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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 61x61 Magic Square

Scroll down to see a magic square
in which all rows, columns, and both diagonals
add up to the same magic sum (113521)

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 61 by 61 magic square shown below has a magic sum equal to 113521.

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 61-by-61 Magic Square

(Magic Sum = 113521)

Books about
Magic Squares
at Amazon.co.UK


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3691 33 96 159 222 285 348 411 474 537 600 663 726 789 852 915 917 980 1043 1106 1169 1232 1295 1358 1421 1484 1547 1610 1673 1736 1799 1862 1925 1988 2051 2114 2177 2240 2303 2366 2429 2492 2555 2618 2681 2744 2746 2809 2872 2935 2998 3061 3124 3187 3250 3313 3376 3439 3502 3565 3628
32 95 158 221 284 347 410 473 536 599 662 725 788 851 914 916 979 1042 1105 1168 1231 1294 1357 1420 1483 1546 1609 1672 1735 1798 1861 1924 1987 2050 2113 2176 2239 2302 2365 2428 2491 2554 2617 2680 2743 2806 2808 2871 2934 2997 3060 3123 3186 3249 3312 3375 3438 3501 3564 3627 3690
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280 343 406 469 532 595 658 721 784 847 910 973 1036 1038 1101 1164 1227 1290 1353 1416 1479 1542 1605 1668 1731 1794 1857 1920 1983 2046 2109 2172 2235 2298 2361 2424 2487 2550 2613 2676 2739 2802 2865 2928 2930 2993 3056 3119 3182 3245 3308 3371 3434 3497 3560 3623 3686 28 91 154 217
342 405 468 531 594 657 720 783 846 909 972 1035 1098 1100 1163 1226 1289 1352 1415 1478 1541 1604 1667 1730 1793 1856 1919 1982 2045 2108 2171 2234 2297 2360 2423 2486 2549 2612 2675 2738 2801 2864 2927 2929 2992 3055 3118 3181 3244 3307 3370 3433 3496 3559 3622 3685 27 90 153 216 279
404 467 530 593 656 719 782 845 908 971 1034 1097 1099 1162 1225 1288 1351 1414 1477 1540 1603 1666 1729 1792 1855 1918 1981 2044 2107 2170 2233 2296 2359 2422 2485 2548 2611 2674 2737 2800 2863 2926 2989 2991 3054 3117 3180 3243 3306 3369 3432 3495 3558 3621 3684 26 89 152 215 278 341
466 529 592 655 718 781 844 907 970 1033 1096 1159 1161 1224 1287 1350 1413 1476 1539 1602 1665 1728 1791 1854 1917 1980 2043 2106 2169 2232 2295 2358 2421 2484 2547 2610 2673 2736 2799 2862 2925 2988 2990 3053 3116 3179 3242 3305 3368 3431 3494 3557 3620 3683 25 88 151 214 277 340 403
528 591 654 717 780 843 906 969 1032 1095 1158 1160 1223 1286 1349 1412 1475 1538 1601 1664 1727 1790 1853 1916 1979 2042 2105 2168 2231 2294 2357 2420 2483 2546 2609 2672 2735 2798 2861 2924 2987 3050 3052 3115 3178 3241 3304 3367 3430 3493 3556 3619 3682 24 87 150 213 276 339 402 465
590 653 716 779 842 905 968 1031 1094 1157 1220 1222 1285 1348 1411 1474 1537 1600 1663 1726 1789 1852 1915 1978 2041 2104 2167 2230 2293 2356 2419 2482 2545 2608 2671 2734 2797 2860 2923 2986 3049 3051 3114 3177 3240 3303 3366 3429 3492 3555 3618 3681 23 86 149 212 275 338 401 464 527
