This page presents all the 280 Euler's magic squares . The solutions are obtained from 10 origins on the chess-board which allow by rotation and symmetry to obtain all the possibilities. These 10 boxes are called "starting zone ".

The squares of the chess-board are numbered from 0 to 63, from left to right and upwards. The starting zone includes squares 0,1,2,3,9,10,11,18,19,27

By several rotations and/or symmetries, each found solution generates 7 other solutions.

Moreover, each solution S, obtained from a given start, has a dual solution , whose origin is the last square of S, and whose way is exactly the opposite way of S. By a series of rotations and/or symmetries, the last square of a solution can be placed in the starting zone. From this square, one can thus obtain a solution which was already found: each solution is thus found 2 times. (there is no self-dual solution). There are thus only 140 independent solutions.

Among the 280 solutions, 126 are made of closed loops. A closed loop is a path whose last square, containing index 64 is far of one knight step from the first square containing index 1. In a closed loop, one can thus shift each square, the first square of index 1 becoming 2, the 2 becoming 3� square 63 becoming 64 and 64 becoming the first square of index 1. It is possible to make this operation 63 times to obtain 63 other solutions. All these "new" solutions do no form magic squares systematically.

However, it is noted that for certain solutions and particular values of shift, one obtains new magic squares. If the algorithm is correct, each of these "new" magic squares should no be new but must be after geometrical transformation, an already found solution. A simple tool was developed to correlate the whole of the solutions of closed loops. It produced the following tables.

The first line (8,16,24,32,40,48,56) indicates the shift values of the closed loops. Only multiple of 8 provide other magic squares. The solutions are noted x_y with x is the first square (in the starting zone) and y the rank of the solution found by the algorithm, i.e. its order of discovery. Ranks of the various solutions are presented thereafter in the summary tables. The first column lists the solutions of closed loops. Each line indicates the other associated solutions, the column indicating the value of the shift. When a solution was already associated, it does no appear any more in the first column. One obtains thus by line, families of solutions of closed loops. The results are presented in two tables: each line of the first table identifies the solutions (from one to five solutions), the dual solutions being indicated on the same line of the second table. Each table has 31 lines: there are only 31 distinct solutions among the 126 closed loop solutions.

It is checked well that all the solutions obtained by shift had already been found directly. By taking account of these particular relations between closed loops, there are only 108 independent solutions, i.e. 108 solutions which do no have relations of symmetry or shift. These 108 solutions consist of 77 open solutions (the start square is no away from a jump to the last square) and of 31 closed solutions.

The following tables present all the magic solutions of Euler's Magic Kinght Tour. The research time for each solution is indicated in seconds. It's worth noting that there is no solution with unresolved dual associated, what tends to show that the algorithm traverses the totality of the tree of research (may be no bug). The algorithm is no very long but a bug is still possible.

The information contained in the tables of results is as follows:

Start : Start square

Sol. : Index of the solution

Time : Computing time to obtain the solution

DP Valeur of the principal diagonal A1-H8 (there is no magic square with magic diagonals)

DS : Value of the secondary diagonal H1-A8

SM : Semi Magic square (each 1/4 of the square is magic). There are 4 of them.

CT : Close Tour. The last square is from one step to start square.

End : Last square of the way.

Dual : Start square of the Duale solution, i.e. last square of the way placed in the starting zone.

Sol. : Index of the solution on the basis of the dual's start square solution

Time : Computing time (starting from the dual solution)


The evolutions of the search times starting from the 10 possible start squares are indicated in another page.

Solutions are grouped by starting square. The time of the complete course of the chess-board, for each starting square is indicated to the beginning of each table. The solutions are classified by order of discovery. Times are expressed in seconds.

After each table of results, two chess-boards are presented. The position of the start square is green. The position of the last squares of the way is yellow. Each number indicates the number of distinct solutions leading to this square. When the way is a closed loop (closed tower), the square is violet (given provided by Harold Cataquet [ cataquet@ntlworld.com ]). The second chess-board presents all the arrivals brought back in the starting zone. It is noted that it is possible to obtain ways whose square of arrival is with some rotations and symmetries, the same one as the starting box. When the departure is on the diagonal (start 0, 9, 18, 27), all the second boxes of the found ways are located in the low half of the chess-board, on the right of the principal diagonal (A1-H8). The symmetrical ways neither are sought, nor presented.

