Shift methode (2)
René Chrétien created a 3x3x3 magic rectangle to contruct a 27x27 panmagic square with extra special magic features.
|
Matrix 3x3x3 |
||||
|
7 |
9 |
23 |
39 |
|
|
4 |
11 |
24 |
39 |
|
|
26 |
1 |
12 |
39 |
|
|
15 |
22 |
2 |
39 |
|
|
19 |
17 |
3 |
39 |
|
|
14 |
0 |
25 |
39 |
|
|
6 |
20 |
13 |
39 |
|
|
18 |
16 |
5 |
39 |
|
|
8 |
21 |
10 |
39 |
|
|
117 |
117 |
117 |
Take 1x number from first grid +1 (shift 2 to the left)
|
9 |
22 |
20 |
11 |
19 |
18 |
26 |
14 |
21 |
23 |
2 |
13 |
24 |
3 |
5 |
12 |
0 |
8 |
7 |
15 |
6 |
4 |
17 |
16 |
1 |
25 |
10 |
|
20 |
11 |
19 |
18 |
26 |
14 |
21 |
23 |
2 |
13 |
24 |
3 |
5 |
12 |
0 |
8 |
7 |
15 |
6 |
4 |
17 |
16 |
1 |
25 |
10 |
9 |
22 |
|
19 |
18 |
26 |
14 |
21 |
23 |
2 |
13 |
24 |
3 |
5 |
12 |
0 |
8 |
7 |
15 |
6 |
4 |
17 |
16 |
1 |
25 |
10 |
9 |
22 |
20 |
11 |
|
26 |
14 |
21 |
23 |
2 |
13 |
24 |
3 |
5 |
12 |
0 |
8 |
7 |
15 |
6 |
4 |
17 |
16 |
1 |
25 |
10 |
9 |
22 |
20 |
11 |
19 |
18 |
|
21 |
23 |
2 |
13 |
24 |
3 |
5 |
12 |
0 |
8 |
7 |
15 |
6 |
4 |
17 |
16 |
1 |
25 |
10 |
9 |
22 |
20 |
11 |
19 |
18 |
26 |
14 |
|
2 |
13 |
24 |
3 |
5 |
12 |
0 |
8 |
7 |
15 |
6 |
4 |
17 |
16 |
1 |
25 |
10 |
9 |
22 |
20 |
11 |
19 |
18 |
26 |
14 |
21 |
23 |
|
24 |
3 |
5 |
12 |
0 |
8 |
7 |
15 |
6 |
4 |
17 |
16 |
1 |
25 |
10 |
9 |
22 |
20 |
11 |
19 |
18 |
26 |
14 |
21 |
23 |
2 |
13 |
|
5 |
12 |
0 |
8 |
7 |
15 |
6 |
4 |
17 |
16 |
1 |
25 |
10 |
9 |
22 |
20 |
11 |
19 |
18 |
26 |
14 |
21 |
23 |
2 |
13 |
24 |
3 |
|
0 |
8 |
7 |
15 |
6 |
4 |
17 |
16 |
1 |
25 |
10 |
9 |
22 |
20 |
11 |
19 |
18 |
26 |
14 |
21 |
23 |
2 |
13 |
24 |
3 |
5 |
12 |
|
7 |
15 |
6 |
4 |
17 |
16 |
1 |
25 |
10 |
9 |
22 |
20 |
11 |
19 |
18 |
26 |
14 |
21 |
23 |
2 |
13 |
24 |
3 |
5 |
12 |
0 |
8 |
|
6 |
4 |
17 |
16 |
1 |
25 |
10 |
9 |
22 |
20 |
11 |
19 |
18 |
26 |
14 |
21 |
23 |
2 |
13 |
24 |
3 |
5 |
12 |
0 |
8 |
7 |
15 |
|
17 |
16 |
1 |
25 |
10 |
9 |
22 |
20 |
11 |
19 |
18 |
26 |
14 |
21 |
23 |
2 |
13 |
24 |
3 |
5 |
12 |
0 |
8 |
7 |
15 |
6 |
4 |
|
1 |
25 |
10 |
9 |
22 |
20 |
11 |
19 |
18 |
26 |
14 |
21 |
23 |
2 |
13 |
