The problem of constructing magic squares with prime numbers only was first discussed by myself in The Weekly Dispatch for 22nd July and 5th August 1900; but during the last three or four years it has received great attention from American mathematicians. First, they have sought to form these squares with the lowest possible constants. Thus, the first nine prime numbers, $1$ to $23$ inclusive, sum to $99,$ which (being divisible by $3$) is theoretically a suitable series; yet it has been demonstrated that the lowest possible constant is $111,$ and the required series as follows: $1, 7, 13, 31,$ $37, 43, 61,$ $67,$ and $73.$ Similarly, in the case of the fourth order, the lowest series of primes that are "theoretically suitable" will not serve. But in every other order, up to the $12$th inclusive, magic squares have been constructed with the lowest series of primes theoretically possible. And the $12$th is the lowest order in which a straight series of prime numbers, unbroken, from $1$ upwards has been made to work. In other words, the first $144$ odd prime numbers have actually been arranged in magic form. The following summary is taken from The Monist (Chicago) for October 1913:—
| Order of Square | Totals of Series | Lowest Constants | Squares made by |
|---|---|---|---|
| 3rd | 333 | 111 | Henry E. Dudeney (1900). |
| 4th | 408 | 102 | Ernest Bergholt and C. D. Shuldham. |
| 5th | 1065 | 213 | H. A. Sayles. |
| 6th | 2448 | 408 | C. D. Shuldham and J. N. Muncey. |
| 7th | 4893 | 699 | do. |
| 8th | 8912 | 1114 | do. |
| 9th | 15129 | 1681 | do. |
| 10th | 24160 | 2416 | J. N. Muncey. |
| 11th | 36095 | 3355 | do. |
| 12th | 54168 | 4514 | do. |
For further details the reader should consult the article itself, by W. S. Andrews and H. A. Sayles.
These same investigators have also performed notable feats in constructing associated and bordered prime magics, and Mr. Shuldham has sent me a remarkable paper in which he gives examples of Nasik squares constructed with primes for all orders from the $4$th to the $10$th, with the exception of the $3$rd (which is clearly impossible) and the 9th, which, up to the time of writing, has baffled all attempts.
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