One can decrease the total search time for all the magic squares starting from the 10 start squares of the starting zone while avoiding again seeking ways providing a last square already taken as start square, in a preceding research. Thus, whenever all ways are found from a given start square, the tree of search for the other solutions (from other start squares) are smaller. The new times obtained by applying this rule are indicated in this new table.

All solutions starting from each departure are grouped on one page. The indexes of these solutions are different from the indexes in the table because the table was constructed in the early 2000s using a single-threaded program, while the page groups the solutions using a multi-threaded program developed a few years later. Of course, the solutions are identical, only the numbering is different.

    All Magic solutions from start 0

start 0 - # Solutions 9 ==== 1 146 284 Sec. ====
Start
Sol.
 Time
DP
DS
 SM CT End Dual Sol. Time
0 1 7 453 292 236 no no 58 2 31 200 511
0 2 60 370 292 206 yes no 58 2 7 132 561
0 3 60 853 292 236 yes no 58 2 8 132 561
0 4 703 085 282 210 yes no 58 2 6 132 561
0 5 706 269 264 228 no no 58 2 10 137 354
0 6 706 792 280 212 no no 58 2 11 138 011
0 7 709 898 264 228 no no 58 2 14 145 995
0 8 716 845 338 154 no no 58 2 17 166 195
0 9 719 798 266 226 yes no 58 2 21 175 572

    All Magic solutions from start 1

start 1 - # Solutions 7 ==== 695 006 Sec. ====
Start
Sol.
 Time
DP
DS
 SM CT End Dual Sol. Time
1 1 306 872 322 246 no no 59 3 33 128 594
1 2 306 872 322 216 no no 59 3 38 129 053
1 3 307 175 352 246 no no 59 3 34 128 594
1 4 307 175 352 216 no no 59 3 39 129 053
1 5 378 530 300 370 no no 59 3 35 128 594
1 6 378 530 300 340 no no 59 3 40 129 054
1 7 489 757 152 376 no no 59 3 1 84 607

    All Magic solutions from start 2

start 2 - # Solutions 37 ==== 658 633 sec. ====
Start
Sol.
 Time
DP
DS
 SM CT End Dual Sol. Time
2 1 131 958 374 224 no no 60 3 36 128 597
2 2 131 958 344 224 no no 60 3 41 129 059
2 3 132 536 252 248 no yes 12 11 18 74 718
2 4 132 538 222 184 no yes 12 11 20 77 004
2 5 132 538 252 232 no yes 12 11 19 77 000
2 6 132 561 310 238 yes no 56 0 4 703 085
2 7 132 561 314 228 yes no 56 0 2 60 370
2 8 132 561 284 228 yes no 56 0 3 60 853
2 9 132 684 286 312 no yes 12 11 31 93 902
2 10 137 354 292 256 no no 56 0 5 706 269
2 11 138 011 308 240 no no 56 0 6 706 792
2 12 138 755 374 258 no no 60 3 37 128 597
2 13 138 755 344 258 no no 60 3 42 129 059
2 14 145 995 292 256 no no 56 0 7 709 898
2 15 150 405 204 324 no no 58 2 16 150 405
2 16 150 405 196 316 no no 58 2 15 150 405
2 17 166 195 366 182 no no 56 0 8 716 845
2 18 166 281 288 312 no no 21 18 6 54 058
2 19 167 247 352 248 no no 21 18 1 52 662
2 20 167 483 208 234 no yes 12 11 36 105 566
2 21 175 572 294 254 yes no 56 0 9 719 798
2 22 181 453 272 278 no no 23 2 35 602 068
2 23 181 453 272 318 no no 28 27 6 13 655
2 24 181 453 258 318 no no 28 27 11 20 511
2 25 181 574 272 328 no no 21 18 2 52 667
2 26 181 575 272 328 no no 28 27 7 13 655
2 27 181 575 258 328 no no 28 27 12 20 511
2 28 196 639 320 200 no yes 17 10 44 188 149
2 29 196 646 306 208 no no 3 3 53 273 280
2 30 197 907 256 264 no yes 12 11 3 68 137
2 31 200 511 284 228 no no 56 0 1 7 453
2 32 203 761 276 258 no no 35 27 27 138 295
2 33 203 829 264 248 no yes 12 11 12 68 750
2 34 203 840 264 256 no yes 12 11 32 101 843
2 35 602 068 242 248 no no 40 2 22 181 453
2 36 605 912 228 344 no no 35 27 18 39 993
2 37 624 529 184 336 no no 49 9 6 64 182

