij)
n=7i=1j=2 PV[T[nij]]=Probability Values(nij)
n=8i=1j=2 PV[T[nij]]=Probability Values(nij)
n=8i=1j=3 PV[T[nij]]=Probability Values(nij)
PV119879311
= [
1
3
1
3
1
3
] PV119879412
= [
1
2
1
4
1
4
] PV119879512
= [
2
5
1
5
2
5
] PV119879612
= [
1
2
1
6
1
3
]
PV119879712
= [
2
7
4
7
1
7
] PV119879812
= [
5
8
1
8
1
4
] PV119879813
= [
1
2
1
8
3
8
]
Box 1
gt restart with(linalg)
Probability Values=proc(nposintiposintjposintkposint)
local AbFrvC global PV
A[F]=matrix([[1111]]) b=vector([1]) C=sum(V[w]w=13)
linsolve(A[F]blsquor1015840) r linsolve(A[F]blsquor1015840v)PV[S[nijk]]=subs(v[1]=inv[2]=jnv[3]=knv[4]=1-C)
end proc
n=4 i=1j=1k=1 PV[F[nijk]]=Probability Values(nijk)
n=5 i=1j=2k=1 PV[F[nijk]]=Probability Values(nijk)
n=6 i=1j=2k=1 PV[F[nijk]]=Probability Values(nijk)
n=7 i=1j=2k=1 PV[F[nijk]]=Probability Values(nijk)
n=8 i=1j=2k=1 PV[F[nijk]]=Probability Values(nijk)
n=8 i=1j=2k=3 PV[F[nijk]]=Probability Values(nijk)
n=8 i=1j=3k=1 PV[F[nijk]]=Probability Values(nijk)
n=9 i=1j=2k=1 PV[F[nijk]]=Probability Values(nijk)
n=9 i=1j=4k=3 PV[F[nijk]]=Probability Values(nijk)
n=13i=1j=2k=1 PV[F[nijk]]=Probability Values(nijk)
n=13i=1j=3k=4 PV[F[nijk]]=Probability Values(nijk)
n=13i=1j=1k=6 PV[F[nijk]]=Probability Values(nijk)
n=13i=1j=1k=7 PV[F[nijk]]=Probability Values(nijk)
n=13i=1j=1k=8 PV[F[nijk]]=Probability Values(nijk)
n=14i=1j=1k=2 PV[F[nijk]]=Probability Values(nijk)
n=14i=2j=3k=4 PV[F[nijk]]=Probability Values(nijk)
n=14i=1j=5k=6 PV[F[nijk]]=Probability Values(nijk)
n=14i=1j=2k=7 PV[F[nijk]]=Probability Values(nijk)
n=14i=1j=2k=8 PV[F[nijk]]=Probability Values(nijk)
n=14i=2j=2k=9 PV[F[nijk]]=Probability Values(nijk)
PV1198654111
= [
1
4
1
4
1
4
1
4
] PV1198655121
= [
1
5
1
5
2
5
1
5
] PV1198656121
= [
1
6
1
3
1
6
1
3
] PV1198657121
= [
3
7
1
7
2
7
1
7
]
PV1198658121
= [
1
8
1
4
1
2
1
8
] PV1198658123
= [
1
4
1
8
1
4
3
8
] PV1198658131
= [
3
8
1
8
3
8
1
8
] PV1198659121
= [
5
9
1
9
2
9
1
9
]
PV1198659143
= [
1
3
4
9
1
9
1
9
] PV11986513121
= [
9
13
1
13
1
13
2
13
] PV11986513134
= [
5
13
4
13
3
13
1
13
] PV11986513116
= [
1
13
6
13
5
13
1
13
]
PV11986513117
= [
1
13
7
13
4
13
1
13
] PV11986513118
= [
8
13
1
13
1
13
3
13
] PV11986514112
= [
1
7
5
7
1
14
1
14
] PV11986514234
= [
1
7
3
14
2
7
5
14
]
PV11986514156
= [
1
14
5
14
3
7
1
7
] PV11986514127
= [
2
7
1
14
1
7
1
4
] PV11986514128
= [
1
14
1
7
4
7
3
14
] PV11986514229
= [
1
14
1
7
1
7
9
14
]
Box 2
8 Abstract and Applied Analysis
gt restart with(linalg)
Probability Values=proc(nposintiposintjposintkposintlposintmposint)
local AbSrvC global PV
A[S]=matrix([[111111]]) b=vector([1]) C=sum(v[w]w=15)
linsolve(A[S]blsquor1015840) r linsolve(A[S]blsquor1015840v)PV[S[nijklm]]=subs(v[1]=inv[2]=jnv[3]=knv[4]=lnv[5]=mnv[6]=1-C)
