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Other Magic Square Sizes: 3 * 5 * 7 * 9 * 11 * 13 * 15 * 17 * 19 *
21 * 23 * 25 * 27 * 29 * 31 * 33 * 35 * 37 * 39 * 41 * 43 * 45 * 47 * 49 * 51 * 53 * 55 * 57 * 59 *
61 * 63 * 65 * 67 * 69 * 71 * 73 * 75 * 77 * 79 * 81 * 83 * 85 * 87 * 89 * 91 * 93 * 95 * 97 * 99 *

A 77x77 Magic Square

Scroll down to see a magic square
in which all rows, columns, and both diagonals
add up to the same magic sum (228305)

A Magic Square is a square of numbers in which every row, every column, and both diagonals add up to the same number. This number is often called the magic sum. The 77 by 77 magic square shown below has a magic sum equal to 228305.

Though magic squares can be made with non-consecutive and non-regular sequences, they are usually seen made up of consecutive numbers. To the best of my knowledge, the middle number in the sequence must be in the center square, and the magic sum will be equal to this middle number times the number of rows in the square. (In the case of 3x3 squares, I have proved that the middle number MUST be equal to one-third of the sum. See proof here.)

All odd-size Magic Squares can be made with the method shown below, which can be summarized as "start in the middle of the top and keep moving up and to the right except when you get blocked, in which case drop down and continue." The more detailed directions about creating these squares can be found further below. There are other methods for creating magic squares, but this is the easiest to learn. (There are basically only two methods of creating a 3x3 square but the larger squares have a large number of irregular variants, and probably other regular methods.)

Read more about Magic Squares

If you'd like to read more about magic squares, click one of these links to look for books about Magic Squares at Amazon.com, at Amazon.canada, and at Amazon.co.UK.

A 77-by-77 Magic Square

(Magic Sum = 228305)

