# Amusements in Mathematics

408.—MAGIC SQUARES OF TWO DEGREES.

While reading a French mathematical work I happened to come across, the following

statement: "A very remarkable magic square of 8, in two degrees, has been constructed

by M. Pfeffermann. In other words, he has managed to dispose the sixty-four first

numbers on the squares of a chessboard in such a way that the sum of the numbers in

every line, every column, and in each of the two diagonals, shall be the same; and more,

that if one substitutes for all the numbers their squares, the square still remains magic." I

at once set to work to solve this problem, and, although it proved a very hard nut, one

was rewarded by the discovery of some curious and beautiful laws that govern it. The

reader may like to try his hand at the puzzle.

MAGIC SQUARES OF PRIMES.

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 12th inclusive,

magic squares have been constructed with the lowest series of primes theoretically

possible. And the 12th 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

Dudeney (1900).

E.

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.

Henry