end proc
n=6 i=1j=1k=1l=1m=1 PV[S[nijklm]]=Probability Values(nijklm)
n=7 i=1j=2k=1l=1m=1 PV[S[nijklm]]=Probability Values(nijklm)
n=8 i=1j=2k=1l=2m=1 PV[S[nijklm]]=Probability Values(nijklm)
n=9 i=1j=2k=1l=2m=1 PV[S[nijklm]]=Probability Values(nijklm)
n=18i=1j=2k=3l=4m=6 PV[S[nijklm]]=Probability Values(nijklm)
n=25i=1j=2k=3l=4m=5 PV[S[nijklm]]=Probability Values(nijklm)
n=25i=1j=2k=3l=4m=5 PV[S[nijklm]]=Probability Values(nijklm)
PV119878611111
= [
1
6
1
6
1
6
1
6
1
6
1
6
] PV119878712111
= [
1
7
1
7
2
7
1
7
1
7
1
7
] PV119878812121
= [
1
4
1
8
1
4
1
8
1
8
1
8
]
PV119878912121
= [
2
9
1
9
2
9
1
9
2
9
1
9
] PV1198781812346
= [
1
9
1
18
1
9
1
6
2
9
1
3
] PV1198782512345
= [
1
5
4
25
3
25
2
25
1
25
2
25
]
PV1198782512345
= [
1
5
4
25
2
5
3
25
2
25
1
25
]
Box 3
Case A3 (JPD-coloring of the 6-monohedral tiling) SeeBox 3
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] B Grunbaum and G C Shephard ldquoTiling by regular polygonsrdquoMathematics Magazine vol 50 no 5 pp 227ndash247 1977
[2] B Grunbaum and G C Shephard ldquoPerfect colorings of transi-tive tilings and patterns in the planerdquoDiscrete Mathematics vol20 no 3 pp 235ndash247 1977
[3] S Eigen J Navarro andV S Prasad ldquoAn aperiodic tiling using adynamical system and Beatty sequencesrdquo in Dynamics ErgodicTheory andGeometry vol 54 ofMathematical Sciences ResearchInstitute pp 223ndash241 Cambridge University Press 2007
[4] C Mann L Asaro J Hyde M Jensen and T Schroeder ldquoUni-form edge-119888-colorings of the Archimedean tilingsrdquo DiscreteMathematics vol 338 no 1 pp 10ndash22 2015
[5] R Santos andR Felix ldquoPerfect precise colorings of plane regulartilingsrdquo Zeitschrift fur Kristallographie vol 226 no 9 pp 726ndash730 2011
[6] B Grunbaum P Mani-Levitska and G C Shephard ldquoTilingthree-dimensional space with polyhedral tiles of a given iso-morphism typerdquo Journal of the London Mathematical Societyvol 29 no 1 pp 181ndash191 1984
[7] M E Basher ldquo120590-Coloring of the monohedral tilingrdquo Interna-tional Journal of Mathematical Combinatorics vol 2 pp 46ndash522009
[8] J E Freund IMiller andMMillerMathematical Statistics withApplications Prentice Hall PTR 7th edition 2003
[9] D C Montgomery and G C Runger Applied Statistics andProbability for Engineers John Wiley amp Sons New York NYUSA 2nd edition 1999
[10] L Bernardin P Chin P DeMarco et al Maple ProgrammingGuide Maplesoft 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Abstract and Applied Analysis
125
125
425
325
325
225
625
225
425
425
324 325 225
225225
225
225
725 325 325
225625225
125 825
525
525525
725225
325425
425625
625
925
525 325 825
725425
825
325225 625 425
