68 .— The Postage Stamps Puzzles.

circulation are these:—½ d. , 1 d. , 1-½ d. , 2 d. , 2-½ d. , 3 d. , 4 d. , 5 d. , 6 d. , 9 d. , 10 d. , 1 s. , 2 s. 6 d. , 5 s. , 10 s. , £1, and £5. In the first solution the numbers are in arithmetical progression—1, 1-½, 2, 2-½, 3, 3-½, 4, 4-½, 5. But any nine numbers will form a magic square if we can write them thus:—

where the horizontal differences are all alike and the vertical differences all alike, but not necessarily the same as the horizontal. This happens in the case of the second solution, the numbers of which may be written:—

Also in the case of the solution to No. 67, the Coinage Puzzle, the numbers are, in shillings:—

If there are to be nine different numbers, 0 may occur once (as in the solution to No. 22). Yet one might construct squares with negative numbers, as follows:—