Dudeney, Amusements in Mathematics Problem 6 A man left instructions to his executors to distribute once a year exactly fifty-five shillings among the poor of his parish; but they were only to continue the gift so long as they could make it in different ways, always giving eighteen pence each to a number of women and half a crown each to men. During how many years could the charity be administered? Of course, by "different ways" is meant a different number of men and women every time.
Suppose in a given year w women and m men receive the charity. Then 55.12 = 18w + 30m, so 110 = 3w + 5m. We must have 110 - 5m divisible by 3 and hence 2 - 5m divisible by 3. So m must be a multiple of 3 plus 1. That gives m = 1, 4, 7, 10, 13, 16, 19, 22. However, Dudeney rules out m = 1, because the problem requires men plural, and similarly rules out m = 22, because the problem requires women . Thus there are 6 possibilities. So 6 years. Dudeney 1-25 (C) John Scholes jscholes@kalva.demon.co.uk 25 Oct 2002