The smallest possible pentamagic square

What is the smallest possible pentamagic (5-multimagic) square? Today, the smallest known is the 729th-order pentamagic square of Li Wen. It is also possible, with the method published in Pour La Science, to build other pentamagic squares of 1024th-order or higher orders, but not of a smaller order. I am however convinced that pentamagic squares, of an order smaller than 729, exists.

We must point out a 36th-order "quasi" pentamagic square, created by the American David M. Collison who communicated it in 1991 to John R Hendricks. David Collison unfortunately died a short time later, during the same year 1991. His square is only "quasi" pentamagic since:

• it uses different numbers, but not consecutive
• the 2 diagonals are only trimagic

John R. Hendricks has published this square in his Magic Square Course book. Here are the magic sums of the different powers:

• S1 = 374 940
• S2 = 5 811 077 364
• S3 = 100 225 960 155 180
• S4 = 1 815 549 271 049 335 956
• S5 = 33 830 849 951 944 563 638 700

In August 2008, still using distinct not consecutive integers, Li Wen has constructed a pentamagic square of the same 36th-order. But his square is better than the Collison's square: this time its two diagonals are pentamagic, and its sums are smaller.

• S1 = 242 172
• S2 = 2 404 039 764
• S3 = 26 598 135 571 308
• S4 = 309 190 059 075 899 988
• S5 = 3 697 835 586 559 070 794 572

• Download the 36th-order pentamagic square by Li Wen (Excel file of 23Kb)

An excellent square! And this 36th-order square, when its numbers are raised to the 5th power, is also the the smallest known magic square of 5th powers: look at this summary and tables.

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