325
225
325 325
125
125125
125125 125
325
325
325
325
325
525525
525625
325125
125
125
125
125
125
125125
125125
125
125125
125
125
125325
325
625
725
125
125 125
125125
125
125
125
125125125
125125
225
225 225
225 225
225
225
225
225
225225
225
225225525
425
225
225225325
325
425
425
425
125
125
125
1025
1325
1625
1224
1425
1025
10251025 1725
1225
1325
1225
1125
1225
1525
1825
1325 525
Figure 8 The JPD values at 119899 = 25 to color the six vertices of the regular hexagon tiling
the conditions in Definition 4 for an integer number 119894 ge
7 This implies that the available total probability values toobtain the mesh under consideration of the conditions inDefinition 4 are 1119899 2119899 3119899 (119899 minus 119894 minus 1)119899 where 7 le
119894 le 119899 minus 2 Avoiding the repetition we obtain the following
(i) for 119899 = 9 the total probability values are 19 29 and119888 equals 2
(ii) for 119899 = 18 the total probability values are 118 218318 418 518 618 718 818 918 1018 1118 and119888 equals 11 (see Figure 7)
(iii) for 119899 = 25 the total probability values are 125 225325 425 525 625 725 825 925 1025 11251225 1325 1425 1525 1625 1725 1825 and 119888
equals 18 (see Figure 8)
In this case the greatest number of colors 119888 equals 119899 minus 7
Corollary 12 The relation between 119896 and 119899 can be shown inTable 3
Table 3The relation between 119896 and 119899 of the regular hexagon tiling
119896 119899
2 6119903 + 3 119903 ge 1
3 6119903 + 3 6119903 + 4 6119903 + 5 119903 ge 1
6119903 6119903 + 1 6119903 + 2 119903 ge 2
4 6119903 6119903 + 1 6119903 + 2 6119903 + 3 6119903 + 4 6119903 + 5 119903 ge 2
5 6119903 + 5 119903 ge 2
6119903 6119903 + 1 6119903 + 2 6119903 + 3 6119903 + 4 119903 ge 3
6 6119903 + 3 6119903 + 4 6119903 + 5 119903 ge 3
6119903 6119903 + 1 6119903 + 2 119903 ge 4
Appendix
As a MAPLE programming guide see [10]
Determination of the Probability Values by Using the MAPLEProgram
Case A1 (JPD-coloring of the 3-monohedral tiling) SeeBox 1
Case A2 (JPD-coloring of the 4-monohedral tiling) SeeBox 2
Abstract and Applied Analysis 7
gt restart with(linalg)
Probability Values=proc(nposintiposintjposint)
local AbTrvC global PV
A[T]= matrix([[111]]) b= vector([1]) C=sum(V[w]w=12)
linsolve(A[T]blsquor1015840) r linsolve(A[T]blsquor1015840v)PV[T[nij]]=subs(v[1]=inv[2]=jnv[3]=1-C)
end proc
n=3i=1j=1 PV[T[nij]]=Probability Values(nij)
n=4i=1j=2 PV[T[nij]]=Probability Values(nij)
n=5i=1j=2 PV[T[nij]]=Probability Values(nij)
n=6i=1j=2 PV[T[nij]]=Probability Values(nij)
n=7i=1j=2 PV[T[nij]]=Probability Values(nij)
n=8i=1j=2 PV[T[nij]]=Probability Values(nij)
n=8i=1j=3 PV[T[nij]]=Probability Values(nij)
PV119879311
