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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 73x73 Magic Square

Scroll down to see a magic square
in which all rows, columns, and both diagonals
add up to the same magic sum (194545)

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 73 by 73 magic square shown below has a magic sum equal to 194545.

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 73-by-73 Magic Square

(Magic Sum = 194545)

Books about
Magic Squares
at Amazon.co.UK


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1296 1371 1446 1521 1596 1671 1746 1821 1896 1971 1973 2048 2123 2198 2273 2348 2423 2498 2573 2648 2723 2798 2873 2948 3023 3098 3173 3248 3323 3398 3473 3548 3623 3698 3773 3848 3923 3998 4073 4148 4223 4298 4373 4448 4523 4598 4600 4675 4750 4825 4900 4975 5050 5125 5200 5275 21 96 171 246 321 396 471 546 621 696 771 846 921 996 1071 1146 1221
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1444 1519 1594 1669 1744 1819 1894 1969 2044 2046 2121 2196 2271 2346 2421 2496 2571 2646 2721 2796 2871 2946 3021 3096 3171 3246 3321 3396 3471 3546 3621 3696 3771 3846 3921 3996 4071 4146 4221 4296 4371 4446 4521 4596 4671 4673 4748 4823 4898 4973 5048 5123 5198 5273 19 94 169 244 319 394 469 544 619 694 769 844 919 994 1069 1144 1219 1294 1369
1518 1593 1668 1743 1818 1893 1968 2043 2045 2120 2195 2270 2345 2420 2495 2570 2645 2720 2795 2870 2945 3020 3095 3170 3245 3320 3395 3470 3545 3620 3695 3770 3845 3920 3995 4070 4145 4220 4295 4370 4445 4520 4595 4670 4745 4747 4822 4897 4972 5047 5122 5197 5272 18 93 168 243 318 393 468 543 618 693 768 843 918 993 1068 1143 1218 1293 1368 1443
1592 1667 1742 1817 1892 1967 2042 2117 2119 2194 2269 2344 2419 2494 2569 2644 2719 2794 2869 2944 3019 3094 3169 3244 3319 3394 3469 3544 3619 3694 3769 3844 3919 3994 4069 4144 4219 4294 4369 4444 4519 4594 4669 4744 4746 4821 4896 4971 5046 5121 5196 5271 17 92 167 242 317 392 467 542 617 692 767 842 917 992 1067 1142 1217 1292 1367 1442 1517
1666 1741 1816 1891 1966 2041 2116 2118 2193 2268 2343 2418 2493 2568 2643 2718 2793 2868 2943 3018 3093 3168 3243 3318 3393 3468 3543 3618 3693 3768 3843 3918 3993 4068 4143 4218 4293 4368 4443 4518 4593 4668 4743 4818 4820 4895 4970 5045 5120 5195 5270 16 91 166 241 316 391 466 541 616 691 766 841 916 991 1066 1141 1216 1291 1366 1441 1516 1591
1740 1815 1890 1965 2040 2115 2190 2192 2267 2342 2417 2492 2567 2642 2717 2792 2867 2942 3017 3092 3167 3242 3317 3392 3467 3542 3617 3692 3767 3842 3917 3992 4067 4142 4217 4292 4367 4442 4517 4592 4667 4742 4817 4819 4894 4969 5044 5119 5194 5269 15 90 165 240 315 390 465 540 615 690 765 840 915 990 1065 1140 1215 1290 1365 1440 1515 1590 1665
1814 1889 1964 2039 2114 2189 2191 2266 2341 2416 2491 2566 2641 2716 2791 2866 2941 3016 3091 3166 3241 3316 3391 3466 3541 3616 3691 3766 3841 3916 3991 4066 4141 4216 4291 4366 4441 4516 4591 4666 4741 4816 4891 4893 4968 5043 5118 5193 5268 14 89 164 239 314 389 464 539 614 689 764 839 914 989 1064 1139 1214 1289 1364 1439 1514 1589 1664 1739
1888 1963 2038 2113 2188 2263 2265 2340 2415 2490 2565 2640 2715 2790 2865 2940 3015 3090 3165 3240 3315 3390 3465 3540 3615 3690 3765 3840 3915 3990 4065 4140 4215 4290 4365 4440 4515 4590 4665 4740 4815 4890 4892 4967 5042 5117 5192 5267 13 88 163 238 313 388 463 538 613 688 763 838 913 988 1063 1138 1213 1288 1363 1438 1513 1588 1663 1738 1813
1962 2037 2112 2187 2262 2264 2339 2414 2489 2564 2639 2714 2789 2864 2939 3014 3089 3164 3239 3314 3389 3464 3539 3614 3689 3764 3839 3914 3989 4064 4139 4214 4289 4364 4439 4514 4589 4664 4739 4814 4889 4964 4966 5041 5116 5191 5266 12 87 162 237 312 387 462 537 612 687 762 837 912 987 1062 1137 1212 1287 1362 1437 1512 1587 1662 1737 1812 1887
