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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 67x67 Magic Square

Scroll down to see a magic square
in which all rows, columns, and both diagonals
add up to the same magic sum (150415)

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 67 by 67 magic square shown below has a magic sum equal to 150415.

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 67-by-67 Magic Square

(Magic Sum = 150415)

Books about
Magic Squares
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1871 1940 2009 2011 2080 2149 2218 2287 2356 2425 2494 2563 2632 2701 2770 2839 2908 2977 3046 3115 3184 3253 3322 3391 3460 3529 3598 3667 3736 3805 3874 3943 4012 4081 4150 4219 4288 4290 4359 4428 8 77 146 215 284 353 422 491 560 629 698 767 836 905 974 1043 1112 1181 1250 1319 1388 1457 1526 1595 1664 1733 1802
1939 2008 2077 2079 2148 2217 2286 2355 2424 2493 2562 2631 2700 2769 2838 2907 2976 3045 3114 3183 3252 3321 3390 3459 3528 3597 3666 3735 3804 3873 3942 4011 4080 4149 4218 4287 4289 4358 4427 7 76 145 214 283 352 421 490 559 628 697 766 835 904 973 1042 1111 1180 1249 1318 1387 1456 1525 1594 1663 1732 1801 1870
2007 2076 2078 2147 2216 2285 2354 2423 2492 2561 2630 2699 2768 2837 2906 2975 3044 3113 3182 3251 3320 3389 3458 3527 3596 3665 3734 3803 3872 3941 4010 4079 4148 4217 4286 4355 4357 4426 6 75 144 213 282 351 420 489 558 627 696 765 834 903 972 1041 1110 1179 1248 1317 1386 1455 1524 1593 1662 1731 1800 1869 1938
2075 2144 2146 2215 2284 2353 2422 2491 2560 2629 2698 2767 2836 2905 2974 3043 3112 3181 3250 3319 3388 3457 3526 3595 3664 3733 3802 3871 3940 4009 4078 4147 4216 4285 4354 4356 4425 5 74 143 212 281 350 419 488 557 626 695 764 833 902 971 1040 1109 1178 1247 1316 1385 1454 1523 1592 1661 1730 1799 1868 1937 2006
2143 2145 2214 2283 2352 2421 2490 2559 2628 2697 2766 2835 2904 2973 3042 3111 3180 3249 3318 3387 3456 3525 3594 3663 3732 3801 3870 3939 4008 4077 4146 4215 4284 4353 4422 4424 4 73 142 211 280 349 418 487 556 625 694 763 832 901 970 1039 1108 1177 1246 1315 1384 1453 1522 1591 1660 1729 1798 1867 1936 2005 2074
2211 2213 2282 2351 2420 2489 2558 2627 2696 2765 2834 2903 2972 3041 3110 3179 3248 3317 3386 3455 3524 3593 3662 3731 3800 3869 3938 4007 4076 4145 4214 4283 4352 4421 4423 3 72 141 210 279 348 417 486 555 624 693 762 831 900 969 1038 1107 1176 1245 1314 1383 1452 1521 1590 1659 1728 1797 1866 1935 2004 2073 2142
2212 2281 2350 2419 2488 2557 2626 2695 2764 2833 2902 2971 3040 3109 3178 3247 3316 3385 3454 3523 3592 3661 3730 3799 3868 3937 4006 4075 4144 4213 4282 4351 4420 4489 2 71 140 209 278 347 416 485 554 623 692 761 830 899 968 1037 1106 1175 1244 1313 1382 1451 1520 1589 1658 1727 1796 1865 1934 2003 2072 2141 2210


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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 (34) and (35) 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 67 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 (2278) 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
www.MAZES.com
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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