The Hymn Board Puzzle

Thi is a two part puzzle that may need a notepad and calculator to work out.

A church has three hymn boards like the above and all three are used at services where hymns are sung by the congregation. The churchwarden has noticed that the metal plates with the numbers painted on are in need of replacing.

A generous church member has volunteered to pay for the new plates but she has said that the smallest number of plates must be bought at the minimum cost. You need to know that there are 700 hymns in the hymn book, and that plates are less expensive if they have a number painted on each side. Also the boards hold up to 15 plates.

The first question is how many is the smallest number of plates needed?

The second question is how much will it cost?

Each plate costs 40p to buy. Painting the numbers depends on how many identical plates are ordered. One plate with one number only costs 31p. One plate with two numbers costs 62p, two plates with two numbers each the same cost 61p each, three plates 60p each and so on (as an example, if you order 25 plates with a number on both sides, you would only pay 40p for each plate) So how much is the total cost of supplying the finished plates?


The answers are 149 plates and the cost is £120.79. You would probably like to know how to get the answers.

Well, the most of each number needed is as follows: numbers 0, 7, 8 and 9 — 30, numbers 1,2,3,4 and 5 — 33 and number 6 — 42 (using 6 upside down for 9 so you don’t need the 9’s). This means a total of 3x30 + 5x33 +42 = 297 numbers are needed, with two numbers on 148 plates, and one number on one plate, that makes 149 plates. This picture shows you how 33 number 1’s, 42 number 6’s and 30 number 7’s might be needed:

However there is a twist. If we take all the ones and put twos on the back, we use a total of 24 ones on the boards below. We have another 9 more twos (with ones on the back) but we need another 12 twos which do not have one on the back. And this is the same for all number combinations except 0 (obviously) cannot be in the first column and 8 also as there are only 700 hymns.

This means juggling the number combinations. This is one arrangement that works:

numbers   £
5 & 6 31 plates these cost 32p each =   9.92
7 & 8 30 plates these cost 33p each =   9.90
1 & 2 21 plates these cost 42p each =   8.82
0 & 3 21 plates these cost 42p each =   8.82
1 & 3 12 plates these cost 51p each =   6.12
2 & 4 12 plates these cost 51p each =   6.12
4 & 6 11 plates these cost 51p each =   5.61
0 & 4 9 plates these cost 55p each =   4.95
4 & 5 1 plate this costs 62p =   0.62
5 only 1 plate this costs 31p =   0.31

You will have spotted that this adds up to 149 plates. The plates will cost 149x40p = £59.60. If my arithmetic is correct, the painting cost adds up to £61.19, so the total cost for the church member to pay is £120.79.

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