= [
1
3
1
3
1
3
] PV119879412
= [
1
2
1
4
1
4
] PV119879512
= [
2
5
1
5
2
5
] PV119879612
= [
1
2
1
6
1
3
]
PV119879712
= [
2
7
4
7
1
7
] PV119879812
= [
5
8
1
8
1
4
] PV119879813
= [
1
2
1
8
3
8
]
Box 1
gt restart with(linalg)
Probability Values=proc(nposintiposintjposintkposint)
local AbFrvC global PV
A[F]=matrix([[1111]]) b=vector([1]) C=sum(V[w]w=13)
linsolve(A[F]blsquor1015840) r linsolve(A[F]blsquor1015840v)PV[S[nijk]]=subs(v[1]=inv[2]=jnv[3]=knv[4]=1-C)
end proc
n=4 i=1j=1k=1 PV[F[nijk]]=Probability Values(nijk)
n=5 i=1j=2k=1 PV[F[nijk]]=Probability Values(nijk)
n=6 i=1j=2k=1 PV[F[nijk]]=Probability Values(nijk)
n=7 i=1j=2k=1 PV[F[nijk]]=Probability Values(nijk)
n=8 i=1j=2k=1 PV[F[nijk]]=Probability Values(nijk)
n=8 i=1j=2k=3 PV[F[nijk]]=Probability Values(nijk)
n=8 i=1j=3k=1 PV[F[nijk]]=Probability Values(nijk)
n=9 i=1j=2k=1 PV[F[nijk]]=Probability Values(nijk)
n=9 i=1j=4k=3 PV[F[nijk]]=Probability Values(nijk)
n=13i=1j=2k=1 PV[F[nijk]]=Probability Values(nijk)
n=13i=1j=3k=4 PV[F[nijk]]=Probability Values(nijk)
n=13i=1j=1k=6 PV[F[nijk]]=Probability Values(nijk)
n=13i=1j=1k=7 PV[F[nijk]]=Probability Values(nijk)
n=13i=1j=1k=8 PV[F[nijk]]=Probability Values(nijk)
n=14i=1j=1k=2 PV[F[nijk]]=Probability Values(nijk)
n=14i=2j=3k=4 PV[F[nijk]]=Probability Values(nijk)
n=14i=1j=5k=6 PV[F[nijk]]=Probability Values(nijk)
n=14i=1j=2k=7 PV[F[nijk]]=Probability Values(nijk)
n=14i=1j=2k=8 PV[F[nijk]]=Probability Values(nijk)
n=14i=2j=2k=9 PV[F[nijk]]=Probability Values(nijk)
PV1198654111
= [
1
4
1
4
1
4
1
4
] PV1198655121
= [
1
5
1
5
2
5
1
5
] PV1198656121
= [
1
6
1
3
1
6
1
3
] PV1198657121
= [
3
7
1
7
2
7
1
7
]
PV1198658121
= [
1
8
1
4
1
2
1
8
] PV1198658123
= [
1
4
1
8
1
4
3
8
] PV1198658131
= [
3
8
1
8
3
8
1
8
] PV1198659121
= [
5
9
1
9
2
9
1
9
]
PV1198659143
= [
1
3
4
9
1
9
1
9
] PV11986513121
= [
9
13
1
13
1
13
2
13
] PV11986513134
= [
5
13
4
13
3
13
1
13
] PV11986513116
= [
1
13
6
13
5
13
1
13
]
PV11986513117
= [
1
13
7
13
4
13
1
13
] PV11986513118
= [
8
13
1
13
1
13
3
13
] PV11986514112
= [
1
7
5
7
1
14
1
14
] PV11986514234
= [
1
7
3
14
2
7
5
14
]
PV11986514156
= [
1
14
5
14
3
7
1
7
] PV11986514127
= [
2
7
1
14
1
7
1
4
] PV11986514128
= [
1
14
1
7
4
7
3
14
] PV11986514229
= [
1
14
1
7
1
7
9
14
]
Box 2
8 Abstract and Applied Analysis
gt restart with(linalg)
Probability Values=proc(nposintiposintjposintkposintlposintmposint)
local AbSrvC global PV
A[S]=matrix([[111111]]) b=vector([1]) C=sum(v[w]w=15)
linsolve(A[S]blsquor1015840) r linsolve(A[S]blsquor1015840v)PV[S[nijklm]]=subs(v[1]=inv[2]=jnv[3]=knv[4]=lnv[5]=mnv[6]=1-C)
end proc
n=6 i=1j=1k=1l=1m=1 PV[S[nijklm]]=Probability Values(nijklm)
n=7 i=1j=2k=1l=1m=1 PV[S[nijklm]]=Probability Values(nijklm)