    All Magic solutions from start 3

start 3 - # Solutions 61 ==== 682 646 Sec. ====
Start
Sol.
 Time
DP
DS
 SM CT End Dual Sol. Time
3 1 84 607 144 368 no no 57 1 7 489 757
3 2 108 007 192 328 no yes 13 10 8 155 864
3 3 108 076 202 318 no yes 13 10 9 155 938
3 4 108 393 158 338 no yes 13 10 37 167 243
3 5 108 393 158 338 no no 4 3 27 119 708
3 6 108 394 202 318 no yes 13 10 14 156 272
3 7 108 394 210 296 no no 25 11 27 82 890
3 8 108 394 258 296 no no 27 27 9 13 761
3 9 108 395 246 234 no yes 20 19 6 14 540
3 10 108 845 216 304 no yes 13 10 21 157 324
3 11 108 845 272 296 no no 27 27 4 13 338
3 12 108 852 216 304 no yes 13 10 22 157 324
3 13 109 043 224 296 no no 25 11 10 68 666
3 14 109 486 216 304 no yes 13 10 19 156 731
3 15 109 486 216 304 no yes 13 10 20 156 736
3 16 109 486 216 288 no yes 20 19 8 18 402
3 17 109 554 208 312 no yes 13 10 23 157 406
3 18 110 621 168 284 no no 4 3 58 278 855
3 19 110 621 272 246 no no 27 27 13 20 605
3 20 110 622 248 244 no yes 20 19 9 21 290
3 21 117 711 200 320 no yes 13 10 33 165 465
3 22 119 598 168 348 no yes 13 10 13 156 271
3 23 119 600 200 320 no no 38 11 1 68 137
3 24 119 600 200 306 no no 38 11 13 72 510
3 25 119 601 136 384 no yes 13 10 46 188 941
3 26 119 708 182 362 no yes 13 10 11 156 271
3 27 119 708 182 362 no no 4 3 5 108 393
3 28 120 037 144 384 no yes 13 10 38 167 589
3 29 120 037 144 384 no no 4 3 30 120 037
3 30 120 037 136 376 no no 4 3 29 120 037
3 31 121 214 208 320 no yes 13 10 34 165 560
3 32 121 226 232 288 no no 38 11 11 68 691
3 33 128 594 274 198 no no 57 1 1 306 872
3 34 128 594 274 168 no no 57 1 3 307 175
3 35 128 594 150 220 no no 57 1 5 378 530
3 36 128 597 146 296 no no 61 2 1 131 958
3 37 128 597 146 262 no no 61 2 12 138 755
3 38 129 053 304 198 no no 57 1 2 306 872
3 39 129 053 304 168 no no 57 1 4 307 175
3 40 129 054 180 220 no no 57 1 6 378 530
3 41 129 059 176 296 no no 61 2 2 131 958
3 42 129 059 176 262 no no 61 2 13 138 755
3 43 136 421 216 234 no no 29 19 10 21 563
3 44 136 421 216 224 no no 29 19 12 21 992
3 45 136 423 248 288 no no 36 27 5 13 655
3 46 136 423 248 274 no no 36 27 10 20 511
3 47 136 472 236 268 no no 36 27 8 13 655
3 48 136 546 230 234 no no 29 19 11 21 563
3 49 136 546 230 224 no no 29 19 13 21 992
3 50 136 567 170 272 no yes 13 10 43 182 823
3 51 137 207 216 274 no no 32 3 61 503 264
3 52 137 207 216 264 no no 34 19 15 22 236
3 53 273 280 214 312 no no 2 2 29 196 646
3 54 278 744 222 338 no yes 13 10 25 158 343
3 55 278 746 222 320 no no 38 11 2 68 137
3 56 278 746 222 306 no no 38 11 14 72 510
3 57 278 855 236 352 no yes 13 10 24 158 343
3 58 278 855 236 352 no no 4 3 18 110 621
3 59 279 481 222 338 no yes 13 10 30 158 356
3 60 279 777 330 264 no no 45 18 4 53 153
3 61 503 264 246 304 no no 31 3 51 137 207

