Transcript
Who killed Mr. X?Who killed Mr. X?
THODORIS ANDRIOPOULOSTHODORIS ANDRIOPOULOS
The investigation of a crime by a The investigation of a crime by a
detective mathematician!detective mathematician!
The story takes place in 1900 in Paris …
… where one of the most significant
mathematics conferences is being held.
At the entrance of the hotel conference
room there is an inscription:
Let no one ignorant of Geometry Let no one ignorant of Geometry Let no one ignorant of Geometry Let no one ignorant of Geometry
enter these doorsenter these doorsenter these doorsenter these doors
Mathematics is the Absolute Truth.
Sooner or later, it can prove whether a theory is right or wrong or it can characterize a
sentence true or false.
From the podium the famous
mathematician, Mr. X, states at the
closing of his speech:
The work of the first day of the
conference is done.
Mr. X is alone in the dining room
reading.
A waiter enters … We must know the truth
The truth is we have to close the
bar. Do you need
anything?
Mr. X asks for a glass of water and the waiter
leaves to bring it.
The mathematicians are totally crazy
The waiter returns.
Mr. X is dead…
The waiter is interrogated by detective
Kurt. He describes the preceding scene
and adds that one second before he
entered the dining room for the second
time, he heard the central hotel clock
striking.
The detective confirms that the
murderer had 20 seconds at his disposal
to commit the crime, which is the time
between the waiter’s exit and his re-
entrance to the dining room.
Mr. Kurt talks with his assistant.
Let us assume that the murderer committed the crime immediately after the waiter left the dining room. That
gave him 20 seconds to move around until the clock struck.
I believe the murderer was walking as he left the dining room in order not to
look suspicious, so, estimating someone walks one meter per second, then the murderer could not have
been more than 20 meters away from the scene of the crime when the clock
struck.
Detective Kurt questions the suspects…
Here is the hotel ground plan
S3
S5 S4
S2
S1
M
Α
D
Β
Η
Ζ
C
Ρ
W
Ν
Ι
J
Ο
L
ΚF
V
Q
G Ε
U
After the first suspects S1, S2, S3, S4, and S5 testify, their positions are
placed on a diagram (a ground plan of the hotel), where M is the spot
where the murder took place.
The suspects could only move on the lines that represent the hotel
corridors shown below.
EVIDENCEMr. Karl Friedrich (S1) states that he was at point A when
the clock struck.
2w 9−
3w 1−
w 8+
2w 2w 20− −
S1
Α
ΒC
DM
SOLUTION-ANSWER
The triangles ABC and ADM are similar soC C
M DM
Α Β=
Α
2 2
w 8 3w 1
w w 12 w 9
+ −=
− − −
KNOWN
ELEMENTS:
ΒC // DM
ΑC=w+8
ΒC=3w-1
CM=w2-2w-20
DM=w2-9
ΑM=30m, so Mr. Karl Friedrich cannot be the murderer.
and the root is w=7 (We don’t accept the root w=2 since it produces DM= -5)
w 8 3w 1or
(w 4)(w 3) (w 3)(w 3)
+ −=
− + − +
w 8 3w 1or
w 4 w 3
+ −=
− −( ) ( )or w 8 (w 3) 3w 1 (w 4)+ − = − −
2or 2w 18w 28 0− + =
SOLUTION-ANSWER
EVIDENCEMr. Constantin (S2) states that
he was at point H when the clock struck.x+6y-8
3x-y-3
2x+y+5
x+2y+10S2
Ζ
D
Η
M
G Ε
ΗG=DΖ and ΗM=ΕΖ
3x-y-3= x+6y-8 and
2x+y+5=x+2y+10
x=8 and y=3
ΗM=24m , so Mr. Constantin cannot be the murderer.
KNOWN ELEMENTS:
DΖ= x+6y-8
ΕΖ=x+2y+10
ΗG=3x-y-3
ΗM=2x+y+5
oZ H 90∧ ∧
= =
2x-7y= -5 and
x-y=5
2x-7y= -5 and
x=y+5
2(y+5)-7y= -5 and
x=y+5
2y+10-7y= -5 and
x=y+5
-5y= -15 and
x=y+5
The triangles ΗGM and DΕΖ are equal so
Let us calculate the distance KF+FM.
KL//JΙ since K and L are the mid-points of the sides NI and NJ in triangle JNI.
F is the mid-point of NO since in the triangle ONI K is the mid-point of NI and KF//OI.
We have since K, F and L are mid-points of NI, NO and NJ.
NF=KF=LF since KNL is a right-angle triangle and NF is the median.
Therefore: KF+FM=NF+FM=NM=JΙ=24 (The diagonals of a rectangle are equal).
We must prove that the shortest distance from point K to point M is KF+FM=NM and not KI+IM.
