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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 75x75 Magic Square

Scroll down to see a magic square
in which all rows, columns, and both diagonals
add up to the same magic sum (210975)

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 75 by 75 magic square shown below has a magic sum equal to 210975.

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 75-by-75 Magic Square

(Magic Sum = 210975)

Books about
Magic Squares
at Amazon.co.UK


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1255 1332 1409 1486 1563 1640 1717 1794 1871 1948 2025 2027 2104 2181 2258 2335 2412 2489 2566 2643 2720 2797 2874 2951 3028 3105 3182 3259 3336 3413 3490 3567 3644 3721 3798 3875 3952 4029 4106 4183 4260 4337 4414 4491 4568 4645 4722 4799 4801 4878 4955 5032 5109 5186 5263 5340 5417 5494 5571 23 100 177 254 331 408 485 562 639 716 793 870 947 1024 1101 1178
1331 1408 1485 1562 1639 1716 1793 1870 1947 2024 2026 2103 2180 2257 2334 2411 2488 2565 2642 2719 2796 2873 2950 3027 3104 3181 3258 3335 3412 3489 3566 3643 3720 3797 3874 3951 4028 4105 4182 4259 4336 4413 4490 4567 4644 4721 4798 4875 4877 4954 5031 5108 5185 5262 5339 5416 5493 5570 22 99 176 253 330 407 484 561 638 715 792 869 946 1023 1100 1177 1254
1407 1484 1561 1638 1715 1792 1869 1946 2023 2100 2102 2179 2256 2333 2410 2487 2564 2641 2718 2795 2872 2949 3026 3103 3180 3257 3334 3411 3488 3565 3642 3719 3796 3873 3950 4027 4104 4181 4258 4335 4412 4489 4566 4643 4720 4797 4874 4876 4953 5030 5107 5184 5261 5338 5415 5492 5569 21 98 175 252 329 406 483 560 637 714 791 868 945 1022 1099 1176 1253 1330
1483 1560 1637 1714 1791 1868 1945 2022 2099 2101 2178 2255 2332 2409 2486 2563 2640 2717 2794 2871 2948 3025 3102 3179 3256 3333 3410 3487 3564 3641 3718 3795 3872 3949 4026 4103 4180 4257 4334 4411 4488 4565 4642 4719 4796 4873 4950 4952 5029 5106 5183 5260 5337 5414 5491 5568 20 97 174 251 328 405 482 559 636 713 790 867 944 1021 1098 1175 1252 1329 1406
1559 1636 1713 1790 1867 1944 2021 2098 2175 2177 2254 2331 2408 2485 2562 2639 2716 2793 2870 2947 3024 3101 3178 3255 3332 3409 3486 3563 3640 3717 3794 3871 3948 4025 4102 4179 4256 4333 4410 4487 4564 4641 4718 4795 4872 4949 4951 5028 5105 5182 5259 5336 5413 5490 5567 19 96 173 250 327 404 481 558 635 712 789 866 943 1020 1097 1174 1251 1328 1405 1482
1635 1712 1789 1866 1943 2020 2097 2174 2176 2253 2330 2407 2484 2561 2638 2715 2792 2869 2946 3023 3100 3177 3254 3331 3408 3485 3562 3639 3716 3793 3870 3947 4024 4101 4178 4255 4332 4409 4486 4563 4640 4717 4794 4871 4948 5025 5027 5104 5181 5258 5335 5412 5489 5566 18 95 172 249 326 403 480 557 634 711 788 865 942 1019 1096 1173 1250 1327 1404 1481 1558
1711 1788 1865 1942 2019 2096 2173 2250 2252 2329 2406 2483 2560 2637 2714 2791 2868 2945 3022 3099 3176 3253 3330 3407 3484 3561 3638 3715 3792 3869 3946 4023 4100 4177 4254 4331 4408 4485 4562 4639 4716 4793 4870 4947 5024 5026 5103 5180 5257 5334 5411 5488 5565 17 94 171 248 325 402 479 556 633 710 787 864 941 1018 1095 1172 1249 1326 1403 1480 1557 1634