    All Magic solutions from start 9

start 9 - # Solutions 6 ==== 190 851 Sec. ====
Start
Sol.
 Time
DP
DS
 SM CT End Dual Sol. Time
9 1 56 965 184 208 no yes 19 19 2 5 448
9 2 57 015 184 208 no yes 19 19 1 5 418
9 3 57 258 240 280 no no 14 9 4 57 778
9 4 57 778 240 280 no no 14 9 3 57 258
9 5 60 780 208 256 no no 37 19 16 31 528
9 6 64 182 184 336 no no 58 2 37 624 529

    All Magic solutions from start 10

start 10 - # Solutions 51 ==== 324 546 Sec. ====
Start
Sol.
 Time
DP
DS
 SM CT End Dual Sol. Time
10 1 56 410 208 216 no no 29 19 23 201 583
10 2 68 924 256 264 no no 11 11 37 164 766
10 3 87 458 264 256 no yes 27 27 19 91 999
10 4 87 458 264 256 no yes 27 27 20 91 999
10 5 87 533 228 292 no yes 27 27 23 124 318
10 6 87 533 228 292 no yes 27 27 24 124 318
10 7 120 327 248 256 no yes 25 11 22 80 277
10 8 155 864 192 328 no yes 4 3 2 108 007
10 9 155 938 202 318 no yes 4 3 3 108 076
10 10 156 271 158 338 no no 13 10 36 167 243
10 11 156 271 158 338 no yes 4 3 26 119 708
10 12 156 271 172 352 no no 13 10 35 167 127
10 13 156 271 172 352 no yes 4 3 22 119 598
10 14 156 272 202 318 no yes 4 3 6 108 394
10 15 156 272 258 296 no no 18 18 3 52 744
10 16 156 272 246 234 no yes 20 19 5 14 522
10 17 156 272 236 224 no yes 20 19 4 14 484
10 18 156 283 212 272 no yes 20 19 3 14 206
10 19 156 731 216 304 no yes 4 3 14 109 486
10 20 156 736 216 304 no yes 4 3 15 109 486
10 21 157 324 216 304 no yes 4 3 10 108 845
10 22 157 324 216 304 no yes 4 3 12 108 852
10 23 157 406 208 312 no yes 4 3 17 109 554
10 24 158 343 168 284 no yes 4 3 57 278 855
10 25 158 343 182 298 no yes 4 3 54 278 744
10 26 158 343 202 254 no yes 25 11 15 72 511
10 27 158 343 296 250 no no 13 10 47 194 526
10 28 158 343 296 220 no yes 20 19 19 31 767
10 29 158 343 286 210 no yes 20 19 18 31 731
10 30 158 356 182 298 no yes 4 3 59 279 481
10 31 158 356 202 254 no yes 25 11 17 72 556
10 32 158 357 286 210 no yes 20 19 14 22 007
10 33 165 465 200 320 no yes 4 3 21 117 711
10 34 165 560 200 312 no yes 4 3 31 121 214
10 35 167 127 168 348 no no 13 10 12 156 271
10 36 167 243 182 362 no no 13 10 10 156 271
10 37 167 243 182 362 no yes 4 3 4 108 393
10 38 167 589 136 376 no yes 4 3 28 120 037
10 39 177 351 200 240 no no 29 19 21 199 522
10 40 177 351 192 200 no yes 25 11 34 105 408
10 41 177 386 200 240 no no 29 19 22 200 503
10 42 177 386 192 200 no yes 25 11 35 105 488
10 43 182 823 248 350 no yes 4 3 50 136 567
10 44 188 149 200 320 no yes 16 2 28 196 639
10 45 188 291 262 306 no no 11 11 9 68 648
10 46 188 941 136 384 no yes 4 3 25 119 601
10 47 194 526 270 224 no no 13 10 27 158 343
10 48 211 636 264 256 no yes 27 27 21 92 030
10 49 211 636 264 256 no yes 27 27 22 92 030
10 50 211 717 228 292 no yes 27 27 25 124 338
10 51 211 717 228 292 no yes 27 27 26 124 338

