THE HYMN-BOARD POSER.
(
Unclassified Problems.)
The worthy vicar of Chumpley St. Winifred is in great distress. A little
church difficulty has arisen that all the combined intelligence of the
parish seems unable to surmount. What this difficulty is I will state
hereafter, but it may add to the interest of the problem if I first give
a short account of the curious position that has been brought about. It
all has to do with the church hymn-boards, the plates of which have
become so damaged that they have ceased to fulfil the purpose for which
they were devised. A generous parishioner has promised to pay for a new
set of plates at a certain rate of cost; but strange as it may seem, no
agreement can be come to as to what that cost should be. The proposed
maker of the plates has named a price which the donor declares to be
absurd. The good vicar thinks they are both wrong, so he asks the
schoolmaster to work out the little sum. But this individual declares
that he can find no rule bearing on the subject in any of his arithmetic
books. An application having been made to the local medical
practitioner, as a man of more than average intellect at Chumpley, he
has assured the vicar that his practice is so heavy that he has not had
time even to look at it, though his assistant whispers that the doctor
has been sitting up unusually late for several nights past. Widow Wilson
has a smart son, who is reputed to have once won a prize for
puzzle-solving. He asserts that as he cannot find any solution to the
problem it must have something to do with the squaring of the circle,
the duplication of the cube, or the trisection of an angle; at any rate,
he has never before seen a puzzle on the principle, and he gives it up.
This was the state of affairs when the assistant curate (who, I should
say, had frankly confessed from the first that a profound study of
theology had knocked out of his head all the knowledge of mathematics he
ever possessed) kindly sent me the puzzle.
A church has three hymn-boards, each to indicate the numbers of five
different hymns to be sung at a service. All the boards are in use at
the same service. The hymn-book contains 700 hymns. A new set of numbers
is required, and a kind parishioner offers to present a set painted on
metal plates, but stipulates that only the smallest number of plates
necessary shall be purchased. The cost of each plate is to be 6d., and
for the painting of each plate the charges are to be: For one plate,
1s.; for two plates alike, 113/4d. each; for three plates alike,
111/2d. each, and so on, the charge being one farthing less per plate
for each similarly painted plate. Now, what should be the lowest cost?
Readers will note that they are required to use every legitimate and
practical method of economy. The illustration will make clear the nature
of the three hymn-boards and plates. The five hymns are here indicated
by means of twelve plates. These plates slide in separately at the back,
and in the illustration there is room, of course, for three more plates.
Answer:
This puzzle is not nearly so easy as it looks at first sight. It was
required to find the smallest possible number of plates that would be
necessary to form a set for three hymn-boards, each of which would show
the five hymns sung at any particular service, and then to discover the
lowest possible cost for the same. The hymn-book contains 700 hymns, and
therefore no higher number than 700 could possibly be needed.
Now, as we are required to use every legitimate and practical method of
economy, it should at once occur to us that the plates must be painted
on both sides; indeed, this is such a common practice in cases of this
kind that it would readily occur to most solvers. We should also
remember that some of the figures may possibly be reversed to form other
figures; but as we were given a sketch of the actual shapes of these
figures when painted on the plates, it would be seen that though the 6's
may be turned upside down to make 9's, none of the other figures can be
so treated.
It will be found that in the case of the figures 1, 2, 3, 4, and 5,
thirty-three of each will be required in order to provide for every
possible emergency; in the case of 7, 8, and 0, we can only need thirty
of each; while in the case of the figure 6 (which may be reversed for
the figure 9) it is necessary to provide exactly forty-two.
It is therefore clear that the total number of figures necessary is 297;
but as the figures are painted on both sides of the plates, only 149
such plates are required. At first it would appear as if one of the
plates need only have a number on one side, the other side being left
blank. But here we come to a rather subtle point in the problem.
Readers may have remarked that in real life it is sometimes cheaper when
making a purchase to buy more articles than we require, on the principle
of a reduction on taking a quantity: we get more articles and we pay
less. Thus, if we want to buy ten apples, and the price asked is a
penny each if bought singly, or ninepence a dozen, we should both save a
penny and get two apples more than we wanted by buying the full twelve.
In the same way, since there is a regular scale of reduction for plates
painted alike, we actually save by having two figures painted on that
odd plate. Supposing, for example, that we have thirty plates painted
alike with 5 on one side and 6 on the other. The rate would be 43/4d., and
the cost 11s. 101/2d. But if the odd plate with, say, only a 5 on one side
of it have a 6 painted on the other side, we get thirty-one plates at
the reduced rate of 41/2d., thus saving a farthing on each of the previous
thirty, and reducing the cost of the last one from 1s. to 41/2d.
But even after these points are all seen there comes in a new
difficulty: for although it will be found that all the 8's may be on the
backs of the 7's, we cannot have all the 2's on the backs of the 1's,
nor all the 4 on the backs of the 3's, etc. There is a great danger, in
our attempts to get as many as possible painted alike, of our so
adjusting the figures that some particular combination of hymns cannot
be represented.
Here is the solution of the difficulty that was sent to the vicar of
Chumpley St. Winifred. Where the sign X is placed between two figures,
it implies that one of these figures is on one side of the plate and the
other on the other side.
d. L s. d.
31 plates painted 5 X 9 @ 41/2 = 0 11 71/2
30 " 7 X 8 @ 43/4 = 0 11 101/2
21 " 1 X 2 @ 7 = 0 12 3
21 " 3 X 0 @ 7 = 0 12 3
12 " 1 X 3 @ 91/4 = 0 9 3
12 " 2 X 4 @ 91/4 = 0 9 3
12 " 9 X 4 @ 91/4 = 0 9 3
8 " 4 X 0 @ 101/4 = 0 6 10
1 " 5 X 4 @ 12 = 0 1 0
1 " 5 X 0 @ 12 = 0 1 0
149 plates @ 6d. each = 3 14 6
----------
L7 19 1
Of course, if we could increase the number of plates, we might get the
painting done for nothing, but such a contingency is prevented by the
condition that the fewest possible plates must be provided.
This puzzle appeared in _Tit-Bits_, and the following remarks, made by
me in the issue for 11th December 1897, may be of interest.
The "Hymn-Board Poser" seems to have created extraordinary interest. The
immense number of attempts at its solution sent to me from all parts of
the United Kingdom and from several Continental countries show a very
kind disposition amongst our readers to help the worthy vicar of
Chumpley St. Winifred over his parochial difficulty. Every conceivable
estimate, from a few shillings up to as high a sum as L1,347, 10s.,
seems to have come to hand. But the astonishing part of it is that,
after going carefully through the tremendous pile of correspondence, I
find that only one competitor has succeeded in maintaining the
reputation of the _Tit-Bits_ solvers for their capacity to solve
anything, and his solution is substantially the same as the one given
above, the cost being identical. Some of his figures are differently
combined, but his grouping of the plates, as shown in the first column,
is exactly the same. Though a large majority of competitors clearly hit
upon all the essential points of the puzzle, they completely collapsed
in the actual arrangement of the figures. According to their methods,
some possible selection of hymns, such as 111, 112, 121, 122,211, cannot
be set up. A few correspondents suggested that it might be possible so
to paint the 7's that upside down they would appear as 2's or 4's; but
this would, of course, be barred out by the fact that a representation
of the actual figures to be used was given.
