Page:Notes and Queries - Series 12 - Volume 6.djvu/115

 s. vi. APRIL s, 1920.] NOTES AND QUERIES.

91

THEODORUS OF GYRENE. In ' Safe ^Studies,' p. 142, the late Mr. Tollemache says of George Grote, the historian :

" He had a sort of timeo Danaos feeling about "the authors of this half-way movement [Clerical

Rationalism] and he had only a partial sympathy even with Sterling .... His view was that o"f

Tkeodorus of Cyrene ; and he regarded the opposite view as containing the root and germ of

every form of superstition."

Who was Theodorus of Cyrene and what -was his " view " ?

H. E. G. EVANS.

St. Mary's House, Tenby.

CHESS: THE KNIGHT'S TOUR. (12 S. v. 92, 136, 325.)

A CORRESPONDENT asks (12 S. v. 325) how -are startling arithmetical combinations ar- rived at. I cannot say exactly, but I can give a specimen, in which the total of every rank and of every file is 260, and on any .straight line through the centre of the board the difference between the numbers on two -squares equidistant from the centre is 32. Here are the figures in order : 10, 35, 48, 23, 38, 29, 50, 27 ; 47, 22, 11, 36, 49, 26, 39, 30 ; 34, 9, 24, 45, 32, 37, 28, 51 ; 21, 46, 33, 12, .25, 52, 31, 40 ; 8, 63, 20, 57, 41, 1, 14, 53 ; 19, 60, 5, 64, 13, 56, 41, 2 ; 62, 7, 58, 17, 4, 43, 54, 15 ; 59, 18, 61, 6, 55, 16, 3, 42.

I do not think a square in which each diagonal (as well as every rank and file) totals 260 can be made by the knight's tour ; but leaving the knight's tour aside, many
 * such squares can be made, with this fact

added, that every pair of adjacent numbers (taking them in pairs from the edge) totals 5, with the consequence that if the board fee regarded as one of 16 great squares, each .great square consisting of 4 chess squares, then the figures on every great square total 1 30. Here is a specimen, in which the odd numbers are all on white squares on the outer two ranks and files, and all on black squares of the middle 16 : 1, 64, 25, 40, 43, 22, 51, 14 ; 32, 33, 8, 57, 54, 11, 46, 19 ; 21, 44, 52, 13, 2, 63, 39, 26 ; 12, 53, 45, 20, 31, 34, 58, 7 ; -59, 6, 30, 35, 48, 17, 9, 56 ; 38, 27, 3, 62, 49, 16, 24, 41 ; 47, 18, 55, 10, 5, 60, 29, 36 ; 50, 15, 42, 23, 28, 37, 4, 61.

It is quite easy to make a magic square fcy rule of thumb on the square of any odd number. The middle number takes the middle square, and the sum of each pair of slumbers equidistant from the centre on

opposite sides twice the middle number. Squares of even numbers are difficult, and I know of no rule for constructing them.

A. M. B. IRWIN.

Some of the readers of ' N. & Q.' interested in this problem may not have access to Tomlinson's ' Amusements in Chess,' as it has long been out of print, or to other more modern works on chess which deal with it ; I therefore offer them the key to its solution as enunciated by Dr. Roget.

The solution consists in the right applica- tion of certain geometrical figures executed by the knight in the course of his tour. These figures are the " diamond " and the " square," and their right application is dominated throughout by the " cross," and conditioned by a law of alternation.

To cover the board in 63 leaps, starting from any square, the knight has to resort to two classes of moves, viz. : the diamond and the square. Hence arise two systems of moves, comprising 32 squares each. These two systems are again divisable into four of 16 squares each, giving two diamond and two square systems, the alternation of the use of which, offering a prescribed law, fur- nishes an unfailing solution of the problem under all conditions.

Below is a diagram of the board as apportioned out into its two diamond and two square systems :

KEY BOARD.

b

y

X

a

b

y

X

a

X

a

b

y

X

a,

b

y

y

b

a

X

y

b

a

X

a,

X

y

b

a

X

y

b

b

y

X

a,

b

y

X

a

X

a

b

y

X

a

b

'

y

11

b

a

X

y

b

a

X

a,

X

y

b

a,

X

y

b

Let a and 6 enumerate the two diamond systems, and x and y the two square,

Before applying this key to specific cases, the following facts must be observed : that