# Solution

(related to Problem: The Excursion Ticket Puzzle)

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 labor 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 $$\frac{2n^3+15n^2+33n}{18} + 1$$

ways. If sixpences are also used there are $$\frac{n^4+22n^3+159n^2+414n+216}{216}$$

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.

Github: non-Github:
@H-Dudeney

### References

#### Project Gutenberg

1. Dudeney, H. E.: "Amusements in Mathematics", The Authors' Club, 1917

This eBook is for the use of anyone anywhere in the United States and most other parts of the world at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this edition or online at http://www.gutenberg.org. If you are not located in the United States, you'll have to check the laws of the country where you are located before using this ebook.