Smallest magic squares of triangular numbers (and of polygonal numbers)

First polygonal numbers

In 1941, Royal Vale Heath proposed this short problem E 496 in The American Mathematical Monthly:

What is the smallest value of n for which the n² triangular numbers 0, 1, 3, 6, 10, …, n²(n² - 1)/2 can be arranged to form a magic square?

With only the proof of n £ 8, this problem remained unsolved. "Better late than never", here is the solution found in April 2007... 66 years later:

n = 6.

My solution was published in the October 2007 issue of The American Mathematical Monthly, including this example:

 0 406 120 528 105 136 1 300 435 378 171 10 66 276 496 15 91 351 595 78 153 28 210 231 3 190 55 21 465 561 630 45 36 325 253 6

On this subject, read also the Mathematical Tourist article written in November 2007 by Ivars Peterson:

My solution was also published in the January 2008 issues of Pour La Science (page 31) and Sciences et Avenir  (page 20).

 p Magic squares using consecutivep-polygonal numbers using distinctp-polygonal numbers Smallest possible order n Smallest possible order n Smallest known order n 3 of triangular numbers 6 (**, by C. Boyer, 2007) Unknown! (maybe 3?) 4 (**, by C. Boyer, 2007) 4 of squares 7 (*, by C. Boyer, 2007) Unknown! (maybe 3?) 4 (*, by L. Euler, 1770) 5 of pentagonal numbers 7 (**, by C. Boyer, 2007) Unknown! (maybe 3?) 4 (**, by L. Morgenstern, 2007) 6 of hexagonal numbers 7 (***, by C. Boyer, 2007) Unknown! (maybe 3?) 4 (***, by L. Morgenstern, 2013) 7 of heptagonal numbers Unknown! (maybe 3?) 8 of octagonal numbers Unknown! (maybe 3?) 4 (***, by L. Morgenstern, 2014) 9 of nonagonal numbers Unknown! (maybe 3?) 10 of decagonal numbers Unknown! (maybe 3?)

(*) Examples of such squares are given here
(**) Examples of such squares are given in my expanded solution below
(***) Examples of such squares are given below

A more detailed text than the version published in the Monthly is available in two formats:

In this expanded solution, you will see:

• The relationship to bimagic squares
• Smallest magic squares of other polygonal numbers (with the examples of magic squares of pentagonal numbers)
• Two unsolved problems, using distinct numbers instead of consecutive numbers
• the smallest magic square of distinct triangular numbers
• the smallest magic square of distinct pentagonal numbers
• My acknowledgements to Sebastião A. da Silva (Brazil), Doug Hensley and Douglas B. West (USA), George P. H. Styan and Evelyn Matheson Styan (Canada)

Smallest magic squares of hexagonal to decagonal numbers

Charles W. Trigg published in School Science and Mathematics and in the Journal of Recreational Mathematics his magic squares of order 8 of hexagonal to decagonal numbers, see references.
But it is possible to construct smaller magic squares. In the extended solution seen above, my magic squares of ordre 7 of hexagonal to decagonal numbers were announced but not given, because "it might be boring to give all my samples". After questions from some readers of this website, here are revealed my squares of order 7, the smallest possible order for squares of consecutive hexagonal to decagonal numbers.

 4560 703 3321 0 120 561 1431 2850 28 630 1 3160 2701 1326 231 15 2016 4371 1540 378 2145 91 45 3003 276 861 4005 2415 2278 3828 66 2556 1035 153 780 190 4186 435 3486 325 1128 946 496 1891 1225 6 3655 1770 1653
 0 235 4995 5452 1782 81 783 4141 3186 286 148 469 4558 540 3940 34 403 189 2356 4347 2059 5221 55 874 2673 2512 342 1651 1 4774 112 616 3744 1071 3010 18 2839 970 3553 1177 1404 3367 7 2205 5688 697 1288 1525 1918
 0 40 1160 4961 4485 1281 4033 1 6816 96 3201 4720 645 481 8 225 4256 833 1825 3605 5208 341 5985 5720 560 1541 1680 133 6533 21 176 2296 1408 5461 65 2821 2465 3816 2133 1045 280 3400 6256 408 736 1976 936 3008 2640
 7291 2871 261 0 1956 3729 2484 325 4699 111 396 2125 3961 6975 1350 6364 856 4446 3286 651 1639 46 1216 6666 1 5500 204 4959 154 1794 3075 5781 750 6069 969 4200 559 9 7944 2301 3504 75 5226 1089 7614 24 2674 474 1491
 976 0 6280 9072 1387 232 3277 7965 27 3510 855 2835 370 5662 637 1105 4000 5076 8326 540 1540 85 8695 10 3751 52 6930 1701 7267 4795 4257 2047 2232 175 451 3052 1 2425 297 1870 7612 5967 1242 6601 742 126 4522 5365 2626

In January 2013, Lee Morgenstern constructed these magic squares of order 4, the smallest known order for squares of distinct hexagonal and heptagonal numbers. Squares of order 4 of octogonal to decagonal numbers are unknown.

 984906 246051 3044278 72010 1540 3194128 141246 1010331 115921 889111 122760 3219453 3244878 17955 1038961 45451
 57233385 4294836 5312223 28962934 21707602 37727235 7592508 28776033 7359066 49744611 739024 37960677 9503325 4036696 82159623 103734

And in November 2014, he added these squares of octagonal, nonagonal and decagonal numbers:

 2896901 469656 150080 1301525 857605 1686000 1297576 976981 446216 1601621 232965 2537360 617440 1060885 3137541 2296
 49366986 5875416 432784 105284575 88188850 10861849 13249341 48659721 19339326 38361246 98487325 4771864 4064599 105861250 48790311 2243601
 43893937 6478297 503035 105252210 93881565 7835800 8911717 45498397 14177107 37788682 101591280 2570410 4174870 104024700 45121447 2806462