    All Magic solutions from start 11

start 11 - # Solutions 37 ==== 290 238 Sec. ====
Start
Sol.
 Time
DP
DS
 SM CT End Dual Sol. Time
11 1 68 137 320 200 no no 39 3 23 119 600
11 2 68 137 298 200 no no 39 3 55 278 746
11 3 68 137 256 264 no yes 5 2 30 197 907
11 4 68 140 342 178 no no 51 11 23 82 842
11 5 68 156 342 178 no no 51 11 24 82 857
11 6 68 158 320 200 no no 51 11 25 82 858
11 7 68 201 312 208 no no 51 11 28 82 897
11 8 68 577 216 304 no no 35 27 2 2 088
11 9 68 648 258 214 no no 10 10 45 188 291
11 10 68 666 296 224 no no 24 3 13 109 043
11 11 68 691 288 232 no no 39 3 32 121 226
11 12 68 750 272 256 no yes 5 2 33 203 829
11 13 72 510 320 214 no no 39 3 24 119 600
11 14 72 510 298 214 no no 39 3 56 278 746
11 15 72 511 318 266 no yes 17 10 26 158 343
11 16 72 531 320 218 no no 51 11 26 82 858
11 17 72 556 318 266 no yes 17 10 31 158 356
11 18 74 718 272 268 no yes 5 2 3 132 536
11 19 77 000 288 268 no yes 5 2 5 132 538
11 20 77 004 336 298 no yes 5 2 4 132 538
11 21 80 276 272 264 no yes 28 27 28 164 417
11 22 80 277 272 264 no yes 17 10 7 120 327
11 23 82 842 342 178 no no 51 11 4 68 140
11 24 82 857 342 178 no no 51 11 5 68 156
11 25 82 858 320 200 no no 51 11 6 68 158
11 26 82 858 302 200 no no 51 11 16 72 531
11 27 82 890 310 224 no no 24 3 7 108 394
11 28 82 897 312 208 no no 51 11 7 68 201
11 29 87 243 240 280 no no 51 11 30 87 681
11 30 87 681 240 280 no no 51 11 29 87 243
11 31 93 902 208 234 no yes 5 2 9 132 684
11 32 101 843 264 256 no yes 5 2 34 203 840
11 33 101 914 272 264 no yes 28 27 30 164 437
11 34 105 408 328 320 no yes 17 10 40 177 351
11 35 105 488 328 320 no yes 17 10 42 177 386
11 36 105 566 286 312 no yes 5 2 20 167 483
11 37 164 766 264 256 no no 10 10 2 68 924

    All Magic solutions from start 18

start 18 - # Solutions 14 ==== 128 139 Sec. ====
Start
Sol.
 Time
DP
DS
 SM CT End Dual Sol. Time
18 1 52 662 272 168 no no 5 2 19 167 247
18 2 52 667 192 248 no no 40 2 25 181 574
18 3 52 744 262 224 no no 17 10 15 156 272
18 4 53 153 190 256 no no 39 3 60 279 777
18 5 53 375 296 168 no no 19 19 24 211 340
18 6 54 058 208 232 no no 5 2 18 166 281
18 7 54 080 176 248 no no 37 19 27 239 076
18 8 57 385 192 328 no yes 35 27 1 2 088
18 9 57 410 200 320 no yes 35 27 14 38 834
18 10 57 509 190 330 no yes 35 27 15 38 931
18 11 57 704 190 330 no yes 35 27 16 39 117
18 12 58 199 184 336 no yes 35 27 17 39 584
18 13 58 601 128 392 no yes 35 27 3 2 351
18 14 114 610 176 232 no no 26 19 28 239 077