n=8 i=1j=2k=1l=2m=1 PV[S[nijklm]]=Probability Values(nijklm)
n=9 i=1j=2k=1l=2m=1 PV[S[nijklm]]=Probability Values(nijklm)
n=18i=1j=2k=3l=4m=6 PV[S[nijklm]]=Probability Values(nijklm)
n=25i=1j=2k=3l=4m=5 PV[S[nijklm]]=Probability Values(nijklm)
n=25i=1j=2k=3l=4m=5 PV[S[nijklm]]=Probability Values(nijklm)
PV119878611111
= [
1
6
1
6
1
6
1
6
1
6
1
6
] PV119878712111
= [
1
7
1
7
2
7
1
7
1
7
1
7
] PV119878812121
= [
1
4
1
8
1
4
1
8
1
8
1
8
]
PV119878912121
= [
2
9
1
9
2
9
1
9
2
9
1
9
] PV1198781812346
= [
1
9
1
18
1
9
1
6
2
9
1
3
] PV1198782512345
= [
1
5
4
25
3
25
2
25
1
25
2
25
]
PV1198782512345
= [
1
5
4
25
2
5
3
25
2
25
1
25
]
Box 3
Case A3 (JPD-coloring of the 6-monohedral tiling) SeeBox 3
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] B Grunbaum and G C Shephard ldquoTiling by regular polygonsrdquoMathematics Magazine vol 50 no 5 pp 227ndash247 1977
[2] B Grunbaum and G C Shephard ldquoPerfect colorings of transi-tive tilings and patterns in the planerdquoDiscrete Mathematics vol20 no 3 pp 235ndash247 1977
[3] S Eigen J Navarro andV S Prasad ldquoAn aperiodic tiling using adynamical system and Beatty sequencesrdquo in Dynamics ErgodicTheory andGeometry vol 54 ofMathematical Sciences ResearchInstitute pp 223ndash241 Cambridge University Press 2007
[4] C Mann L Asaro J Hyde M Jensen and T Schroeder ldquoUni-form edge-119888-colorings of the Archimedean tilingsrdquo DiscreteMathematics vol 338 no 1 pp 10ndash22 2015
[5] R Santos andR Felix ldquoPerfect precise colorings of plane regulartilingsrdquo Zeitschrift fur Kristallographie vol 226 no 9 pp 726ndash730 2011
[6] B Grunbaum P Mani-Levitska and G C Shephard ldquoTilingthree-dimensional space with polyhedral tiles of a given iso-morphism typerdquo Journal of the London Mathematical Societyvol 29 no 1 pp 181ndash191 1984
[7] M E Basher ldquo120590-Coloring of the monohedral tilingrdquo Interna-tional Journal of Mathematical Combinatorics vol 2 pp 46ndash522009
[8] J E Freund IMiller andMMillerMathematical Statistics withApplications Prentice Hall PTR 7th edition 2003
[9] D C Montgomery and G C Runger Applied Statistics andProbability for Engineers John Wiley amp Sons New York NYUSA 2nd edition 1999
[10] L Bernardin P Chin P DeMarco et al Maple ProgrammingGuide Maplesoft 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 7
gt restart with(linalg)
Probability Values=proc(nposintiposintjposint)
local AbTrvC global PV
A[T]= matrix([[111]]) b= vector([1]) C=sum(V[w]w=12)
linsolve(A[T]blsquor1015840) r linsolve(A[T]blsquor1015840v)PV[T[nij]]=subs(v[1]=inv[2]=jnv[3]=1-C)
end proc
n=3i=1j=1 PV[T[nij]]=Probability Values(nij)
n=4i=1j=2 PV[T[nij]]=Probability Values(nij)