1787 1864 1941 2018 2095 2172 2249 2251 2328 2405 2482 2559 2636 2713 2790 2867 2944 3021 3098 3175 3252 3329 3406 3483 3560 3637 3714 3791 3868 3945 4022 4099 4176 4253 4330 4407 4484 4561 4638 4715 4792 4869 4946 5023 5100 5102 5179 5256 5333 5410 5487 5564 16 93 170 247 324 401 478 555 632 709 786 863 940 1017 1094 1171 1248 1325 1402 1479 1556 1633 1710
1863 1940 2017 2094 2171 2248 2325 2327 2404 2481 2558 2635 2712 2789 2866 2943 3020 3097 3174 3251 3328 3405 3482 3559 3636 3713 3790 3867 3944 4021 4098 4175 4252 4329 4406 4483 4560 4637 4714 4791 4868 4945 5022 5099 5101 5178 5255 5332 5409 5486 5563 15 92 169 246 323 400 477 554 631 708 785 862 939 1016 1093 1170 1247 1324 1401 1478 1555 1632 1709 1786
1939 2016 2093 2170 2247 2324 2326 2403 2480 2557 2634 2711 2788 2865 2942 3019 3096 3173 3250 3327 3404 3481 3558 3635 3712 3789 3866 3943 4020 4097 4174 4251 4328 4405 4482 4559 4636 4713 4790 4867 4944 5021 5098 5175 5177 5254 5331 5408 5485 5562 14 91 168 245 322 399 476 553 630 707 784 861 938 1015 1092 1169 1246 1323 1400 1477 1554 1631 1708 1785 1862
2015 2092 2169 2246 2323 2400 2402 2479 2556 2633 2710 2787 2864 2941 3018 3095 3172 3249 3326 3403 3480 3557 3634 3711 3788 3865 3942 4019 4096 4173 4250 4327 4404 4481 4558 4635 4712 4789 4866 4943 5020 5097 5174 5176 5253 5330 5407 5484 5561 13 90 167 244 321 398 475 552 629 706 783 860 937 1014 1091 1168 1245 1322 1399 1476 1553 1630 1707 1784 1861 1938
2091 2168 2245 2322 2399 2401 2478 2555 2632 2709 2786 2863 2940 3017 3094 3171 3248 3325 3402 3479 3556 3633 3710 3787 3864 3941 4018 4095 4172 4249 4326 4403 4480 4557 4634 4711 4788 4865 4942 5019 5096 5173 5250 5252 5329 5406 5483 5560 12 89 166 243 320 397 474 551 628 705 782 859 936 1013 1090 1167 1244 1321 1398 1475 1552 1629 1706 1783 1860 1937 2014
2167 2244 2321 2398 2475 2477 2554 2631 2708 2785 2862 2939 3016 3093 3170 3247 3324 3401 3478 3555 3632 3709 3786 3863 3940 4017 4094 4171 4248 4325 4402 4479 4556 4633 4710 4787 4864 4941 5018 5095 5172 5249 5251 5328 5405 5482 5559 11 88 165 242 319 396 473 550 627 704 781 858 935 1012 1089 1166 1243 1320 1397 1474 1551 1628 1705 1782 1859 1936 2013 2090
2243 2320 2397 2474 2476 2553 2630 2707 2784 2861 2938 3015 3092 3169 3246 3323 3400 3477 3554 3631 3708 3785 3862 3939 4016 4093 4170 4247 4324 4401 4478 4555 4632 4709 4786 4863 4940 5017 5094 5171 5248 5325 5327 5404 5481 5558 10 87 164 241 318 395 472 549 626 703 780 857 934 1011 1088 1165 1242 1319 1396 1473 1550 1627 1704 1781 1858 1935 2012 2089 2166
2319 2396 2473 2550 2552 2629 2706 2783 2860 2937 3014 3091 3168 3245 3322 3399 3476 3553 3630 3707 3784 3861 3938 4015 4092 4169 4246 4323 4400 4477 4554 4631 4708 4785 4862 4939 5016 5093 5170 5247 5324 5326 5403 5480 5557 9 86 163 240 317 394 471 548 625 702 779 856 933 1010 1087 1164 1241 1318 1395 1472 1549 1626 1703 1780 1857 1934 2011 2088 2165 2242
