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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 57x57 Magic Square

Scroll down to see a magic square
in which all rows, columns, and both diagonals
add up to the same magic sum (92625)

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 57 by 57 magic square shown below has a magic sum equal to 92625.

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 57-by-57 Magic Square

(Magic Sum = 92625)

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Magic Squares
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958 1017 1076 1135 1194 1253 1255 1314 1373 1432 1491 1550 1609 1668 1727 1786 1845 1904 1963 2022 2081 2140 2199 2258 2317 2376 2435 2494 2553 2612 2671 2730 2789 2848 2907 2909 2968 3027 3086 3145 3204 14 73 132 191 250 309 368 427 486 545 604 663 722 781 840 899
1016 1075 1134 1193 1252 1311 1313 1372 1431 1490 1549 1608 1667 1726 1785 1844 1903 1962 2021 2080 2139 2198 2257 2316 2375 2434 2493 2552 2611 2670 2729 2788 2847 2906 2908 2967 3026 3085 3144 3203 13 72 131 190 249 308 367 426 485 544 603 662 721 780 839 898 957
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1132 1191 1250 1309 1368 1370 1429 1488 1547 1606 1665 1724 1783 1842 1901 1960 2019 2078 2137 2196 2255 2314 2373 2432 2491 2550 2609 2668 2727 2786 2845 2904 2963 2965 3024 3083 3142 3201 11 70 129 188 247 306 365 424 483 542 601 660 719 778 837 896 955 1014 1073
1190 1249 1308 1367 1369 1428 1487 1546 1605 1664 1723 1782 1841 1900 1959 2018 2077 2136 2195 2254 2313 2372 2431 2490 2549 2608 2667 2726 2785 2844 2903 2962 3021 3023 3082 3141 3200 10 69 128 187 246 305 364 423 482 541 600 659 718 777 836 895 954 1013 1072 1131
1248 1307 1366 1425 1427 1486 1545 1604 1663 1722 1781 1840 1899 1958 2017 2076 2135 2194 2253 2312 2371 2430 2489 2548 2607 2666 2725 2784 2843 2902 2961 3020 3022 3081 3140 3199 9 68 127 186 245 304 363 422 481 540 599 658 717 776 835 894 953 1012 1071 1130 1189
1306 1365 1424 1426 1485 1544 1603 1662 1721 1780 1839 1898 1957 2016 2075 2134 2193 2252 2311 2370 2429 2488 2547 2606 2665 2724 2783 2842 2901 2960 3019 3078 3080 3139 3198 8 67 126 185 244 303 362 421 480 539 598 657 716 775 834 893 952 1011 1070 1129 1188 1247
1364 1423 1482 1484 1543 1602 1661 1720 1779 1838 1897 1956 2015 2074 2133 2192 2251 2310 2369 2428 2487 2546 2605 2664 2723 2782 2841 2900 2959 3018 3077 3079 3138 3197 7 66 125 184 243 302 361 420 479 538 597 656 715 774 833 892 951 1010 1069 1128 1187 1246 1305
1422 1481 1483 1542 1601 1660 1719 1778 1837 1896 1955 2014 2073 2132 2191 2250 2309 2368 2427 2486 2545 2604 2663 2722 2781 2840 2899 2958 3017 3076 3135 3137 3196 6 65 124 183 242 301 360 419 478 537 596 655 714 773 832 891 950 1009 1068 1127 1186 1245 1304 1363
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1538 1540 1599 1658 1717 1776 1835 1894 1953 2012 2071 2130 2189 2248 2307 2366 2425 2484 2543 2602 2661 2720 2779 2838 2897 2956 3015 3074 3133 3192 3194 4 63 122 181 240 299 358 417 476 535 594 653 712 771 830 889 948 1007 1066 1125 1184 1243 1302 1361 1420 1479
1596 1598 1657 1716 1775 1834 1893 1952 2011 2070 2129 2188 2247 2306 2365 2424 2483 2542 2601 2660 2719 2778 2837 2896 2955 3014 3073 3132 3191 3193 3 62 121 180 239 298 357 416 475 534 593 652 711 770 829 888 947 1006 1065 1124 1183 1242 1301 1360 1419 1478 1537
1597 1656 1715 1774 1833 1892 1951 2010 2069 2128 2187 2246 2305 2364 2423 2482 2541 2600 2659 2718 2777 2836 2895 2954 3013 3072 3131 3190 3249 2 61 120 179 238 297 356 415 474 533 592 651 710 769 828 887 946 1005 1064 1123 1182 1241 1300 1359 1418 1477 1536 1595


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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 (29) and (30) 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 57 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 (1653) 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.

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