    All Magic solutions from start 19

start 19 - # Solutions 28 ==== 321 107 Sec. ====
Start
Sol.
 Time
DP
DS
 SM CT End Dual Sol. Time
19 1 5 418 336 312 no yes 9 9 2 57 015
19 2 5 448 336 312 no yes 9 9 1 56 965
19 3 14 206 248 308 no yes 13 10 18 156 283
19 4 14 484 296 284 no yes 13 10 17 156 272
19 5 14 522 286 274 no yes 13 10 16 156 272
19 6 14 540 286 274 no yes 4 3 9 108 395
19 7 16 984 264 272 no yes 36 27 29 164 435
19 8 18 402 232 304 no yes 4 3 16 109 486
19 9 21 290 276 272 no yes 4 3 20 110 622
19 10 21 563 286 304 no no 32 3 43 136 421
19 11 21 563 286 290 no no 32 3 48 136 546
19 12 21 992 296 304 no no 32 3 44 136 421
19 13 21 992 296 290 no no 32 3 49 136 546
19 14 22 007 310 234 no yes 13 10 32 158 357
19 15 22 236 256 304 no no 31 3 52 137 207
19 16 31 528 312 264 no no 54 9 5 60 780
19 17 31 555 264 256 no yes 34 19 25 211 544
19 18 31 731 310 234 no yes 13 10 29 158 343
19 19 31 767 300 224 no yes 13 10 28 158 343
19 20 31 767 264 256 no yes 34 19 26 211 880
19 21 199 522 280 320 no no 41 10 39 177 351
19 22 200 503 280 320 no no 41 10 41 177 386
19 23 201 583 304 312 no no 41 10 1 56 410
19 24 211 340 224 352 no no 18 18 5 53 375
19 25 211 544 264 256 no yes 29 19 17 31 555
19 26 211 880 264 256 no yes 29 19 20 31 767
19 27 239 076 344 272 no no 45 18 7 54 080
19 28 239 077 344 288 no no 18 18 14 114 610

    All Magic solutions from start 27

start 27 - # Solutions 30 ==== 198 763 Sec. ====
Start
Sol.
 Time
DP
DS
 SM CT End Dual Sol. Time
27 1 2 088 192 328 no yes 42 18 8 57 385
27 2 2 088 216 304 no no 30 11 8 68 577
27 3 2 351 128 392 no yes 42 18 13 58 601
27 4 13 338 248 224 no no 24 3 11 108 845
27 5 13 655 272 232 no no 39 3 45 136 423
27 6 13 655 202 248 no no 40 2 23 181 453
27 7 13 655 192 248 no no 40 2 26 181 575
27 8 13 655 284 252 no no 39 3 47 136 472
27 9 13 761 262 224 no no 24 3 8 108 394
27 10 20 511 272 246 no no 39 3 46 136 423
27 11 20 511 202 262 no no 40 2 24 181 453
27 12 20 511 192 262 no no 40 2 27 181 575
27 13 20 605 248 274 no no 24 3 19 110 621
27 14 38 834 200 320 no yes 42 18 9 57 410
27 15 38 931 190 330 no yes 42 18 10 57 509
27 16 39 117 190 330 no yes 42 18 11 57 704
27 17 39 584 184 336 no yes 42 18 12 58 199
27 18 39 993 176 292 no no 58 2 36 605 912
27 19 91 999 256 264 no yes 10 10 3 87 458
27 20 91 999 256 264 no yes 10 10 4 87 458
27 21 92 030 256 264 no yes 10 10 48 211 636
27 22 92 030 256 264 no yes 10 10 49 211 636
27 23 124 318 292 228 no yes 10 10 5 87 533
27 24 124 318 292 228 no yes 10 10 6 87 533
27 25 124 338 292 228 no yes 10 10 50 211 717
27 26 124 338 292 228 no yes 10 10 51 211 717
27 27 138 295 262 244 no no 58 2 32 203 761
27 28 164 417 256 248 no yes 33 11 21 80 276
27 29 164 435 256 248 no yes 44 19 7 16 984
27 30 164 437 256 248 no yes 33 11 33 101 914