2395 2472 2549 2551 2628 2705 2782 2859 2936 3013 3090 3167 3244 3321 3398 3475 3552 3629 3706 3783 3860 3937 4014 4091 4168 4245 4322 4399 4476 4553 4630 4707 4784 4861 4938 5015 5092 5169 5246 5323 5400 5402 5479 5556 8 85 162 239 316 393 470 547 624 701 778 855 932 1009 1086 1163 1240 1317 1394 1471 1548 1625 1702 1779 1856 1933 2010 2087 2164 2241 2318
2471 2548 2625 2627 2704 2781 2858 2935 3012 3089 3166 3243 3320 3397 3474 3551 3628 3705 3782 3859 3936 4013 4090 4167 4244 4321 4398 4475 4552 4629 4706 4783 4860 4937 5014 5091 5168 5245 5322 5399 5401 5478 5555 7 84 161 238 315 392 469 546 623 700 777 854 931 1008 1085 1162 1239 1316 1393 1470 1547 1624 1701 1778 1855 1932 2009 2086 2163 2240 2317 2394
2547 2624 2626 2703 2780 2857 2934 3011 3088 3165 3242 3319 3396 3473 3550 3627 3704 3781 3858 3935 4012 4089 4166 4243 4320 4397 4474 4551 4628 4705 4782 4859 4936 5013 5090 5167 5244 5321 5398 5475 5477 5554 6 83 160 237 314 391 468 545 622 699 776 853 930 1007 1084 1161 1238 1315 1392 1469 1546 1623 1700 1777 1854 1931 2008 2085 2162 2239 2316 2393 2470
2623 2700 2702 2779 2856 2933 3010 3087 3164 3241 3318 3395 3472 3549 3626 3703 3780 3857 3934 4011 4088 4165 4242 4319 4396 4473 4550 4627 4704 4781 4858 4935 5012 5089 5166 5243 5320 5397 5474 5476 5553 5 82 159 236 313 390 467 544 621 698 775 852 929 1006 1083 1160 1237 1314 1391 1468 1545 1622 1699 1776 1853 1930 2007 2084 2161 2238 2315 2392 2469 2546
2699 2701 2778 2855 2932 3009 3086 3163 3240 3317 3394 3471 3548 3625 3702 3779 3856 3933 4010 4087 4164 4241 4318 4395 4472 4549 4626 4703 4780 4857 4934 5011 5088 5165 5242 5319 5396 5473 5550 5552 4 81 158 235 312 389 466 543 620 697 774 851 928 1005 1082 1159 1236 1313 1390 1467 1544 1621 1698 1775 1852 1929 2006 2083 2160 2237 2314 2391 2468 2545 2622
2775 2777 2854 2931 3008 3085 3162 3239 3316 3393 3470 3547 3624 3701 3778 3855 3932 4009 4086 4163 4240 4317 4394 4471 4548 4625 4702 4779 4856 4933 5010 5087 5164 5241 5318 5395 5472 5549 5551 3 80 157 234 311 388 465 542 619 696 773 850 927 1004 1081 1158 1235 1312 1389 1466 1543 1620 1697 1774 1851 1928 2005 2082 2159 2236 2313 2390 2467 2544 2621 2698
2776 2853 2930 3007 3084 3161 3238 3315 3392 3469 3546 3623 3700 3777 3854 3931 4008 4085 4162 4239 4316 4393 4470 4547 4624 4701 4778 4855 4932 5009 5086 5163 5240 5317 5394 5471 5548 5625 2 79 156 233 310 387 464 541 618 695 772 849 926 1003 1080 1157 1234 1311 1388 1465 1542 1619 1696 1773 1850 1927 2004 2081 2158 2235 2312 2389 2466 2543 2620 2697 2774


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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 (38) and (39) 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 75 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 (2850) 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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