24 |
3 |
5 |
12 |
0 |
8 |
7 |
15 |
6 |
4 |
17 |
16 |
|
10 |
9 |
22 |
20 |
11 |
19 |
18 |
26 |
14 |
21 |
23 |
2 |
13 |
24 |
3 |
5 |
12 |
0 |
8 |
7 |
15 |
6 |
4 |
17 |
16 |
1 |
25 |
|
22 |
20 |
11 |
19 |
18 |
26 |
14 |
21 |
23 |
2 |
13 |
24 |
3 |
5 |
12 |
0 |
8 |
7 |
15 |
6 |
4 |
17 |
16 |
1 |
25 |
10 |
9 |
|
11 |
19 |
18 |
26 |
14 |
21 |
23 |
2 |
13 |
24 |
3 |
5 |
12 |
0 |
8 |
7 |
15 |
6 |
4 |
17 |
16 |
1 |
25 |
10 |
9 |
22 |
20 |
|
18 |
26 |
14 |
21 |
23 |
2 |
13 |
24 |
3 |
5 |
12 |
0 |
8 |
7 |
15 |
6 |
4 |
17 |
16 |
1 |
25 |
10 |
9 |
22 |
20 |
11 |
19 |
|
14 |
21 |
23 |
2 |
13 |
24 |
3 |
5 |
12 |
0 |
8 |
7 |
15 |
6 |
4 |
17 |
16 |
1 |
25 |
10 |
9 |
22 |
20 |
11 |
19 |
18 |
26 |
|
23 |
2 |
13 |
24 |
3 |
5 |
12 |
0 |
8 |
7 |
15 |
6 |
4 |
17 |
16 |
1 |
25 |
10 |
9 |
22 |
20 |
11 |
19 |
18 |
26 |
14 |
21 |
|
13 |
24 |
3 |
5 |
12 |
0 |
8 |
7 |
15 |
6 |
4 |
17 |
16 |
1 |
25 |
10 |
9 |
22 |
20 |
11 |
19 |
18 |
26 |
14 |
21 |
23 |
2 |
|
3 |
5 |
12 |
0 |
8 |
7 |
15 |
6 |
4 |
17 |
16 |
1 |
25 |
10 |
9 |
22 |
20 |
11 |
19 |
18 |
26 |
14 |
21 |
23 |
2 |
13 |
24 |
|
12 |
0 |
8 |
7 |
15 |
6 |
4 |
17 |
16 |
1 |
25 |
10 |
9 |
22 |
20 |
11 |
19 |
18 |
26 |
14 |
21 |
23 |
2 |
13 |
24 |
3 |
5 |
|
8 |
7 |
15 |
6 |
4 |
17 |
16 |
1 |
25 |
10 |
9 |
22 |
20 |
11 |
19 |
18 |
26 |
14 |
21 |
23 |
2 |
13 |
24 |
3 |
5 |
12 |
0 |
|
15 |
6 |
4 |
17 |
16 |
1 |
25 |
10 |
9 |
22 |
20 |
11 |
19 |
18 |
26 |
14 |
21 |
23 |
2 |
13 |
24 |
3 |
5 |
12 |
0 |
8 |
7 |
|
4 |
17 |
16 |
1 |
25 |
10 |
9 |
22 |
20 |
11 |
19 |
18 |
26 |
14 |
21 |
23 |
2 |
13 |
24 |
3 |
5 |
12 |
0 |
8 |
7 |
15 |
6 |
|
16 |
1 |
25 |
10 |
9 |
22 |
20 |
11 |
19 |
18 |
26 |
14 |
21 |
23 |
2 |
13 |
24 |
3 |
5 |
12 |
0 |
8 |
7 |
15 |
6 |
4 |
17 |
|
25 |
10 |
9 |
22 |
20 |
11 |
19 |
18 |
26 |
14 |
21 |
23 |
2 |
13 |
24 |
3 |
5 |
12 |
0 |
8 |
7 |
15 |
6 |
4 |
17 |
16 |
1 |
+ 27x number from second grid (shift 2 to the right)
|
9 |
22 |
20 |
11 |
19 |
18 |
26 |
14 |
21 |
23 |
2 |
13 |
24 |
3 |
5 |
12 |
0 |
8 |
7 |
15 |
6 |
4 |
17 |
16 |
1 |
25 |
10 |
|
25 |
10 |
9 |
22 |
20 |