Books about
Magic Squares
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898 977 1056 1135 1214 1293 1372 1451 1530 1609 1688 1767 1846 1925 1927 2006 2085 2164 2243 2322 2401 2480 2559 2638 2717 2796 2875 2954 3033 3112 3191 3270 3349 3428 3507 3586 3665 3744 3823 3902 3981 4060 4139 4218 4297 4376 4455 4534 4613 4692 4771 4850 4852 4931 5010 5089 5168 5247 5326 5405 5484 5563 5642 5721 5800 5879 29 108 187 266 345 424 503 582 661 740 819
976 1055 1134 1213 1292 1371 1450 1529 1608 1687 1766 1845 1924 1926 2005 2084 2163 2242 2321 2400 2479 2558 2637 2716 2795 2874 2953 3032 3111 3190 3269 3348 3427 3506 3585 3664 3743 3822 3901 3980 4059 4138 4217 4296 4375 4454 4533 4612 4691 4770 4849 4928 4930 5009 5088 5167 5246 5325 5404 5483 5562 5641 5720 5799 5878 28 107 186 265 344 423 502 581 660 739 818 897
1054 1133 1212 1291 1370 1449 1528 1607 1686 1765 1844 1923 2002 2004 2083 2162 2241 2320 2399 2478 2557 2636 2715 2794 2873 2952 3031 3110 3189 3268 3347 3426 3505 3584 3663 3742 3821 3900 3979 4058 4137 4216 4295 4374 4453 4532 4611 4690 4769 4848 4927 4929 5008 5087 5166 5245 5324 5403 5482 5561 5640 5719 5798 5877 27 106 185 264 343 422 501 580 659 738 817 896 975
1132 1211 1290 1369 1448 1527 1606 1685 1764 1843 1922 2001 2003 2082 2161 2240 2319 2398 2477 2556 2635 2714 2793 2872 2951 3030 3109 3188 3267 3346 3425 3504 3583 3662 3741 3820 3899 3978 4057 4136 4215 4294 4373 4452 4531 4610 4689 4768 4847 4926 5005 5007 5086 5165 5244 5323 5402 5481 5560 5639 5718 5797 5876 26 105 184 263 342 421 500 579 658 737 816 895 974 1053
1210 1289 1368 1447 1526 1605 1684 1763 1842 1921 2000 2079 2081 2160 2239 2318 2397 2476 2555 2634 2713 2792 2871 2950 3029 3108 3187 3266 3345 3424 3503 3582 3661 3740 3819 3898 3977 4056 4135 4214 4293 4372 4451 4530 4609 4688 4767 4846 4925 5004 5006 5085 5164 5243 5322 5401 5480 5559 5638 5717 5796 5875 25 104 183 262 341 420 499 578 657 736 815 894 973 1052 1131
1288 1367 1446 1525 1604 1683 1762 1841 1920 1999 2078 2080 2159 2238 2317 2396 2475 2554 2633 2712 2791 2870 2949 3028 3107 3186 3265 3344 3423 3502 3581 3660 3739 3818 3897 3976 4055 4134 4213 4292 4371 4450 4529 4608 4687 4766 4845 4924 5003 5082 5084 5163 5242 5321 5400 5479 5558 5637 5716 5795 5874 24 103 182 261 340 419 498 577 656 735 814 893 972 1051 1130 1209
1366 1445 1524 1603 1682 1761 1840 1919 1998 2077 2156 2158 2237 2316 2395 2474 2553 2632 2711 2790 2869 2948 3027 3106 3185 3264 3343 3422 3501 3580 3659 3738 3817 3896 3975 4054 4133 4212 4291 4370 4449 4528 4607 4686 4765 4844 4923 5002 5081 5083 5162 5241 5320 5399 5478 5557 5636 5715 5794 5873 23 102 181 260 339 418 497 576 655 734 813 892 971 1050 1129 1208 1287
1444 1523 1602 1681 1760 1839 1918 1997 2076 2155 2157 2236 2315 2394 2473 2552 2631 2710 2789 2868 2947 3026 3105 3184 3263 3342 3421 3500 3579 3658 3737 3816 3895 3974 4053 4132 4211 4290 4369 4448 4527 4606 4685 4764 4843 4922 5001 5080 5159 5161 5240 5319 5398 5477 5556 5635 5714 5793 5872 22 101 180 259 338 417 496 575 654 733 812 891 970 1049 1128 1207 1286 1365
1522 1601 1680 1759 1838 1917 1996 2075 2154 2233 2235 2314 2393 2472 2551 2630 2709 2788 2867 2946 3025 3104 3183 3262 3341 3420 3499 3578 3657 3736 3815 3894 3973 4052 4131 4210 4289 4368 4447 4526 4605 4684 4763 4842 4921 5000 5079 5158 5160 5239 5318 5397 5476 5555 5634 5713 5792 5871 21 100 179 258 337 416 495 574 653 732 811 890 969 1048 1127 1206 1285 1364 1443
1600 1679 1758 1837 1916 1995 2074 2153 2232 2234 2313 2392 2471 2550 2629 2708 2787 2866 2945 3024 3103 3182 3261 3340 3419 3498 3577 3656 3735 3814 3893 3972 4051 4130 4209 4288 4367 4446 4525 4604 4683 4762 4841 4920 4999 5078 5157 5236 5238 5317 5396 5475 5554 5633 5712 5791 5870 20 99 178 257 336 415 494 573 652 731 810 889 968 1047 1126 1205 1284 1363 1442 1521
1678 1757 1836 1915 1994 2073 2152 2231 2310 2312 2391 2470 2549 2628 2707 2786 2865 2944 3023 3102 3181 3260 3339 3418 3497 3576 3655 3734 3813 3892 3971 4050 4129 4208 4287 4366 4445 4524 4603 4682 4761 4840 4919 4998 5077 5156 5235 5237 5316 5395 5474 5553 5632 5711 5790 5869 19 98 177 256 335 414 493 572 651 730 809 888 967 1046 1125 1204 1283 1362 1441 1520 1599
1756 1835 1914 1993 2072 2151 2230 2309 2311 2390 2469 2548 2627 2706 2785 2864 2943 3022 3101 3180 3259 3338 3417 3496 3575 3654 3733 3812 3891 3970 4049 4128 4207 4286 4365 4444 4523 4602 4681 4760 4839 4918 4997 5076 5155 5234 5313 5315 5394 5473 5552 5631 5710 5789 5868 18 97 176 255 334 413 492 571 650 729 808 887 966 1045 1124 1203 1282 1361 1440 1519 1598 1677
1834 1913 1992 2071 2150 2229 2308 2387 2389 2468 2547 2626 2705 2784 2863 2942 3021 3100 3179 3258 3337 3416 3495 3574 3653 3732 3811 3890 3969 4048 4127 4206 4285 4364 4443 4522 4601 4680 4759 4838 4917 4996 5075 5154 5233 5312 5314 5393 5472 5551 5630 5709 5788 5867 17 96 175 254 333 412 491 570 649 728 807 886 965 1044 1123 1202 1281 1360 1439 1518 1597 1676 1755
1912 1991 2070 2149 2228 2307 2386 2388 2467 2546 2625 2704 2783 2862 2941 3020 3099 3178 3257 3336 3415 3494 3573 3652 3731 3810 3889 3968 4047 4126 4205 4284 4363 4442 4521 4600 4679 4758 4837 4916 4995 5074 5153 5232 5311 5390 5392 5471 5550 5629 5708 5787 5866 16 95 174 253 332 411 490 569 648 727 806 885 964 1043 1122 1201 1280 1359 1438 1517 1596 1675 1754 1833
