Edmund's Brainteasers
Problem 9
Solution
Problem 9: The Hymn Boards. Solution.
We'll first think about one of the hymn boards. No number has 0 in the first place so at most 10 'digit cards' with a 0 are required. But 100, 200, 300, 400, 500 may be the chosen hymns so we need 10 digit cards with a 0. Now consider 1. Only 1 hymn needs 3 number 1s, the hymn 111, but each of the other 4 may need 2 number 1s, for example 11, 211, 311, 411. So we need 11 digit cards with a 1. Similarly we need 11 with a 2, 11 with a 3, 11 with a 4. However, 5 is different. There is no hymn which needs 3 number 5s. Then at most 10 'digit cards' with a 5 are needed, but we may sing hymns 55, 155, 255, 355, 455 so exactly 10 are needed. Similarly we need 10 each of the 7 and 8 digit cards. We also need 10 digit cards with a 6 since it can't be in the first place either as a 6 or as a 9. Therefore 94 digit boards are needed for each hymn board, giving a total of 188 digit cards for the two boards.