11 |
19 |
18 |
26 |
14 |
21 |
23 |
2 |
13 |
24 |
3 |
5 |
12 |
0 |
8 |
7 |
15 |
6 |
4 |
17 |
16 |
1 |
|
16 |
1 |
25 |
10 |
9 |
22 |
20 |
11 |
19 |
18 |
26 |
14 |
21 |
23 |
2 |
13 |
24 |
3 |
5 |
12 |
0 |
8 |
7 |
15 |
6 |
4 |
17 |
|
4 |
17 |
16 |
1 |
25 |
10 |
9 |
22 |
20 |
11 |
19 |
18 |
26 |
14 |
21 |
23 |
2 |
13 |
24 |
3 |
5 |
12 |
0 |
8 |
7 |
15 |
6 |
|
15 |
6 |
4 |
17 |
16 |
1 |
25 |
10 |
9 |
22 |
20 |
11 |
19 |
18 |
26 |
14 |
21 |
23 |
2 |
13 |
24 |
3 |
5 |
12 |
0 |
8 |
7 |
|
8 |
7 |
15 |
6 |
4 |
17 |
16 |
1 |
25 |
10 |
9 |
22 |
20 |
11 |
19 |
18 |
26 |
14 |
21 |
23 |
2 |
13 |
24 |
3 |
5 |
12 |
0 |
|
12 |
0 |
8 |
7 |
15 |
6 |
4 |
17 |
16 |
1 |
25 |
10 |
9 |
22 |
20 |
11 |
19 |
18 |
26 |
14 |
21 |
23 |
2 |
13 |
24 |
3 |
5 |
|
3 |
5 |
12 |
0 |
8 |
7 |
15 |
6 |
4 |
17 |
16 |
1 |
25 |
10 |
9 |
22 |
20 |
11 |
19 |
18 |
26 |
14 |
21 |
23 |
2 |
13 |
24 |
|
13 |
24 |
3 |
5 |
12 |
0 |
8 |
7 |
15 |
6 |
4 |
17 |
16 |
1 |
25 |
10 |
9 |
22 |
20 |
11 |
19 |
18 |
26 |
14 |
21 |
23 |
2 |
|
23 |
2 |
13 |
24 |
3 |
5 |
12 |
0 |
8 |
7 |
15 |
6 |
4 |
17 |
16 |
1 |
25 |
10 |
9 |
22 |
20 |
11 |
19 |
18 |
26 |
14 |
21 |
|
14 |
21 |
23 |
2 |
13 |
24 |
3 |
5 |
12 |
0 |
8 |
7 |
15 |
6 |
4 |
17 |
16 |
1 |
25 |
10 |
9 |
22 |
20 |
11 |
19 |
18 |
26 |
|
18 |
26 |
14 |
21 |
23 |
2 |
13 |
24 |
3 |
5 |
12 |
0 |
8 |
7 |
15 |
6 |
4 |
17 |
16 |
1 |
25 |
10 |
9 |
22 |
20 |
11 |
19 |
|
11 |
19 |
18 |
26 |
14 |
21 |
23 |
2 |
13 |
24 |
3 |
5 |
12 |
0 |
8 |
7 |
15 |
6 |
4 |
17 |
16 |
1 |
25 |
10 |
9 |
22 |
20 |
|
22 |
20 |
11 |
19 |
18 |
26 |
14 |
21 |
23 |
2 |
13 |
24 |
3 |
5 |
12 |
0 |
8 |
7 |
15 |
6 |
4 |
17 |
16 |
1 |
25 |
10 |
9 |
|
10 |
9 |
22 |
20 |
11 |
19 |
18 |
26 |
14 |
21 |
23 |
2 |
13 |
24 |
3 |
5 |
12 |
0 |
8 |
7 |
15 |
6 |
4 |
17 |
16 |
1 |
25 |
|
1 |
25 |
10 |
9 |
22 |
20 |
11 |
19 |
18 |
26 |
14 |
21 |
23 |
2 |
13 |
24 |
3 |
5 |
12 |
0 |
8 |
7 |
15 |
6 |
4 |
17 |
16 |
|
17 |
16 |
1 |
25 |
10 |
9 |
22 |
20 |
11 |
19 |
18 |
26 |
14 |
21 |
23 |
2 |
13 |
24 |
3 |
5 |
12 |
0 |
8 |
7 |
15 |
6 |
4 |
|
6 |
4 |
17 |
16 |
1 |
25 |