n=5i=1j=2 PV[T[nij]]=Probability Values(nij)
n=6i=1j=2 PV[T[nij]]=Probability Values(nij)
n=7i=1j=2 PV[T[nij]]=Probability Values(nij)
n=8i=1j=2 PV[T[nij]]=Probability Values(nij)
n=8i=1j=3 PV[T[nij]]=Probability Values(nij)
PV119879311
= [
1
3
1
3
1
3
] PV119879412
= [
1
2
1
4
1
4
] PV119879512
= [
2
5
1
5
2
5
] PV119879612
= [
1
2
1
6
1
3
]
PV119879712
= [
2
7
4
7
1
7
] PV119879812
= [
5
8
1
8
1
4
] PV119879813
= [
1
2
1
8
3
8
]
Box 1
gt restart with(linalg)
Probability Values=proc(nposintiposintjposintkposint)
local AbFrvC global PV
A[F]=matrix([[1111]]) b=vector([1]) C=sum(V[w]w=13)
linsolve(A[F]blsquor1015840) r linsolve(A[F]blsquor1015840v)PV[S[nijk]]=subs(v[1]=inv[2]=jnv[3]=knv[4]=1-C)
end proc
n=4 i=1j=1k=1 PV[F[nijk]]=Probability Values(nijk)
n=5 i=1j=2k=1 PV[F[nijk]]=Probability Values(nijk)
n=6 i=1j=2k=1 PV[F[nijk]]=Probability Values(nijk)
n=7 i=1j=2k=1 PV[F[nijk]]=Probability Values(nijk)
n=8 i=1j=2k=1 PV[F[nijk]]=Probability Values(nijk)
n=8 i=1j=2k=3 PV[F[nijk]]=Probability Values(nijk)
n=8 i=1j=3k=1 PV[F[nijk]]=Probability Values(nijk)
n=9 i=1j=2k=1 PV[F[nijk]]=Probability Values(nijk)
n=9 i=1j=4k=3 PV[F[nijk]]=Probability Values(nijk)
n=13i=1j=2k=1 PV[F[nijk]]=Probability Values(nijk)
n=13i=1j=3k=4 PV[F[nijk]]=Probability Values(nijk)
n=13i=1j=1k=6 PV[F[nijk]]=Probability Values(nijk)
n=13i=1j=1k=7 PV[F[nijk]]=Probability Values(nijk)
n=13i=1j=1k=8 PV[F[nijk]]=Probability Values(nijk)
n=14i=1j=1k=2 PV[F[nijk]]=Probability Values(nijk)
n=14i=2j=3k=4 PV[F[nijk]]=Probability Values(nijk)
n=14i=1j=5k=6 PV[F[nijk]]=Probability Values(nijk)
n=14i=1j=2k=7 PV[F[nijk]]=Probability Values(nijk)
n=14i=1j=2k=8 PV[F[nijk]]=Probability Values(nijk)
n=14i=2j=2k=9 PV[F[nijk]]=Probability Values(nijk)
PV1198654111
= [
1
4
1
4
1
4
1
4
] PV1198655121
= [
1
5
1
5
2
5
1
5
] PV1198656121
= [
1
6
1
3
1
6
1
3
] PV1198657121
= [
3
7
1
7
2
7
1
7
]
PV1198658121
= [
1
8
1
4
1
2
1
8
] PV1198658123
= [
1
4
1
8
1
4
3
8
] PV1198658131
= [
3
8
1
8
3
8
1
8
] PV1198659121
= [
5
9
1
9
2
9
1
9
]
PV1198659143
= [
1
3
4
9
1
9
1
9
] PV11986513121
= [
9
13
1
13
1
13
2
13
] PV11986513134
= [
5
13
4
13
3
13
1
13
] PV11986513116
= [
1
13
6
13
5
13
1
13
]
PV11986513117
= [
1
13
7
13
4
13
1
13
] PV11986513118
= [
8
13
1
13
1
13
3
13
] PV11986514112
= [
1
7
5
7
1
14
1
14
] PV11986514234
= [
1
7
3
14
2
7
5
14
]
PV11986514156
= [
1
14
5
14
3
7
1
7
] PV11986514127
= [
2
7
1
14
1
7
1
4
] PV11986514128
= [
1
14
1
7
4
7
3
14
] PV11986514229
= [
1
14
1
7
1
7
9
14
]
Box 2
8 Abstract and Applied Analysis
gt restart with(linalg)
Probability Values=proc(nposintiposintjposintkposintlposintmposint)
local AbSrvC global PV
A[S]=matrix([[111111]]) b=vector([1]) C=sum(v[w]w=15)