1990 2069 2148 2227 2306 2385 2464 2466 2545 2624 2703 2782 2861 2940 3019 3098 3177 3256 3335 3414 3493 3572 3651 3730 3809 3888 3967 4046 4125 4204 4283 4362 4441 4520 4599 4678 4757 4836 4915 4994 5073 5152 5231 5310 5389 5391 5470 5549 5628 5707 5786 5865 15 94 173 252 331 410 489 568 647 726 805 884 963 1042 1121 1200 1279 1358 1437 1516 1595 1674 1753 1832 1911
2068 2147 2226 2305 2384 2463 2465 2544 2623 2702 2781 2860 2939 3018 3097 3176 3255 3334 3413 3492 3571 3650 3729 3808 3887 3966 4045 4124 4203 4282 4361 4440 4519 4598 4677 4756 4835 4914 4993 5072 5151 5230 5309 5388 5467 5469 5548 5627 5706 5785 5864 14 93 172 251 330 409 488 567 646 725 804 883 962 1041 1120 1199 1278 1357 1436 1515 1594 1673 1752 1831 1910 1989
2146 2225 2304 2383 2462 2541 2543 2622 2701 2780 2859 2938 3017 3096 3175 3254 3333 3412 3491 3570 3649 3728 3807 3886 3965 4044 4123 4202 4281 4360 4439 4518 4597 4676 4755 4834 4913 4992 5071 5150 5229 5308 5387 5466 5468 5547 5626 5705 5784 5863 13 92 171 250 329 408 487 566 645 724 803 882 961 1040 1119 1198 1277 1356 1435 1514 1593 1672 1751 1830 1909 1988 2067
2224 2303 2382 2461 2540 2542 2621 2700 2779 2858 2937 3016 3095 3174 3253 3332 3411 3490 3569 3648 3727 3806 3885 3964 4043 4122 4201 4280 4359 4438 4517 4596 4675 4754 4833 4912 4991 5070 5149 5228 5307 5386 5465 5544 5546 5625 5704 5783 5862 12 91 170 249 328 407 486 565 644 723 802 881 960 1039 1118 1197 1276 1355 1434 1513 1592 1671 1750 1829 1908 1987 2066 2145
2302 2381 2460 2539 2618 2620 2699 2778 2857 2936 3015 3094 3173 3252 3331 3410 3489 3568 3647 3726 3805 3884 3963 4042 4121 4200 4279 4358 4437 4516 4595 4674 4753 4832 4911 4990 5069 5148 5227 5306 5385 5464 5543 5545 5624 5703 5782 5861 11 90 169 248 327 406 485 564 643 722 801 880 959 1038 1117 1196 1275 1354 1433 1512 1591 1670 1749 1828 1907 1986 2065 2144 2223
2380 2459 2538 2617 2619 2698 2777 2856 2935 3014 3093 3172 3251 3330 3409 3488 3567 3646 3725 3804 3883 3962 4041 4120 4199 4278 4357 4436 4515 4594 4673 4752 4831 4910 4989 5068 5147 5226 5305 5384 5463 5542 5621 5623 5702 5781 5860 10 89 168 247 326 405 484 563 642 721 800 879 958 1037 1116 1195 1274 1353 1432 1511 1590 1669 1748 1827 1906 1985 2064 2143 2222 2301
2458 2537 2616 2695 2697 2776 2855 2934 3013 3092 3171 3250 3329 3408 3487 3566 3645 3724 3803 3882 3961 4040 4119 4198 4277 4356 4435 4514 4593 4672 4751 4830 4909 4988 5067 5146 5225 5304 5383 5462 5541 5620 5622 5701 5780 5859 9 88 167 246 325 404 483 562 641 720 799 878 957 1036 1115 1194 1273 1352 1431 1510 1589 1668 1747 1826 1905 1984 2063 2142 2221 2300 2379
2536 2615 2694 2696 2775 2854 2933 3012 3091 3170 3249 3328 3407 3486 3565 3644 3723 3802 3881 3960 4039 4118 4197 4276 4355 4434 4513 4592 4671 4750 4829 4908 4987 5066 5145 5224 5303 5382 5461 5540 5619 5698 5700 5779 5858 8 87 166 245 324 403 482 561 640 719 798 877 956 1035 1114 1193 1272 1351 1430 1509 1588 1667 1746 1825 1904 1983 2062 2141 2220 2299 2378 2457
2614 2693 2772 2774 2853 2932 3011 3090 3169 3248 3327 3406 3485 3564 3643 3722 3801 3880 3959 4038 4117 4196 4275 4354 4433 4512 4591 4670 4749 4828 4907 4986 5065 5144 5223 5302 5381 5460 5539 5618 5697 5699 5778 5857 7 86 165 244 323 402 481 560 639 718 797 876 955 1034 1113 1192 1271 1350 1429 1508 1587 1666 1745 1824 1903 1982 2061 2140 2219 2298 2377 2456 2535
2692 2771 2773 2852 2931 3010 3089 3168 3247 3326 3405 3484 3563 3642 3721 3800 3879 3958 4037 4116 4195 4274 4353 4432 4511 4590 4669 4748 4827 4906 4985 5064 5143 5222 5301 5380 5459 5538 5617 5696 5775 5777 5856 6 85 164 243 322 401 480 559 638 717 796 875 954 1033 1112 1191 1270 1349 1428 1507 1586 1665 1744 1823 1902 1981 2060 2139 2218 2297 2376 2455 2534 2613
2770 2849 2851 2930 3009 3088 3167 3246 3325 3404 3483 3562 3641 3720 3799 3878 3957 4036 4115 4194 4273 4352 4431 4510 4589 4668 4747 4826 4905 4984 5063 5142 5221 5300 5379 5458 5537 5616 5695 5774 5776 5855 5 84 163 242 321 400 479 558 637 716 795 874 953 1032 1111 1190 1269 1348 1427 1506 1585 1664 1743 1822 1901 1980 2059 2138 2217 2296 2375 2454 2533 2612 2691
2848 2850 2929 3008 3087 3166 3245 3324 3403 3482 3561 3640 3719 3798 3877 3956 4035 4114 4193 4272 4351 4430 4509 4588 4667 4746 4825 4904 4983 5062 5141 5220 5299 5378 5457 5536 5615 5694 5773 5852 5854 4 83 162 241 320 399 478 557 636 715 794 873 952 1031 1110 1189 1268 1347 1426 1505 1584 1663 1742 1821 1900 1979 2058 2137 2216 2295 2374 2453 2532 2611 2690 2769
2926 2928 3007 3086 3165 3244 3323 3402 3481 3560 3639 3718 3797 3876 3955 4034 4113 4192 4271 4350 4429 4508 4587 4666 4745 4824 4903 4982 5061 5140 5219 5298 5377 5456 5535 5614 5693 5772 5851 5853 3 82 161 240 319 398 477 556 635 714 793 872 951 1030 1109 1188 1267 1346 1425 1504 1583 1662 1741 1820 1899 1978 2057 2136 2215 2294 2373 2452 2531 2610 2689 2768 2847
2927 3006 3085 3164 3243 3322 3401 3480 3559 3638 3717 3796 3875 3954 4033 4112 4191 4270 4349 4428 4507 4586 4665 4744 4823 4902 4981 5060 5139 5218 5297 5376 5455 5534 5613 5692 5771 5850 5929 2 81 160 239 318 397 476 555 634 713 792 871 950 1029 1108 1187 1266 1345 1424 1503 1582 1661 1740 1819 1898 1977 2056 2135 2214 2293 2372 2451 2530 2609 2688 2767 2846 2925