10 |
9 |
22 |
20 |
11 |
19 |
18 |
26 |
14 |
21 |
23 |
2 |
13 |
24 |
3 |
5 |
12 |
0 |
8 |
7 |
15 |
|
7 |
15 |
6 |
4 |
17 |
16 |
1 |
25 |
10 |
9 |
22 |
20 |
11 |
19 |
18 |
26 |
14 |
21 |
23 |
2 |
13 |
24 |
3 |
5 |
12 |
0 |
8 |
|
0 |
8 |
7 |
15 |
6 |
4 |
17 |
16 |
1 |
25 |
10 |
9 |
22 |
20 |
11 |
19 |
18 |
26 |
14 |
21 |
23 |
2 |
13 |
24 |
3 |
5 |
12 |
|
5 |
12 |
0 |
8 |
7 |
15 |
6 |
4 |
17 |
16 |
1 |
25 |
10 |
9 |
22 |
20 |
11 |
19 |
18 |
26 |
14 |
21 |
23 |
2 |
13 |
24 |
3 |
|
24 |
3 |
5 |
12 |
0 |
8 |
7 |
15 |
6 |
4 |
17 |
16 |
1 |
25 |
10 |
9 |
22 |
20 |
11 |
19 |
18 |
26 |
14 |
21 |
23 |
2 |
13 |
|
2 |
13 |
24 |
3 |
5 |
12 |
0 |
8 |
7 |
15 |
6 |
4 |
17 |
16 |
1 |
25 |
10 |
9 |
22 |
20 |
11 |
19 |
18 |
26 |
14 |
21 |
23 |
|
21 |
23 |
2 |
13 |
24 |
3 |
5 |
12 |
0 |
8 |
7 |
15 |
6 |
4 |
17 |
16 |
1 |
25 |
10 |
9 |
22 |
20 |
11 |
19 |
18 |
26 |
14 |
|
26 |
14 |
21 |
23 |
2 |
13 |
24 |
3 |
5 |
12 |
0 |
8 |
7 |
15 |
6 |
4 |
17 |
16 |
1 |
25 |
10 |
9 |
22 |
20 |
11 |
19 |
18 |
|
19 |
18 |
26 |
14 |
21 |
23 |
2 |
13 |
24 |
3 |
5 |
12 |
0 |
8 |
7 |
15 |
6 |
4 |
17 |
16 |
1 |
25 |
10 |
9 |
22 |
20 |
11 |
|
20 |
11 |
19 |
18 |
26 |
14 |
21 |
23 |
2 |
13 |
24 |
3 |
5 |
12 |
0 |
8 |
7 |
15 |
6 |
4 |
17 |
16 |
1 |
25 |
10 |
9 |
22 |
= 27x27 panmagic square
|
253 |
617 |
561 |
309 |
533 |
505 |
729 |
393 |
589 |
645 |
57 |
365 |
673 |
85 |
141 |
337 |
1 |
225 |
197 |
421 |
169 |
113 |
477 |
449 |
29 |
701 |
281 |
|
696 |
282 |
263 |
613 |
567 |
312 |
535 |
510 |
705 |
392 |
592 |
625 |
60 |
364 |
649 |
90 |
143 |
340 |
7 |
221 |
207 |
422 |
164 |
134 |
470 |
442 |
50 |
|
452 |
46 |
702 |
285 |
265 |
618 |
543 |
311 |
538 |
490 |
708 |
391 |
568 |
630 |
62 |
367 |
655 |
86 |
153 |
341 |
2 |
242 |
200 |
415 |
185 |
129 |
471 |
|
135 |
474 |
454 |
51 |
678 |
284 |
268 |
598 |
546 |
310 |
514 |
495 |
710 |
394 |
574 |
626 |
72 |
368 |
650 |
107 |
146 |
334 |
23 |
237 |
201 |
425 |
181 |
|
427 |
186 |
111 |
473 |
457 |
31 |
681 |
283 |
244 |
603 |
548 |
313 |
520 |
491 |
720 |
395 |
569 |
647 |
65 |
361 |
671 |
102 |
