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 |
| 18 |
ways. If sixpences are also used there are
| n4+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.