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.