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
ways. If sixpences are also used there are
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.
click here to go to my blog.
See more interesting puzzles at http://puzzles.50webs.org