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http://www.scctmprogram.org/2014-Sept-MathMate.pdf
CRYPTARITHMETIC RULES
Find which digits should replace each of the letters in each of the following word sums. For each problem, each of the letters stands for a different digit. The first digit in a number is never zero. Replace the letters so that each sum is correct.
Are these possible? If so, can you find more than one solution? Are there restrictions on which digits are possible for any of the letters?
CCSS CONTENT STANDARDS ADDRESSED
Students should :
• Understand the difference between face value and place values of digits in multi-digit numbers (2.NBT.1, 5.NBT.1)
• Use their place value understanding and properties of operations to perform multi-digit arithmetic (3.NBT.2, 4.NBT.4)
• Evaluate expressions in which letters stand for numbers (6.EE.2)
• Determine whether a given number makes an equation true (6.EE.5)
• Understand that a variable can represent an unknown number or any number in a specified set (6.EE.6).
CCSS STANDARDS FOR MATHEMATICAL PRACTICE ADDRESSEDMake sense of problems and persevere in solving them Understand what the problem is asking –
what is the nature of the solutions which exist?
Construct viable arguments & critique the reasoning of others Describe the constraints on variables. Listen to others’ solution methods Evaluate efficiencies of different solution
methods.Attend to precision Make sure the numbers add correctly. Make sure that each different letter is
represented by a different digit. Keep track of the digits they have already
used & the digits that are remaining. Find ways to record their valid & invalid
solutions.
Look for and express regularity in repeated reasoning Look for patterns. Reason abstractly and quantitatively Substitute letters for digits. Understand that the digit chosen for each
letter is a face value. Understand the place value of the digit &
its corresponding letter.Use appropriate tools strategically Decide what tools to useLook for & make use of structure Create a system to record addends which
result in a carry. Determine when a carry to the next higher
place value will be used.Model with mathematics Letter-number substitution problems are
used in cryptography, a real-world application.
W I N + T H ER A C E
Find which digits should replace each of the letters in each of the following word sums. For each problem, each of the letters stands for a different digit. The first digit in a number is never zero. Replace the letters so that each sum is correct.
Can you find more than one solution? Are there restrictions on which digits are possible for any of the letters? Explain how you know.
Find which digits should replace each of the letters in each of the following word sums. For each problem, each of the letters stands for a different digit. The first digit in a number is never zero. Replace the letters so that each sum is correct.
Can you find more than one solution? Are there restrictions on which digits are possible for any of the letters? Explain how you know.
+
REDUCING THE PROBLEM
F O U R
+ O N E
F I V E
• R + E = E, so R = 0 (identity property of addition)
• O + O = I with no carry, so O = 1, 2, 3, 4
EXTENSION QUESTIONSQuestion 1: Did you notice any shortcuts to
finding solutions, and if so, why are these valid shortcuts?
Question 2: Given a solution to the FOUR+ONE=FIVE problem, 5130 + 167 = 5297, find as many different solutions as you can using the same digits. How will you know when you have found all of the possible solutions using these digits?
Question 3: Find the total number of solutions there are to this word sum. Describe how you found this number of solutions.
Question 4: Answer the question, “If I find all of the solutions to this problem for F = 1, can I multiply that number of solutions by 9 to find the total number of solutions to this problem? Why or why not?”
Question 5: Create your own letter-number substitution problem and find its solution(s).
