THE EXCURSION TICKET PUZZLE.
(
Money Puzzles)
When the big flaming placards were exhibited at the little provincial
railway station, announcing that the Great ---- Company would run cheap
excursion trains to London for the Christmas holidays, the inhabitants
of Mudley-cum-Turmits were in quite a flutter of excitement. Half an
hour before the train came in the little booking office was crowded with
country passengers, all bent on visiting their friends in the great
Metropolis. The booking clerk was unaccustomed to dealing with crowds of
such a dimension, and he told me afterwards, while wiping his manly
brow, that what caused him so much trouble was the fact that these
rustics paid their fares in such a lot of small money.
He said that he had enough farthings to supply a West End draper with
change for a week, and a sufficient number of threepenny pieces for the
congregations of three parish churches. "That excursion fare," said he,
"is nineteen shillings and ninepence, and I should like to know in just
how many different ways it is possible for such an amount to be paid in
the current coin of this realm."
Here, then, is a puzzle: In how many different ways may nineteen
shillings and ninepence be paid in our current coin? Remember that the
fourpenny-piece is not now current.
Answer:
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) squared ways where n always represents the number of
pence. If threepenny-pieces are admitted, there are
2n cubed + 15n squared + 33n
--------------------- + 1 ways.
18
If sixpences are also used there are
n^{4} + 22n cubed + 159n squared + 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.