652 715 778 841 904 967 1030 1093 1156 1219 1221 1284 1347 1410 1473 1536 1599 1662 1725 1788 1851 1914 1977 2040 2103 2166 2229 2292 2355 2418 2481 2544 2607 2670 2733 2796 2859 2922 2985 3048 3111 3113 3176 3239 3302 3365 3428 3491 3554 3617 3680 22 85 148 211 274 337 400 463 526 589
714 777 840 903 966 1029 1092 1155 1218 1281 1283 1346 1409 1472 1535 1598 1661 1724 1787 1850 1913 1976 2039 2102 2165 2228 2291 2354 2417 2480 2543 2606 2669 2732 2795 2858 2921 2984 3047 3110 3112 3175 3238 3301 3364 3427 3490 3553 3616 3679 21 84 147 210 273 336 399 462 525 588 651
776 839 902 965 1028 1091 1154 1217 1280 1282 1345 1408 1471 1534 1597 1660 1723 1786 1849 1912 1975 2038 2101 2164 2227 2290 2353 2416 2479 2542 2605 2668 2731 2794 2857 2920 2983 3046 3109 3172 3174 3237 3300 3363 3426 3489 3552 3615 3678 20 83 146 209 272 335 398 461 524 587 650 713
838 901 964 1027 1090 1153 1216 1279 1342 1344 1407 1470 1533 1596 1659 1722 1785 1848 1911 1974 2037 2100 2163 2226 2289 2352 2415 2478 2541 2604 2667 2730 2793 2856 2919 2982 3045 3108 3171 3173 3236 3299 3362 3425 3488 3551 3614 3677 19 82 145 208 271 334 397 460 523 586 649 712 775
900 963 1026 1089 1152 1215 1278 1341 1343 1406 1469 1532 1595 1658 1721 1784 1847 1910 1973 2036 2099 2162 2225 2288 2351 2414 2477 2540 2603 2666 2729 2792 2855 2918 2981 3044 3107 3170 3233 3235 3298 3361 3424 3487 3550 3613 3676 18 81 144 207 270 333 396 459 522 585 648 711 774 837
962 1025 1088 1151 1214 1277 1340 1403 1405 1468 1531 1594 1657 1720 1783 1846 1909 1972 2035 2098 2161 2224 2287 2350 2413 2476 2539 2602 2665 2728 2791 2854 2917 2980 3043 3106 3169 3232 3234 3297 3360 3423 3486 3549 3612 3675 17 80 143 206 269 332 395 458 521 584 647 710 773 836 899
1024 1087 1150 1213 1276 1339 1402 1404 1467 1530 1593 1656 1719 1782 1845 1908 1971 2034 2097 2160 2223 2286 2349 2412 2475 2538 2601 2664 2727 2790 2853 2916 2979 3042 3105 3168 3231 3294 3296 3359 3422 3485 3548 3611 3674 16 79 142 205 268 331 394 457 520 583 646 709 772 835 898 961
1086 1149 1212 1275 1338 1401 1464 1466 1529 1592 1655 1718 1781 1844 1907 1970 2033 2096 2159 2222 2285 2348 2411 2474 2537 2600 2663 2726 2789 2852 2915 2978 3041 3104 3167 3230 3293 3295 3358 3421 3484 3547 3610 3673 15 78 141 204 267 330 393 456 519 582 645 708 771 834 897 960 1023
1148 1211 1274 1337 1400 1463 1465 1528 1591 1654 1717 1780 1843 1906 1969 2032 2095 2158 2221 2284 2347 2410 2473 2536 2599 2662 2725 2788 2851 2914 2977 3040 3103 3166 3229 3292 3355 3357 3420 3483 3546 3609 3672 14 77 140 203 266 329 392 455 518 581 644 707 770 833 896 959 1022 1085
1210 1273 1336 1399 1462 1525 1527 1590 1653 1716 1779 1842 1905 1968 2031 2094 2157 2220 2283 2346 2409 2472 2535 2598 2661 2724 2787 2850 2913 2976 3039 3102 3165 3228 3291 3354 3356 3419 3482 3545 3608 3671 13 76 139 202 265 328 391 454 517 580 643 706 769 832 895 958 1021 1084 1147