SOLUTION METHOD #1: PLACE VALUES
F1O1UR = F*1000+O1*100+U*10+R O2NE = O*100+N*10+E
F O1 U R
1000 100 10 0
2000 200 20
3000 300 30
4000 400 40
5000 50
6000 60
7000 70
8000 80
9000 90
O2 N E100 10 1200 20 2300 30 3400 40 4
50 5 60 6 70 7 80 8 90 9
DISTINCT DIGITS FOR F1, O1, U, R
(219 SETS)
F1 = 1
F1 = 2
F1= 3
F1 = 4
F1 = 5
F1 = 6
F1 = 7
F1 = 8
F1 = 9
1230 2130 3120 4120 5120 6120 7120 8120 91201240 2140 3140 4130 5130 6130 7130 8130 91301250 2150 3150 4150 5140 6140 7140 8140 91401260 2160 3160 4160 5160 6150 7150 8150 91501270 2170 3170 4170 5170 6170 7160 8160 91601280 2180 3180 4180 5180 6180 7180 8170 91701290 2190 3190 4190 5190 6190 7190 8190 91801320 2310 3210 4210 5210 6210 7210 8210 92101340 2340 3240 4230 5230 6230 7230 8230 92301350 2350 3250 4250 5240 6240 7240 8240 92401360 2360 3260 4260 5260 6250 7250 8250 92501370 2370 3270 4270 5270 6270 7260 8260 92601380 2380 3280 4280 5280 6280 7280 8270 92701390 2390 3290 4290 5290 6290 7290 8290 92801420 2410 3410 4310 5310 6310 7310 8310 93101430 2430 3420 4320 5320 6320 7320 8320 93201450 2450 3450 5340 6340 7340 8340 93401460 2460 3460 5360 6350 7350 8350 93501470 2470 3470 5370 6370 7360 8360 93601480 2480 3480 5380 6380 7380 8370 93701490 2490 3490 5390 6390 7390 8390 9380
5410 6410 7410 8410 9410 5420 6420 7420 8420 9420 5430 6430 7430 8430 9430 5460 6450 7450 8450 9450 5470 6470 7460 8460 9460 5480 6480 7480 8470 9470 5490 6490 7490 8490 9480
O2 = 1 O2= 2 O2= 3 O2= 4120 165 201 256 301 356 401 456
123 167 203 257 302 357 402 457
124 168 204 258 304 358 403 458
125 169 205 259 305 359 405 459
126 170 206 260 306 360 406 460
127 172 207 261 307 361 407 461
128 173 208 263 308 362 408 462
129 174 209 264 309 364 409 463
130 175 210 265 310 365 410 465
132 176 213 267 312 367 412 467
134 178 214 268 314 368 413 468
135 179 215 269 315 369 415 469
136 180 216 270 316 370 416 470
137 182 217 271 317 371 417 471
138 183 218 273 318 372 418 472
139 184 219 274 319 374 419 473
140 185 230 275 320 375 420 475
142 186 231 276 321 376 421 476
143 187 234 278 324 378 423 478
145 189 235 279 325 379 425 479
146 190 236 280 326 380 426 480
147 192 237 281 327 381 427 481
148 193 238 283 328 382 428 482
149 194 239 284 329 384 429 483
150 195 240 285 340 385 430 485
152 196 241 286 341 386 431 486
153 197 243 287 342 387 432 487
154 198 245 289 345 389 435 489
156 246 290 346 390 436 490
157 247 291 347 391 437 491
158 248 293 348 392 438 492
159 249 294 349 394 439 493
160 250 295 350 395 450 495
162 251 296 351 396 451 496
163 253 297 352 397 452 497
164 254 298 354 398 453 498
Distinct digits for O2, N, E
(280 sets)
SUMS FOR EACH COMBINATION OF F1O1UR AND O2NE
(219 * 280 = 61,320 SETS)
O2NE
120 123 124 125 126
F1O
1UR
1230 1230+120=13
501230+123=1353
1230+124=1354
1230+125=1355
1230+126=1356
1240 1240+120=13
601240+123=1363
1240+124=1364
1240+125=1365
1240+126=1366
1250 1250+120=13
701250+123=1373
1250+124=1374
1250+125=1375
1250+126=1376
1260 1260+120=13
801260+123=1383
1260+124=1384
1260+125=1385
1260+126=1386
1270 1270+120=13
901270+123=1393
1270+124=1394
1270+125=1395
1270+126=1396
1280 1280+120=14
001280+123=1403
1280+124=1404
1280+125=1405
1280+126=1406
1290 1290+120=14
101290+123=1413
1290+124=1414
1290+125=1415
1290+126=1416
1320 1320+120=14
401320+123=1443
1320+124=1444
1320+125=1445
1320+126=1446
1340 1340+120=14
601340+123=1463
1340+124=1464
1340+125=1465
1340+126=1466
1350 1350+120=14
701350+123=1473
1350+124=1474
1350+125=1475
1350+126=1476
1360 1360+120=14
801360+123=1483
1360+124=1484
1360+125=1485
1360+126=1486
1370 1370+120=14
901370+123=1493
1370+124=1494
1370+125=1495
1370+126=1496
1380 1380+120=15
001380+123=1503
1380+124=1504
1380+125=1505
1380+126=1506
1390 1390+120=15
101390+123=1513
1390+124=1514
1390+125=1515
1390+126=1516
LOGICAL STATEMENT CHECKS
For each of 61,320 possibilities for “F1O1UR + O2NE = F2IVE,” check 10 logic statements:
Whether F1 is the same as F2 Whether O1 is the same as O2 Whether F1 is distinct from N, E, I, VWhether O1 is distinct from N, E, I, VWhether U is distinct from N, E, I, VWhether R is distinct from N, E, I, V Whether N is distinct from I, VWhether E is distinct from I, VWhether I is distinct from V
If and only if all of the above are true, then the solution is valid for “FOUR + ONE = FIVE.”
