**THE
EXCURSION TICKET PUZZLE.— solution**

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

2n^{3}+15n^{2}+33n |
+ 1 |

18 |

ways. If sixpences are also used there are

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.

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