147 |
344 |
19 |
243 |
204 |
|
219 |
203 |
430 |
166 |
114 |
472 |
433 |
36 |
683 |
286 |
250 |
599 |
558 |
314 |
515 |
512 |
713 |
388 |
590 |
642 |
66 |
371 |
667 |
108 |
150 |
346 |
24 |
|
349 |
4 |
222 |
202 |
406 |
171 |
116 |
475 |
439 |
32 |
693 |
287 |
245 |
620 |
551 |
307 |
536 |
507 |
714 |
398 |
586 |
648 |
69 |
373 |
672 |
84 |
149 |
|
87 |
148 |
325 |
9 |
224 |
205 |
412 |
167 |
126 |
476 |
434 |
53 |
686 |
280 |
266 |
615 |
552 |
317 |
532 |
513 |
717 |
400 |
591 |
624 |
68 |
376 |
652 |
|
352 |
657 |
89 |
151 |
331 |
5 |
234 |
206 |
407 |
188 |
119 |
469 |
455 |
48 |
687 |
290 |
262 |
621 |
555 |
319 |
537 |
489 |
716 |
403 |
571 |
627 |
67 |
|
629 |
70 |
358 |
653 |
99 |
152 |
326 |
26 |
227 |
199 |
428 |
183 |
120 |
479 |
451 |
54 |
690 |
292 |
267 |
597 |
554 |
322 |
517 |
492 |
715 |
379 |
576 |
|
385 |
572 |
639 |
71 |
353 |
674 |
92 |
145 |
347 |
21 |
228 |
209 |
424 |
189 |
123 |
481 |
456 |
30 |
689 |
295 |
247 |
600 |
553 |
298 |
522 |
494 |
718 |
|
504 |
719 |
380 |
593 |
632 |
64 |
374 |
669 |
93 |
155 |
343 |
27 |
231 |
211 |
429 |
165 |
122 |
484 |
436 |
33 |
688 |
271 |
252 |
602 |
556 |
304 |
518 |
|
299 |
539 |
497 |
712 |
401 |
588 |
633 |
74 |
370 |
675 |
96 |
157 |
348 |
3 |
230 |
214 |
409 |
168 |
121 |
460 |
441 |
35 |
691 |
277 |
248 |
612 |
557 |
|
605 |
550 |
320 |
534 |
498 |
722 |
397 |
594 |
636 |
76 |
375 |
651 |
95 |
160 |
328 |
6 |
229 |
190 |
414 |
170 |
124 |
466 |
437 |
45 |
692 |
272 |
269 |
|
293 |
264 |
606 |
560 |
316 |
540 |
501 |
724 |
402 |
570 |
635 |
79 |
355 |
654 |
94 |
136 |
333 |
8 |
232 |
196 |
410 |
180 |
125 |
461 |
458 |
38 |
685 |
|
39 |
695 |
289 |
270 |
609 |
562 |
321 |
516 |
500 |
727 |
382 |
573 |
634 |
55 |
360 |
656 |
97 |
142 |
329 |
18 |
233 |
191 |
431 |
173 |
118 |
482 |
453 |
|
478 |
459 |
42 |
697 |
294 |
246 |
608 |
565 |
301 |
519 |
499 |
703 |
387 |
575 |
637 |
61 |
356 |
666 |
98 |
137 |
350 |
11 |
226 |
212 |
426 |
174 |
128 |
|
177 |
130 |
483 |
435 |
41 |
700 |
274 |
249 |
607 |
541 |
306 |
521 |
502 |
709 |
383 |
585 |
