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 consecutive 
using distinct 

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:
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 
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