Nineteen shillings and ninepence may be paid in 458,908,622 different ways.
I do not propose to give my method of solution. Any such explanation would occupy an amount of space out of proportion to its interest or value. If I could give within reasonable limits a general solution for all money payments, I would strain a point to find room; but such a solution would be extremely complex and cumbersome, and I do not consider it worth the labour of working out.
Just to give an idea of what such a solution would involve, I will merely say that I find that, dealing only with those sums of money that are multiples of threepence, if we only use bronze coins any sum can be paid in (n + 1)² ways where n always represents the number of pence. If threepenny-pieces are admitted, there are
2n³ + 15n² + 33n --------------------- + 1 ways. 18
If sixpences are also used there are
n^{4} + 22n³ + 159n² + 414n + 216 --------------------------------- 216
ways, when the sum is a multiple of sixpence, and the constant, 216, changes to 324 when the money is not such a multiple. And so the formulas increase in complexity in an accelerating ratio as we go on to the other coins.
I will, however, add an interesting little table of the possible ways of changing our current coins which I believe has never been given in a book before. Change may be given for a
Farthing in 0 way. Halfpenny in 1 way. Penny in 3 ways. Threepenny-piece in 16 ways. Sixpence in 66 ways. Shilling in 402 ways. Florin in 3,818 ways. Half-crown in 8,709 ways. Double florin in 60,239 ways. Crown in 166,651 ways. Half-sovereign in 6,261,622 ways. Sovereign in 500,291,833 ways.
It is a little surprising to find that a sovereign may be changed in over five hundred million different ways. But I have no doubt as to the correctness of my figures.