2036 2111 2186 2261 2336 2338 2413 2488 2563 2638 2713 2788 2863 2938 3013 3088 3163 3238 3313 3388 3463 3538 3613 3688 3763 3838 3913 3988 4063 4138 4213 4288 4363 4438 4513 4588 4663 4738 4813 4888 4963 4965 5040 5115 5190 5265 11 86 161 236 311 386 461 536 611 686 761 836 911 986 1061 1136 1211 1286 1361 1436 1511 1586 1661 1736 1811 1886 1961
2110 2185 2260 2335 2337 2412 2487 2562 2637 2712 2787 2862 2937 3012 3087 3162 3237 3312 3387 3462 3537 3612 3687 3762 3837 3912 3987 4062 4137 4212 4287 4362 4437 4512 4587 4662 4737 4812 4887 4962 5037 5039 5114 5189 5264 10 85 160 235 310 385 460 535 610 685 760 835 910 985 1060 1135 1210 1285 1360 1435 1510 1585 1660 1735 1810 1885 1960 2035
2184 2259 2334 2409 2411 2486 2561 2636 2711 2786 2861 2936 3011 3086 3161 3236 3311 3386 3461 3536 3611 3686 3761 3836 3911 3986 4061 4136 4211 4286 4361 4436 4511 4586 4661 4736 4811 4886 4961 5036 5038 5113 5188 5263 9 84 159 234 309 384 459 534 609 684 759 834 909 984 1059 1134 1209 1284 1359 1434 1509 1584 1659 1734 1809 1884 1959 2034 2109
2258 2333 2408 2410 2485 2560 2635 2710 2785 2860 2935 3010 3085 3160 3235 3310 3385 3460 3535 3610 3685 3760 3835 3910 3985 4060 4135 4210 4285 4360 4435 4510 4585 4660 4735 4810 4885 4960 5035 5110 5112 5187 5262 8 83 158 233 308 383 458 533 608 683 758 833 908 983 1058 1133 1208 1283 1358 1433 1508 1583 1658 1733 1808 1883 1958 2033 2108 2183
2332 2407 2482 2484 2559 2634 2709 2784 2859 2934 3009 3084 3159 3234 3309 3384 3459 3534 3609 3684 3759 3834 3909 3984 4059 4134 4209 4284 4359 4434 4509 4584 4659 4734 4809 4884 4959 5034 5109 5111 5186 5261 7 82 157 232 307 382 457 532 607 682 757 832 907 982 1057 1132 1207 1282 1357 1432 1507 1582 1657 1732 1807 1882 1957 2032 2107 2182 2257
2406 2481 2483 2558 2633 2708 2783 2858 2933 3008 3083 3158 3233 3308 3383 3458 3533 3608 3683 3758 3833 3908 3983 4058 4133 4208 4283 4358 4433 4508 4583 4658 4733 4808 4883 4958 5033 5108 5183 5185 5260 6 81 156 231 306 381 456 531 606 681 756 831 906 981 1056 1131 1206 1281 1356 1431 1506 1581 1656 1731 1806 1881 1956 2031 2106 2181 2256 2331
2480 2555 2557 2632 2707 2782 2857 2932 3007 3082 3157 3232 3307 3382 3457 3532 3607 3682 3757 3832 3907 3982 4057 4132 4207 4282 4357 4432 4507 4582 4657 4732 4807 4882 4957 5032 5107 5182 5184 5259 5 80 155 230 305 380 455 530 605 680 755 830 905 980 1055 1130 1205 1280 1355 1430 1505 1580 1655 1730 1805 1880 1955 2030 2105 2180 2255 2330 2405
2554 2556 2631 2706 2781 2856 2931 3006 3081 3156 3231 3306 3381 3456 3531 3606 3681 3756 3831 3906 3981 4056 4131 4206 4281 4356 4431 4506 4581 4656 4731 4806 4881 4956 5031 5106 5181 5256 5258 4 79 154 229 304 379 454 529 604 679 754 829 904 979 1054 1129 1204 1279 1354 1429 1504 1579 1654 1729 1804 1879 1954 2029 2104 2179 2254 2329 2404 2479
2628 2630 2705 2780 2855 2930 3005 3080 3155 3230 3305 3380 3455 3530 3605 3680 3755 3830 3905 3980 4055 4130 4205 4280 4355 4430 4505 4580 4655 4730 4805 4880 4955 5030 5105 5180 5255 5257 3 78 153 228 303 378 453 528 603 678 753 828 903 978 1053 1128 1203 1278 1353 1428 1503 1578 1653 1728 1803 1878 1953 2028 2103 2178 2253 2328 2403 2478 2553
2629 2704 2779 2854 2929 3004 3079 3154 3229 3304 3379 3454 3529 3604 3679 3754 3829 3904 3979 4054 4129 4204 4279 4354 4429 4504 4579 4654 4729 4804 4879 4954 5029 5104 5179 5254 5329 2 77 152 227 302 377 452 527 602 677 752 827 902 977 1052 1127 1202 1277 1352 1427 1502 1577 1652 1727 1802 1877 1952 2027 2102 2177 2252 2327 2402 2477 2552 2627


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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 (37) and (38) 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 73 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 (2701) 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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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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