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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 69x69 Magic Square

Scroll down to see a magic square
in which all rows, columns, and both diagonals
add up to the same magic sum (164289)

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 69 by 69 magic square shown below has a magic sum equal to 164289.

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 69-by-69 Magic Square

(Magic Sum = 164289)

Books about
Magic Squares
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1576 1647 1718 1789 1860 1931 1933 2004 2075 2146 2217 2288 2359 2430 2501 2572 2643 2714 2785 2856 2927 2998 3069 3140 3211 3282 3353 3424 3495 3566 3637 3708 3779 3850 3921 3992 4063 4134 4205 4276 4347 4349 4420 4491 4562 4633 4704 14 85 156 227 298 369 440 511 582 653 724 795 866 937 1008 1079 1150 1221 1292 1363 1434 1505
1646 1717 1788 1859 1930 2001 2003 2074 2145 2216 2287 2358 2429 2500 2571 2642 2713 2784 2855 2926 2997 3068 3139 3210 3281 3352 3423 3494 3565 3636 3707 3778 3849 3920 3991 4062 4133 4204 4275 4346 4348 4419 4490 4561 4632 4703 13 84 155 226 297 368 439 510 581 652 723 794 865 936 1007 1078 1149 1220 1291 1362 1433 1504 1575
1716 1787 1858 1929 2000 2002 2073 2144 2215 2286 2357 2428 2499 2570 2641 2712 2783 2854 2925 2996 3067 3138 3209 3280 3351 3422 3493 3564 3635 3706 3777 3848 3919 3990 4061 4132 4203 4274 4345 4416 4418 4489 4560 4631 4702 12 83 154 225 296 367 438 509 580 651 722 793 864 935 1006 1077 1148 1219 1290 1361 1432 1503 1574 1645
1786 1857 1928 1999 2070 2072 2143 2214 2285 2356 2427 2498 2569 2640 2711 2782 2853 2924 2995 3066 3137 3208 3279 3350 3421 3492 3563 3634 3705 3776 3847 3918 3989 4060 4131 4202 4273 4344 4415 4417 4488 4559 4630 4701 11 82 153 224 295 366 437 508 579 650 721 792 863 934 1005 1076 1147 1218 1289 1360 1431 1502 1573 1644 1715
1856 1927 1998 2069 2071 2142 2213 2284 2355 2426 2497 2568 2639 2710 2781 2852 2923 2994 3065 3136 3207 3278 3349 3420 3491 3562 3633 3704 3775 3846 3917 3988 4059 4130 4201 4272 4343 4414 4485 4487 4558 4629 4700 10 81 152 223 294 365 436 507 578 649 720 791 862 933 1004 1075 1146 1217 1288 1359 1430 1501 1572 1643 1714 1785
1926 1997 2068 2139 2141 2212 2283 2354 2425 2496 2567 2638 2709 2780 2851 2922 2993 3064 3135 3206 3277 3348 3419 3490 3561 3632 3703 3774 3845 3916 3987 4058 4129 4200 4271 4342 4413 4484 4486 4557 4628 4699 9 80 151 222 293 364 435 506 577 648 719 790 861 932 1003 1074 1145 1216 1287 1358 1429 1500 1571 1642 1713 1784 1855
1996 2067 2138 2140 2211 2282 2353 2424 2495 2566 2637 2708 2779 2850 2921 2992 3063 3134 3205 3276 3347 3418 3489 3560 3631 3702 3773 3844 3915 3986 4057 4128 4199 4270 4341 4412 4483 4554 4556 4627 4698 8 79 150 221 292 363 434 505 576 647 718 789 860 931 1002 1073 1144 1215 1286 1357 1428 1499 1570 1641 1712 1783 1854 1925
2066 2137 2208 2210 2281 2352 2423 2494 2565 2636 2707 2778 2849 2920 2991 3062 3133 3204 3275 3346 3417 3488 3559 3630 3701 3772 3843 3914 3985 4056 4127 4198 4269 4340 4411 4482 4553 4555 4626 4697 7 78 149 220 291 362 433 504 575 646 717 788 859 930 1001 1072 1143 1214 1285 1356 1427 1498 1569 1640 1711 1782 1853 1924 1995
2136 2207 2209 2280 2351 2422 2493 2564 2635 2706 2777 2848 2919 2990 3061 3132 3203 3274 3345 3416 3487 3558 3629 3700 3771 3842 3913 3984 4055 4126 4197 4268 4339 4410 4481 4552 4623 4625 4696 6 77 148 219 290 361 432 503 574 645 716 787 858 929 1000 1071 1142 1213 1284 1355 1426 1497 1568 1639 1710 1781 1852 1923 1994 2065
2206 2277 2279 2350 2421 2492 2563 2634 2705 2776 2847 2918 2989 3060 3131 3202 3273 3344 3415 3486 3557 3628 3699 3770 3841 3912 3983 4054 4125 4196 4267 4338 4409 4480 4551 4622 4624 4695 5 76 147 218 289 360 431 502 573 644 715 786 857 928 999 1070 1141 1212 1283 1354 1425 1496 1567 1638 1709 1780 1851 1922 1993 2064 2135
2276 2278 2349 2420 2491 2562 2633 2704 2775 2846 2917 2988 3059 3130 3201 3272 3343 3414 3485 3556 3627 3698 3769 3840 3911 3982 4053 4124 4195 4266 4337 4408 4479 4550 4621 4692 4694 4 75 146 217 288 359 430 501 572 643 714 785 856 927 998 1069 1140 1211 1282 1353 1424 1495 1566 1637 1708 1779 1850 1921 1992 2063 2134 2205
2346 2348 2419 2490 2561 2632 2703 2774 2845 2916 2987 3058 3129 3200 3271 3342 3413 3484 3555 3626 3697 3768 3839 3910 3981 4052 4123 4194 4265 4336 4407 4478 4549 4620 4691 4693 3 74 145 216 287 358 429 500 571 642 713 784 855 926 997 1068 1139 1210 1281 1352 1423 1494 1565 1636 1707 1778 1849 1920 1991 2062 2133 2204 2275
2347 2418 2489 2560 2631 2702 2773 2844 2915 2986 3057 3128 3199 3270 3341 3412 3483 3554 3625 3696 3767 3838 3909 3980 4051 4122 4193 4264 4335 4406 4477 4548 4619 4690 4761 2 73 144 215 286 357 428 499 570 641 712 783 854 925 996 1067 1138 1209 1280 1351 1422 1493 1564 1635 1706 1777 1848 1919 1990 2061 2132 2203 2274 2345


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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 (35) and (36) 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 69 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 (2415) 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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