Readers may have remarked that in real life it is sometimes cheaper when making a purchase to buy more articles than we require, on the principle of a reduction on taking a quantity: we get more articles and we pay less. Thus, if we want to buy ten apples, and the price asked is a penny each if bought singly, or ninepence a dozen, we should both save a penny and get two apples more than we wanted by buying the full twelve. In the same way, since there is a regular scale of reduction for plates painted alike, we actually save by having two figures painted on that odd plate. Supposing, for example, that we have thirty plates painted alike with 5 on one side and 6 on the other. The rate would be 43/4d., and the cost 11s. 101/2d. But if the odd plate with, say, only a 5 on one side of it have a 6 painted on the other side, we get thirty-one plates at the reduced rate of 41/2d., thus saving a farthing on each of the previous thirty, and reducing the cost of the last one from 1s. to 41/2d. But even after these points are all seen there comes in a new difficulty: for although it will be found that all the 8's may be on the backs of the 7's, we cannot have all the 2's on the backs of the 1's, nor all the 4 on the backs of the 3's, etc. There is a great danger, in our attempts to get as many as possible painted alike, of our so adjusting the figures that some particular combination of hymns cannot be represented. Here is the solution of the difficulty that was sent to the vicar of Chumpley St. Winifred. Where the sign X is placed between two figures, it implies that one of these figures is on one side of the plate and the other on the other side. d. L s. d. 31 plates painted 5 X 9 @ 41/2 = 0 11 71/2 30 " 7 X 8 @ 43/4 = 0 11 101/2 21 " 1 X 2 @ 7 = 0 12 3 21 " 3 X 0 @ 7 = 0 12 3 12 " 1 X 3 @ 91/4 = 0 9 3 12 " 2 X 4 @ 91/4 = 0 9 3 12 " 9 X 4 @ 91/4 = 0 9 3 8 " 4 X 0 @ 101/4 = 0 6 10 1 " 5 X 4 @ 12 = 0 1 0 1 " 5 X 0 @ 12 = 0 1 0 149 plates @ 6d. each = 3 14 6 ---------- L7 19 1 Of course, if we could increase the number of plates, we might get the painting done for nothing, but such a contingency is prevented by the condition that the fewest possible plates must be provided. This puzzle appeared in _Tit-Bits_, and the following remarks, made by me in the issue for 11th December 1897, may be of interest.