The Hymn Board Puzzle
This 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?
You need to know that each metal plate costs 40p to buy.
The costy of painting the numbers on depends on how many identical plates you order. Painting one plate with a number on one side costs 31p. Painting a plate with two numbers, one on each side, (the same or different numbers) costs 62p, two plates the same cost less - 61p each, three plates the same would be 60p each and so on. So, for example, if you were to order 25 plates with the same two numbers painted on, you would only pay 40p for the painting of 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) would not be in the first column and 8 and 9 also not needed in the first column 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.