linsolve(A[S]blsquor1015840) r linsolve(A[S]blsquor1015840v)PV[S[nijklm]]=subs(v[1]=inv[2]=jnv[3]=knv[4]=lnv[5]=mnv[6]=1-C)
end proc
n=6 i=1j=1k=1l=1m=1 PV[S[nijklm]]=Probability Values(nijklm)
n=7 i=1j=2k=1l=1m=1 PV[S[nijklm]]=Probability Values(nijklm)
n=8 i=1j=2k=1l=2m=1 PV[S[nijklm]]=Probability Values(nijklm)
n=9 i=1j=2k=1l=2m=1 PV[S[nijklm]]=Probability Values(nijklm)
n=18i=1j=2k=3l=4m=6 PV[S[nijklm]]=Probability Values(nijklm)
n=25i=1j=2k=3l=4m=5 PV[S[nijklm]]=Probability Values(nijklm)
n=25i=1j=2k=3l=4m=5 PV[S[nijklm]]=Probability Values(nijklm)
PV119878611111
= [
1
6
1
6
1
6
1
6
1
6
1
6
] PV119878712111
= [
1
7
1
7
2
7
1
7
1
7
1
7
] PV119878812121
= [
1
4
1
8
1
4
1
8
1
8
1
8
]
PV119878912121
= [
2
9
1
9
2
9
1
9
2
9
1
9
] PV1198781812346
= [
1
9
1
18
1
9
1
6
2
9
1
3
] PV1198782512345
= [
1
5
4
25
3
25
2
25
1
25
2
25
]
PV1198782512345
= [
1
5
4
25
2
5
3
25
2
25
1
25
]
Box 3
Case A3 (JPD-coloring of the 6-monohedral tiling) SeeBox 3
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] B Grunbaum and G C Shephard ldquoTiling by regular polygonsrdquoMathematics Magazine vol 50 no 5 pp 227ndash247 1977
[2] B Grunbaum and G C Shephard ldquoPerfect colorings of transi-tive tilings and patterns in the planerdquoDiscrete Mathematics vol20 no 3 pp 235ndash247 1977
[3] S Eigen J Navarro andV S Prasad ldquoAn aperiodic tiling using adynamical system and Beatty sequencesrdquo in Dynamics ErgodicTheory andGeometry vol 54 ofMathematical Sciences ResearchInstitute pp 223ndash241 Cambridge University Press 2007
[4] C Mann L Asaro J Hyde M Jensen and T Schroeder ldquoUni-form edge-119888-colorings of the Archimedean tilingsrdquo DiscreteMathematics vol 338 no 1 pp 10ndash22 2015
[5] R Santos andR Felix ldquoPerfect precise colorings of plane regulartilingsrdquo Zeitschrift fur Kristallographie vol 226 no 9 pp 726ndash730 2011
[6] B Grunbaum P Mani-Levitska and G C Shephard ldquoTilingthree-dimensional space with polyhedral tiles of a given iso-morphism typerdquo Journal of the London Mathematical Societyvol 29 no 1 pp 181ndash191 1984
[7] M E Basher ldquo120590-Coloring of the monohedral tilingrdquo Interna-tional Journal of Mathematical Combinatorics vol 2 pp 46ndash522009
[8] J E Freund IMiller andMMillerMathematical Statistics withApplications Prentice Hall PTR 7th edition 2003
[9] D C Montgomery and G C Runger Applied Statistics andProbability for Engineers John Wiley amp Sons New York NYUSA 2nd edition 1999
[10] L Bernardin P Chin P DeMarco et al Maple ProgrammingGuide Maplesoft 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Abstract and Applied Analysis
gt restart with(linalg)
Probability Values=proc(nposintiposintjposintkposintlposintmposint)
