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:

A solution of Heath’s Problem E 496.
Order n=6. Magic square with consecutive triangular numbers from 0 to 630. S=1295

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:

•  http://www.maa.org/mathtourist/mathtourist_11_7_07.html

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

Smallest possible magic squares of polygonal numbers

p	Magic squares	using consecutive p-polygonal numbers	using distinct p-polygonal numbers
		Smallest possible order n	Smallest possible order n	Smallest known order n
3	of triangular numbers	6 (**)	Unknown! (maybe 3?)	4 (**)
4	of squares	7 (*)	Unknown! (maybe 3?)	4 (*, by L. Euler, 1770)
5	of pentagonal numbers	7 (**)	Unknown! (maybe 3?)	4 (**, by L. Morgenstern, 2007)

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

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

• See the expanded solution in HTML format
• Download the expanded solution in PDF format (131Kb, 7 pages)

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)
• The explanation of this gift received from Rio:

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