638 |
56 |
377 |
659 |
91 |
158 |
345 |
12 |
236 |
208 |
432 |
|
213 |
408 |
176 |
133 |
463 |
438 |
40 |
676 |
279 |
251 |
610 |
547 |
302 |
531 |
503 |
704 |
404 |
578 |
631 |
77 |
372 |
660 |
101 |
154 |
351 |
15 |
238 |
|
14 |
241 |
193 |
411 |
175 |
109 |
468 |
440 |
43 |
682 |
275 |
261 |
611 |
542 |
323 |
524 |
496 |
725 |
399 |
579 |
641 |
73 |
378 |
663 |
103 |
159 |
327 |
|
139 |
330 |
13 |
217 |
198 |
413 |
178 |
115 |
464 |
450 |
44 |
677 |
296 |
254 |
604 |
563 |
318 |
525 |
506 |
721 |
405 |
582 |
643 |
78 |
354 |
662 |
106 |
|
661 |
82 |
144 |
332 |
16 |
223 |
194 |
423 |
179 |
110 |
485 |
443 |
37 |
698 |
291 |
255 |
614 |
559 |
324 |
528 |
508 |
726 |
381 |
581 |
646 |
58 |
357 |
|
63 |
359 |
664 |
88 |
140 |
342 |
17 |
218 |
215 |
416 |
172 |
131 |
480 |
444 |
47 |
694 |
297 |
258 |
616 |
564 |
300 |
527 |
511 |
706 |
384 |
580 |
622 |
|
583 |
628 |
59 |
369 |
665 |
83 |
161 |
335 |
10 |
239 |
210 |
417 |
182 |
127 |
486 |
447 |
49 |
699 |
273 |
257 |
619 |
544 |
303 |
526 |
487 |
711 |
386 |
|
707 |
396 |
584 |
623 |
80 |
362 |
658 |
104 |
156 |
336 |
20 |
235 |
216 |
420 |
184 |
132 |
462 |
446 |
52 |
679 |
276 |
256 |
595 |
549 |
305 |
529 |
493 |
|
530 |
488 |
728 |
389 |
577 |
644 |
75 |
363 |
668 |
100 |
162 |
339 |
22 |
240 |
192 |
419 |
187 |
112 |
465 |
445 |
28 |
684 |
278 |
259 |
601 |
545 |
315 |
|
566 |
308 |
523 |
509 |
723 |
390 |
587 |
640 |
81 |
366 |
670 |
105 |
138 |
338 |
25 |
220 |
195 |
418 |
163 |
117 |
467 |
448 |
34 |
680 |
288 |
260 |
596 |
N.B.: Extra magic features are: each 9th cell (= position 1+10 +19 = 2+11+20 = ... = 9+18+27) on a row/column gives 1/9 of the magic sum and each 3rd cell (= position 1+4+7+10+13+16+19+22+25 = 2+5+8+11+14+17+20+23+26 = 3+6+9+12+15+18+21+24+27) on a diagonal gives 1/3 of the magic sum.
Use the shift method to construct magic squares of odd order from 5x5 to infinity.
See 5x5, 7x7, 9x9 (1), 9x9 (2), 11x11, 13x13, 15x15 (1), 15x15 (2), 17x17, 19x19, 21x21 (1), 21x21 (2), 23x23, 25x25, 27x27 (1), 27x27 (2), 29x29 and 31x31
