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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 71x71 Magic Square

Scroll down to see a magic square
in which all rows, columns, and both diagonals
add up to the same magic sum (178991)

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 71 by 71 magic square shown below has a magic sum equal to 178991.

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 71-by-71 Magic Square

(Magic Sum = 178991)

Books about
Magic Squares
at Amazon.co.UK


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1549 1622 1695 1768 1841 1914 1987 1989 2062 2135 2208 2281 2354 2427 2500 2573 2646 2719 2792 2865 2938 3011 3084 3157 3230 3303 3376 3449 3522 3595 3668 3741 3814 3887 3960 4033 4106 4179 4252 4325 4398 4471 4544 4546 4619 4692 4765 4838 4911 4984 16 89 162 235 308 381 454 527 600 673 746 819 892 965 1038 1111 1184 1257 1330 1403 1476
1621 1694 1767 1840 1913 1986 2059 2061 2134 2207 2280 2353 2426 2499 2572 2645 2718 2791 2864 2937 3010 3083 3156 3229 3302 3375 3448 3521 3594 3667 3740 3813 3886 3959 4032 4105 4178 4251 4324 4397 4470 4543 4545 4618 4691 4764 4837 4910 4983 15 88 161 234 307 380 453 526 599 672 745 818 891 964 1037 1110 1183 1256 1329 1402 1475 1548
1693 1766 1839 1912 1985 2058 2060 2133 2206 2279 2352 2425 2498 2571 2644 2717 2790 2863 2936 3009 3082 3155 3228 3301 3374 3447 3520 3593 3666 3739 3812 3885 3958 4031 4104 4177 4250 4323 4396 4469 4542 4615 4617 4690 4763 4836 4909 4982 14 87 160 233 306 379 452 525 598 671 744 817 890 963 1036 1109 1182 1255 1328 1401 1474 1547 1620
1765 1838 1911 1984 2057 2130 2132 2205 2278 2351 2424 2497 2570 2643 2716 2789 2862 2935 3008 3081 3154 3227 3300 3373 3446 3519 3592 3665 3738 3811 3884 3957 4030 4103 4176 4249 4322 4395 4468 4541 4614 4616 4689 4762 4835 4908 4981 13 86 159 232 305 378 451 524 597 670 743 816 889 962 1035 1108 1181 1254 1327 1400 1473 1546 1619 1692
1837 1910 1983 2056 2129 2131 2204 2277 2350 2423 2496 2569 2642 2715 2788 2861 2934 3007 3080 3153 3226 3299 3372 3445 3518 3591 3664 3737 3810 3883 3956 4029 4102 4175 4248 4321 4394 4467 4540 4613 4686 4688 4761 4834 4907 4980 12 85 158 231 304 377 450 523 596 669 742 815 888 961 1034 1107 1180 1253 1326 1399 1472 1545 1618 1691 1764
1909 1982 2055 2128 2201 2203 2276 2349 2422 2495 2568 2641 2714 2787 2860 2933 3006 3079 3152 3225 3298 3371 3444 3517 3590 3663 3736 3809 3882 3955 4028 4101 4174 4247 4320 4393 4466 4539 4612 4685 4687 4760 4833 4906 4979 11 84 157 230 303 376 449 522 595 668 741 814 887 960 1033 1106 1179 1252 1325 1398 1471 1544 1617 1690 1763 1836
1981 2054 2127 2200 2202 2275 2348 2421 2494 2567 2640 2713 2786 2859 2932 3005 3078 3151 3224 3297 3370 3443 3516 3589 3662 3735 3808 3881 3954 4027 4100 4173 4246 4319 4392 4465 4538 4611 4684 4757 4759 4832 4905 4978 10 83 156 229 302 375 448 521 594 667 740 813 886 959 1032 1105 1178 1251 1324 1397 1470 1543 1616 1689 1762 1835 1908