local AbSrvC global PV
A[S]=matrix([[111111]]) b=vector([1]) C=sum(v[w]w=15)
linsolve(A[S]blsquor1015840) r linsolve(A[S]blsquor1015840v)PV[S[nijklm]]=subs(v[1]=inv[2]=jnv[3]=knv[4]=lnv[5]=mnv[6]=1-C)
end proc
n=6 i=1j=1k=1l=1m=1 PV[S[nijklm]]=Probability Values(nijklm)
n=7 i=1j=2k=1l=1m=1 PV[S[nijklm]]=Probability Values(nijklm)
n=8 i=1j=2k=1l=2m=1 PV[S[nijklm]]=Probability Values(nijklm)
n=9 i=1j=2k=1l=2m=1 PV[S[nijklm]]=Probability Values(nijklm)
n=18i=1j=2k=3l=4m=6 PV[S[nijklm]]=Probability Values(nijklm)
n=25i=1j=2k=3l=4m=5 PV[S[nijklm]]=Probability Values(nijklm)
n=25i=1j=2k=3l=4m=5 PV[S[nijklm]]=Probability Values(nijklm)
PV119878611111
= [
1
6
1
6
1
6
1
6
1
6
1
6
] PV119878712111
= [
1
7
1
7
2
7
1
7
1
7
1
7
] PV119878812121
= [
1
4
1
8
1
4
1
8
1
8
1
8
]
PV119878912121
= [
2
9
1
9
2
9
1
9
2
9
1
9
] PV1198781812346
= [
1
9
1
18
1
9
1
6
2
9
1
3
] PV1198782512345
= [
1
5
4
25
3
25
2
25
1
25
2
25
]
PV1198782512345
= [
1
5
4
25
2
5
3
25
2
25
1
25
]
Box 3
Case A3 (JPD-coloring of the 6-monohedral tiling) SeeBox 3
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] B Grunbaum and G C Shephard ldquoTiling by regular polygonsrdquoMathematics Magazine vol 50 no 5 pp 227ndash247 1977
[2] B Grunbaum and G C Shephard ldquoPerfect colorings of transi-tive tilings and patterns in the planerdquoDiscrete Mathematics vol20 no 3 pp 235ndash247 1977
[3] S Eigen J Navarro andV S Prasad ldquoAn aperiodic tiling using adynamical system and Beatty sequencesrdquo in Dynamics ErgodicTheory andGeometry vol 54 ofMathematical Sciences ResearchInstitute pp 223ndash241 Cambridge University Press 2007
[4] C Mann L Asaro J Hyde M Jensen and T Schroeder ldquoUni-form edge-119888-colorings of the Archimedean tilingsrdquo DiscreteMathematics vol 338 no 1 pp 10ndash22 2015
[5] R Santos andR Felix ldquoPerfect precise colorings of plane regulartilingsrdquo Zeitschrift fur Kristallographie vol 226 no 9 pp 726ndash730 2011
[6] B Grunbaum P Mani-Levitska and G C Shephard ldquoTilingthree-dimensional space with polyhedral tiles of a given iso-morphism typerdquo Journal of the London Mathematical Societyvol 29 no 1 pp 181ndash191 1984
[7] M E Basher ldquo120590-Coloring of the monohedral tilingrdquo Interna-tional Journal of Mathematical Combinatorics vol 2 pp 46ndash522009
[8] J E Freund IMiller andMMillerMathematical Statistics withApplications Prentice Hall PTR 7th edition 2003
[9] D C Montgomery and G C Runger Applied Statistics andProbability for Engineers John Wiley amp Sons New York NYUSA 2nd edition 1999
[10] L Bernardin P Chin P DeMarco et al Maple ProgrammingGuide Maplesoft 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Related Documents