1272 1335 1398 1461 1524 1526 1589 1652 1715 1778 1841 1904 1967 2030 2093 2156 2219 2282 2345 2408 2471 2534 2597 2660 2723 2786 2849 2912 2975 3038 3101 3164 3227 3290 3353 3416 3418 3481 3544 3607 3670 12 75 138 201 264 327 390 453 516 579 642 705 768 831 894 957 1020 1083 1146 1209
1334 1397 1460 1523 1586 1588 1651 1714 1777 1840 1903 1966 2029 2092 2155 2218 2281 2344 2407 2470 2533 2596 2659 2722 2785 2848 2911 2974 3037 3100 3163 3226 3289 3352 3415 3417 3480 3543 3606 3669 11 74 137 200 263 326 389 452 515 578 641 704 767 830 893 956 1019 1082 1145 1208 1271
1396 1459 1522 1585 1587 1650 1713 1776 1839 1902 1965 2028 2091 2154 2217 2280 2343 2406 2469 2532 2595 2658 2721 2784 2847 2910 2973 3036 3099 3162 3225 3288 3351 3414 3477 3479 3542 3605 3668 10 73 136 199 262 325 388 451 514 577 640 703 766 829 892 955 1018 1081 1144 1207 1270 1333
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1582 1645 1708 1710 1773 1836 1899 1962 2025 2088 2151 2214 2277 2340 2403 2466 2529 2592 2655 2718 2781 2844 2907 2970 3033 3096 3159 3222 3285 3348 3411 3474 3537 3539 3602 3665 7 70 133 196 259 322 385 448 511 574 637 700 763 826 889 952 1015 1078 1141 1204 1267 1330 1393 1456 1519
1644 1707 1709 1772 1835 1898 1961 2024 2087 2150 2213 2276 2339 2402 2465 2528 2591 2654 2717 2780 2843 2906 2969 3032 3095 3158 3221 3284 3347 3410 3473 3536 3599 3601 3664 6 69 132 195 258 321 384 447 510 573 636 699 762 825 888 951 1014 1077 1140 1203 1266 1329 1392 1455 1518 1581
1706 1769 1771 1834 1897 1960 2023 2086 2149 2212 2275 2338 2401 2464 2527 2590 2653 2716 2779 2842 2905 2968 3031 3094 3157 3220 3283 3346 3409 3472 3535 3598 3600 3663 5 68 131 194 257 320 383 446 509 572 635 698 761 824 887 950 1013 1076 1139 1202 1265 1328 1391 1454 1517 1580 1643
1768 1770 1833 1896 1959 2022 2085 2148 2211 2274 2337 2400 2463 2526 2589 2652 2715 2778 2841 2904 2967 3030 3093 3156 3219 3282 3345 3408 3471 3534 3597 3660 3662 4 67 130 193 256 319 382 445 508 571 634 697 760 823 886 949 1012 1075 1138 1201 1264 1327 1390 1453 1516 1579 1642 1705
1830 1832 1895 1958 2021 2084 2147 2210 2273 2336 2399 2462 2525 2588 2651 2714 2777 2840 2903 2966 3029 3092 3155 3218 3281 3344 3407 3470 3533 3596 3659 3661 3 66 129 192 255 318 381 444 507 570 633 696 759 822 885 948 1011 1074 1137 1200 1263 1326 1389 1452 1515 1578 1641 1704 1767
1831 1894 1957 2020 2083 2146 2209 2272 2335 2398 2461 2524 2587 2650 2713 2776 2839 2902 2965 3028 3091 3154 3217 3280 3343 3406 3469 3532 3595 3658 3721 2 65 128 191 254 317 380 443 506 569 632 695 758 821 884 947 1010 1073 1136 1199 1262 1325 1388 1451 1514 1577 1640 1703 1766 1829


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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 (31) and (32) 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 61 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 (1891) 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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