Problems & Puzzles: Puzzles

Puzzle 840. Second follow up to Puzzle 835. Jean Brette sent the following nice puzzle: Find smallest prime matrixes such that all the absolute differences between contiguous elements give distinct even integers. Suppose that every row and column is a circular array of integers. Smallest matrix means here using the smallest larger prime. Example, N=3 Even integers produced: 2, 4, 6, 8, 10, 12, 14, 16, 18, 22, 26, 30, 32, 34, 38, 48, 50, 54 Q1 : Does it exist such a square where the biggest prime is  <61 ? Q2 : Does it exist such squares  n x n  (n > 3) where all the primes and all the even differences are different ? Q3 :  What about rectangles   n x m, with n > 2 and m > n?

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Contributions came from Claudio Meller and Dmitry Kamenetsky

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Claudio wrote:

Q1. Here is a smaller solution:

Largest prime=47. Diffferences : 2    4    6    8    10    12    14    16    18    20    22    24    26    28    30    36    38    42​

 

19	41	17		22	24	2
23	3	11		20	8	12
5	31	47		26	16	42
4	38	6
18	28	36
14	10	30

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Dmitry wrote:

1. There is a slightly better 3x3 solution with largest prime 43:

 

29 7 23 

19 37 11 

5 41 43 

even produced: 2 4 6 8 10 12 14 16 18 20 22 24 26 30 32 34 36 38

 

2. Here are the solutions I found:

 

4x4, largest 71

67 7 29 5 

3 23 37 71 

61 19 47 17 

11 59 41 43 

evens: 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 38 40 42 44 48 50 52 54 56 58 60 62 64 66 68

 

5x5, largest 113

41 103 7 71 17 

97 29 107 19 101 

5 109 59 53 43 

113 3 89 13 11 

23 31 67 83 37 

evens: 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 46 48 50 52 54 56 58 60 62 64 68 70 72 74 76 78 80 82 84 86 88 90 92 96 100 102 104 106 108 110

 

6x6, largest 179

97 43 53 173 61 29 

127 83 139 7 157 17 

67 59 23 167 5 131 

137 31 179 3 107 13 

37 109 11 149 103 89 

71 151 101 19 41 163 

evens: 4 8 10 12 14 20 22 24 26 28 30 32 34 36 40 42 44 46 48 50 52 54 56 60 62 64 68 70 72 74 76 78 80 82 86 90 92 94 96 98 100 102 104 106 108 110 112 114 116 118 120 122 124 126 130 132 134 138 140 144 146 148 150 152 154 156 160 162 164 166 168 176

 

7x7, largest 241

29 193 7 233 31 23 67 

199 11 241 19 131 229 5 

83 223 163 53 73 71 157 

61 191 59 107 17 139 167 

227 103 197 89 113 127 47 

109 149 13 181 97 101 173 

evens: 2 4 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 54 56 58 60 62 64 66 68 72 74 76 78 80 82 84 86 88 90 92 94 96 98 100 102 104 106 108 110 112 114 116 118 120 122 124 126 130 132 136 138 140 142 144 146 148 152 158 160 164 166 168 170 172 174 180 182 184 186 188 190 194 196 198 202 204 206 208 212 214 222 224 226 230 234

 

8x8, largest 349

271 23 269 313 17 349 61 71 

83 229 233 97 331 107 79 241 

7 263 103 311 101 109 227 163 

179 37 347 5 191 67 73 47 

193 337 29 307 53 317 43 139 

281 197 151 251 31 19 211 59 

199 127 89 283 11 223 137 239 

3 293 41 149 277 173 113 181 

evens: 2 4 6 8 10 12 14 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 68 70 72 74 76 78 80 82 84 86 88 90 92 96 100 102 104 108 110 116 118 122 124 128 130 132 134 136 138 140 142 144 146 148 152 154 156 158 160 162 164 166 168 170 172 176 178 180 186 188 192 194 196 200 204 206 208 210 212 214 216 220 222 224 226 228 230 234 242 244 246 248 250 252 254 256 260 264 266 268 270 272 274 278 288 290 296 298 300 302 306 308 310 314 318 332 342

 

9x9, largest 433

17 223 167 397 47 389 251 89 409 

347 139 151 23 227 173 79 311 13 

7 433 281 359 157 269 233 97 401 

379 11 19 383 239 71 283 349 3 

131 103 421 31 197 307 41 37 229 

43 263 137 271 163 5 337 293 191 

353 67 313 241 59 331 61 83 73 

109 367 113 181 419 53 193 317 431 

373 29 199 179 127 101 257 211 149 

evens: 2 4 6 8 10 12 16 20 22 24 26 28 30 32 34 36 38 42 44 46 48 50 52 54 56 60 62 64 66 68 70 72 78 80 82 84 86 88 92 94 96 98 102 104 106 108 110 112 114 118 122 124 126 128 130 132 134 136 138 140 144 148 152 154 156 158 160 162 166 168 170 172 176 180 182 192 194 196 198 200 202 204 206 208 210 212 214 216 218 220 222 224 226 230 232 234 236 238 240 242 244 246 248 252 254 256 258 260 262 264 266 270 272 276 278 280 282 284 286 288 292 294 296 298 300 302 304 310 312 318 320 322 326 330 332 334 336 338 340 342 344 346 350 352 356 358 360 364 366 368 372 374 376 388 390 392 394 396 398 402 422 426

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Claudio Meller wrote on August 10, 2016

Hola encontré una solución distinta a la de Dmitry (con menor suma total de los primos usados).
 

3	5	23		2	18	20
13	41	7		28	34	6
43	19	11		24	8	32
10	36	16
30	22	4
40	14	12

 

Hola, sigo buscando estos usan el 2 y el máximo es el 37

2	7	29		5	22	27
11	31	3		20	28	8
23	37	19		14	18	4
9	24	26
12	6	16
21	30	10

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