Observation #1
F and E do not depend on the other letters. There are 36 pairs of 2-digit combinations: 1,2 2,3 3,4 4,5 5,6 6,7 7,8 8,9 1,3 2,4 3,5 4,6 5,7 6,8 7,9 1,4 2,5 3,6 4,7 5,8 6,9 1,5 2,6 3,7 4,8 5,9 1,6 2,7 3,8 4,9 1,7 2,8 3,9 1,8 2,9 1,9
The possibilities for F and E are interchangeable.
METHOD 2: COMBINATORICS #1 F O U R + O N E F I V E
Observation #2
O + O = I with no carry. Thus, O can only be 1, 2, 3, or 4.
Observation #3
U + N = V can, but does not need to, have a carry.
Observation #4
The possibilities for U and N are interchangeable (commutative property of addition).
* For every combination of F, E, U, N, we can wrote 4 possible arrangements: FEUN, EFUN, FENU, EFNU. Thus the total # of possible solutions is a multiple of four.
F O U R + O N E F I V E
METHOD FOR GENERATING SOLUTIONS – DISTINCT DIGITSFor each of the 36 pairs of (F,E) possibilities, find O, I, U, N, V
possibilities:O + O = I No Carry from U + N = V
1+1=2 1+2=3 2+3=5 3+4=7 4+5=9
2+2=4 1+3=4 2+4=6 3+5=8
3+3=6 1+4=5 2+5=7 3+6=9
4+4=8 1+5=6 2+6=8
1+6=7 2+7=9
1+7=8
1+8=9
O + O = I Carry from U + N = V
1+1 3 2+9 1 4+8 2 5+8 3 6+9 5
2+2 5 3+8 1 4+9 3 5+9 4 7+8 5
3+3 7 3+9 2 5+6 1 6+7 3 7+9 6
4+4 9 4+7 1 5+7 2 6+8 4 8+9 7
There are 300 solutions for (F, E, O, I, U, N, V) with F<E and U<V, so there are 300* 4 = 1200 total solutions
METHOD 3: COMBINATORICS #2
• For every combination of (O, I, V), there are 2 combinations of (U,N).
• For every combination of (O, I, V, U, N, R=0), there are 12 combinations of (F, E):
* Since O, I, V, U, N, R represent 6 distinct digits, there are 4 remaining digits for F and E
* For each of the 4 digits that are possible for the value of F, there are 3 remaining digits possible for the value of E. (4 * 3 = 12)
If we find the number of combinations of
(O, I, U, N, V, R) where U < N,
we have 2*12 = 24 times more
total solutions to (F, E, O, I, U, N, V, R)
F O U R + O N E F I V E
50 COMBINATIONS OF (O, U, N, I, V)
O U N I V O U N I V O U N I V O U N I V
1 3 4 2 7 2 1 5 4 6 3 1 4 6 5 4 1 2 8 3
1 3 5 2 8 2 1 6 4 7 3 1 7 6 8 4 1 5 8 6
1 3 6 2 9 2 1 7 4 8 3 1 8 6 9 4 1 6 8 7
1 4 5 2 9 2 1 8 4 9 3 2 5 6 7 4 2 3 8 5
1 4 8 3 2 2 3 5 4 8 3 2 7 6 9 4 2 5 8 9
1 5 7 3 2 2 3 6 4 9 3 2 9 7 1 4 2 7 8 9
1 5 9 3 4 2 3 8 5 1 3 4 5 6 9 4 3 6 8 9
1 6 8 3 4 2 4 7 5 1 3 4 8 7 2 4 3 8 9 1
1 6 9 3 5 2 4 9 5 3 3 5 6 7 1 4 5 6 9 1
1 7 8 3 5 2 6 7 4 3 3 5 9 7 4 4 5 7 9 2
1 7 9 3 6 2 6 8 5 4 3 6 8 7 4 4 5 8 9 3
1 8 9 3 7 2 7 9 5 6 3 6 9 7 5 4 6 7 9 3
2 8 9 5 7 4 7 8 9 5
F O U R + O N E F I V E
50 * 24 = 1200 TOTAL SOLUTIONS