2053 2126 2199 2272 2274 2347 2420 2493 2566 2639 2712 2785 2858 2931 3004 3077 3150 3223 3296 3369 3442 3515 3588 3661 3734 3807 3880 3953 4026 4099 4172 4245 4318 4391 4464 4537 4610 4683 4756 4758 4831 4904 4977 9 82 155 228 301 374 447 520 593 666 739 812 885 958 1031 1104 1177 1250 1323 1396 1469 1542 1615 1688 1761 1834 1907 1980
2125 2198 2271 2273 2346 2419 2492 2565 2638 2711 2784 2857 2930 3003 3076 3149 3222 3295 3368 3441 3514 3587 3660 3733 3806 3879 3952 4025 4098 4171 4244 4317 4390 4463 4536 4609 4682 4755 4828 4830 4903 4976 8 81 154 227 300 373 446 519 592 665 738 811 884 957 1030 1103 1176 1249 1322 1395 1468 1541 1614 1687 1760 1833 1906 1979 2052
2197 2270 2343 2345 2418 2491 2564 2637 2710 2783 2856 2929 3002 3075 3148 3221 3294 3367 3440 3513 3586 3659 3732 3805 3878 3951 4024 4097 4170 4243 4316 4389 4462 4535 4608 4681 4754 4827 4829 4902 4975 7 80 153 226 299 372 445 518 591 664 737 810 883 956 1029 1102 1175 1248 1321 1394 1467 1540 1613 1686 1759 1832 1905 1978 2051 2124
2269 2342 2344 2417 2490 2563 2636 2709 2782 2855 2928 3001 3074 3147 3220 3293 3366 3439 3512 3585 3658 3731 3804 3877 3950 4023 4096 4169 4242 4315 4388 4461 4534 4607 4680 4753 4826 4899 4901 4974 6 79 152 225 298 371 444 517 590 663 736 809 882 955 1028 1101 1174 1247 1320 1393 1466 1539 1612 1685 1758 1831 1904 1977 2050 2123 2196
2341 2414 2416 2489 2562 2635 2708 2781 2854 2927 3000 3073 3146 3219 3292 3365 3438 3511 3584 3657 3730 3803 3876 3949 4022 4095 4168 4241 4314 4387 4460 4533 4606 4679 4752 4825 4898 4900 4973 5 78 151 224 297 370 443 516 589 662 735 808 881 954 1027 1100 1173 1246 1319 1392 1465 1538 1611 1684 1757 1830 1903 1976 2049 2122 2195 2268
2413 2415 2488 2561 2634 2707 2780 2853 2926 2999 3072 3145 3218 3291 3364 3437 3510 3583 3656 3729 3802 3875 3948 4021 4094 4167 4240 4313 4386 4459 4532 4605 4678 4751 4824 4897 4970 4972 4 77 150 223 296 369 442 515 588 661 734 807 880 953 1026 1099 1172 1245 1318 1391 1464 1537 1610 1683 1756 1829 1902 1975 2048 2121 2194 2267 2340
2485 2487 2560 2633 2706 2779 2852 2925 2998 3071 3144 3217 3290 3363 3436 3509 3582 3655 3728 3801 3874 3947 4020 4093 4166 4239 4312 4385 4458 4531 4604 4677 4750 4823 4896 4969 4971 3 76 149 222 295 368 441 514 587 660 733 806 879 952 1025 1098 1171 1244 1317 1390 1463 1536 1609 1682 1755 1828 1901 1974 2047 2120 2193 2266 2339 2412
2486 2559 2632 2705 2778 2851 2924 2997 3070 3143 3216 3289 3362 3435 3508 3581 3654 3727 3800 3873 3946 4019 4092 4165 4238 4311 4384 4457 4530 4603 4676 4749 4822 4895 4968 5041 2 75 148 221 294 367 440 513 586 659 732 805 878 951 1024 1097 1170 1243 1316 1389 1462 1535 1608 1681 1754 1827 1900 1973 2046 2119 2192 2265 2338 2411 2484


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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 (36) and (37) 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 71 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 (2556) 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:

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P O Box 235
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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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