NM= from the Pythagorean Theorem in triangle MNI.
It is sufficient to prove that KI+IM>NM, or
We square both sides,
which is a valid operation, since is given b>a.
The shortest distance that Mr. Isaac could have covered is KF+FM=24m, so he cannot be the murderer.
EVIDENCEMr. Isaac (S3) states that he was at
point K when the clock struck.
KNOWN
ELEMENTS:
ΚΝ=ΚΙ
LΝ=LJ
ΟΙ=ΟJ=12
a<b
SOLUTION-ANSWER
L
S3
b b
a
a
12
12
J
Ο
M
Ν
FΚ
Ι
2 2(2a) (2b)+
2 2a 2b (2a) (2b)+ > +2 2 2 2 2 2 2 2 3
(a 2b) (2a) (2b) a 4ab 4b 4a 4b 4ab 3a b a ,4
+ > + ⇔ + + > + ⇔ > ⇔ >
KF FL2
ΟΙ= =
G Ε
D
24
Η
M
40
Ρ
Χ
Τ
R
Y
KNOWN
ELEMENTS:
DM=GΕ=40
MΗ=24
90∧
οΗ =
The detective’s assistant cries out:
Mr. Leonhard was at point M, so he is the murderer!
EVIDENCEMr. Leonhard (S4) states that when the clock struck he was in corridor MD at a point such that the distance from corridor ΡD plus the distance from
corridor MΡ was equal to 24m.
SOLUTION-ANSWER
We assume point R on the side MD.
We take the heights RT to MH and RY to ΡD.
RY=TH (1), since RYHT is a rectangle.
We take the height RX to MP.
The triangles RMΧ and RTM are equal, since they are both right, they have a common side RM and
angles , since angles (the triangle MPD is isosceles) and angles
(corresponding angles on RT//DP).
So, RX=TM (2) and from (1) and (2) we have: RX+ RY=ΤM+ΤΗ=MH=24.
This means that the point R can be any point on the side MD, therefore we cannot conclude whether Mr.
Leonhard is guilty or innocent, since he could have been either to the right of the mid-point of MD or to
the left of the mid-point of MD.
RMX TRM∧ ∧
= RMX RDY∧ ∧
= TRM RDY∧ ∧
=
EVIDENCEMr. René (S5) states that when the clock
struck he was at point V and if the rectangle MVQW had an area equal to four times it’s actual area and remained similar to the initial rectangle, then the
distance from point M would have been 60m.
The detective’s assistant cries out: One fourth of 60 is 15, so Mr.
René was at a distance 15m from point M, so he is the murderer!
S5M
W
V
Q
SOLUTION-ANSWER
Let E be the actual area of the rectangle and E΄ the area of the similar rectangle
then its true that
since the ratio of the areas of two similar shapes is equal to the square of the ratio
of their sides.
So, Mr. René cannot be the murderer.
2 2E d 1 d 1 d
o r o r an d fin a lly, d = 3 0 m4 E 6 0 4 6 0 2 6 0
= = =
2E d
E d
= ′ ′
The function d(x) has a minimum value.
Let us calculate this value:
The discriminant is ∆= -200
The minimum value of d(x) is dmin= 254
∆− =
αSo, Mr. Pierre cannot have been the murderer since his minimum distance
from point M was 25m.
SOLUTION-ANSWER
The detective’s assistant cries out:
We can’t find the answer! We have to calculate the value of the function for an
infinite number of values of x.
EVIDENCEMr. Pierre (S6) states that his distance from point M when the clock struck
is given by the function d(x)=2x2-12x+43 for an appropriate value of x
5d 102.5
2d 4
+=
+
So, the equation is indeterminate.
Then, the distance could be 0 or any positive number.
Therefore, we cannot conclude whether Mr. Blaise is guilty or innocent.
SOLUTION-ANSWER
If d is the distance then it holds that:
or 5d 10 2.5(2d 4)+ = + or 5d 10 5d 10+ = +
or 0 d 0.⋅ =
The detective’s assistant cries out:
The answer is 0 then Mr. Blaise was at point M, so he is the murderer!
EVIDENCEMr. Blaise (S7) states: “You can find my distance from point M when the
clock struck, if you know that five times this distance increased by 10 and the whole thing divided by twice this distance increased by 4,
is equal to 2.5m.I know this statement is too long but I didn’t have time to make it shorter”
The assistant informs the detective
that an employee heard
Mr. Pheidias (S8) saying to someone:
“Do as I tell you, and
you will be rewarded with gold.”
The detective calls in Mr. Pheidias to
give evidence.
EVIDENCE
What Mr. Pheidias (S8) said was: “Separate a line segment 10cm
in length into two parts, one with length x and one with length
10-x so that: x2=10.(10-x). Then calculate the ratio
Do as I tell you, and you will be rewarded with gold.”