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How to Make Your Own Odd-Size Magic Square

  1. Draw a square with any odd number of rows and columns.
  2. Put your first number in the MIDDLE square of the TOP row.
  3. Move UP and to the RIGHT to find the position for your next number, but ...
    • if the next "position" is ABOVE a column, go to BOTTOM of that column (you can see a sample of this movement by looking at how we moved from 1 to 2 in the square above).
    • if the next "position" is to the RIGHT of a row, go to LEFT of that row (you can see this movement between (39) and (40) above).
    • if the next cell is already filled in, drop down to the cell below the number you just filled in for the next number (you can see this when you look at how we moved from 77 to the next number above).
    • if the next cell is both ABOVE and to the RIGHT of the entire square, drop down to the cell below the number you just filled in for your next number (you can see this if you look at (3003) and the next number).
  4. Print your NEXT number into the new square you just moved to.
  5. Repeat steps 3-4 until the entire square is full.
  6. You can see how it works by printing this page and drawing arrows to see how it worked when I created this magic square
This Magic Square was created by John Knoderer, a webmaster and computer programmer who is available to telecommute to your location.

If you found this page useful, and especially if you printed this page to distribute to students, we invite you to send a suitable donation to John (address below) or PayPal your donation to Donations (at) Mazes.com.

You can find more of John's puzzling materials at www.MAZES.com and at www.GodLovesEveryone.org. You can reach John at Webmaster (at) Mazes.com.

Fine Print: If you print and duplicate these magic squares for classroom use or other multiple copy use, please send a payment of $(however much you think this is worth) to the author. John puts a GREAT DEAL of time into creating and posting materials to the internet for you to use (and enjoy). Your donations will help pay bills and keep new materials coming your way.

Send your donations to:

John Knoderer
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P O Box 235
Sulphur Springs, AR 72768 USA

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E-mail the webmaster: webmaster (at) mazes.com

The contents of this page are copyright 2005 by the author. However, we recognize that teachers may wish to download and use our materials with their students. We give that permission, as long as the teacher sends what they believe to be a fair donation to the author each time the material is used.

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