SOLUTION-ANSWER
The equation becomes x2+10x-100=0,
whose positive root is the number
Then, , which is the number φ of the Golden Ratio.
Any piece of artwork containing this number offers us the sense of harmony
and beauty.
Therefore, the gold that Mr. Pheidias promised was the “golden number” φ,
so we cannot consider him guilty.
5( 5 1)−
10 5 1
x 2
+=
10.
x
EVIDENCEMr. Evarist (S9) states that he knows who killed Mr. X.
He knows, because they both come from France.
“Mr. Pierre (S6) lied to you. He killed Mr. X!
I know the French very well, and they are all liars.”
SOLUTION-ANSWER
Assistant: If Mr. Evarist is telling the truth, then Mr. Pierre lied
to us and…
Detective: One moment. If Mr. Evarist is telling the truth that
the French always lie, then since Mr. Evarist is also French,
he is also lying that he knows the killer.
Assistant: So Mr. Evarist is lying.
Detective: If Mr. Evarist is lying about the French being liars,
then the French tell the truth and so Mr. Evarist, as a
Frenchman, is telling the truth.
Assistant: If Mr. Evarist is telling the truth, then… he is lying,
on the other hand, if he is lying, then… he is telling the truth.
Detective Kurt confirms the conclusion with a grimace and,
putting his hands to his temples, he thinks while staring out
of the window.
Assistant: After all mathematics
doesn’t have all the answers.
Detective: What did you say?
Assistant: I said Mathematics can’t
solve all problems.
Suddenly the detective’s face lights up
and he mumbles to himself as he
leaves…
That killed him!!!
Everyone is gathered in the conference room.
Detective Court explains:
In the last lines of his notes, Mr. X wrote:Sentence A: “Mathematics can’t prove
sentence A”
If the above sentence is characterized as true, then its meaning is corroborated. So the
conclusion is mathematics can’t prove a true sentence.
However, if sentence A is characterized as false that means mathematics can prove a
false sentence, which is not acceptable.
Conclusion: If sentence A is true mathematics can’t prove it.
Mathematics is the Absolute Truth.
Sooner or later, it can prove whether a theory is right or wrong or it can characterize a sentence true or
false.
It seems that Mr. X found out that
Mathematics is not complete,
in other words, that there will always be sentences or theories for which we cannot
determine whether they are true or false.
Mr. X dedicated his life tο the quest for truth and
when it was revealed to him, it took his own life.
The truth killed Mr. X!!!
THE THEORY
OF INCOMPLETENESS,
as proven by Kurt Gödel in 1931,
is unfortunately…
REAL!!!
The above story is thankfully… imaginary!
The Protagonists who took part unintentionally are:
Detective Kurt Kurt Gödel
Austrian Mathematician (1906-1978)
Mr. X David Hilbert
German Mathematician (1862-1943)
Suspect Karl Friedrich (S1) Karl Friedrich Gauss
German Mathematician (1777-1855)
Suspect Constantin (S2) Constantin Carathéodory
Greek Mathematician (1873-1950)
Suspect Isaac (S3) Isaac Newton
British Mathematician (1642-1727)
Suspect Leonhard (S4) Leonhard Euler
Swiss Mathematician (1707-1783)
Suspect René (S5) René Descartes
French Mathematician (1596-1650)
Suspect Pierre (S6) Pierre de Fermat
French Mathematician (1601-1665)
Suspect Blaise (S7) Blaise Pascal
French Mathematician (1623-1662)
Suspect Pheidias (S8) Pheidias
Greek Sculptor (498 BC – 432 BC)
Suspect Evarist (S9) Evarist Galois
French Mathematician (1811-1832)
THODORIS ANDRIOPOULOS
Project Design
This project is dynamic in design. The following
adjustable parameters can be manipulated in each
application by the teacher:
� Curricular Topics
� Level of Difficult
� Time Duration
� Application Area (Mathematics, Physics, …)
It can be easily adopted for all grade levels and
student abilities
This presentation constitutes a 4 hour review of the 3rd grade junior high school mathematics curriculum, employing a different and hopefully interesting way of teaching, as conducted at
the end of the academic year at the Anatolia College of Thessaloniki.
The topics that are included in this presentation are:
Similar triangles InequalitiesFractional equations IdentitiesQuadratic equations Angles between parallel linesEquality of triangles Isosceles trianglesTheory of parallelograms Areas ratio of similar shapesSystem of equations Max-min points of parabolasTheory of mid-points of triangles Indeterminate equationsMedian of a right triangle The Golden ratio or Golden MeanThe Pythagorean Theorem LogicSquare roots A little… Math history
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