Edmund's Brainteasers
Problem 10
Solution
Problem 10: The Hymn Boards Again. Solution.
Again we'll think first about just one hymn board. The argument given before for the digit cards 0, 1, 2, 3, 4 is unchanged. So we need 10, 11, 11, 11, 11 digit cards for each, respectively. Now 5 may occur in the first place so we also need 11 digit cards with a 5. Although 7 can occur in the first place of hymn 700, since that uses up only 1 number 7, it doesn't occur in a list of hymns requiring the largest number of 7s. We therefore need 10 digit cards 7, and similarly 10 digit cards 8. The digit card 6 is trickier! There are 4 three digit numbers in the new hymnbook made up of 6 or 9 (which is 6 up-side-down). These are 666, 669, 696, and 699. These numbers, together with one number requiring 2 6s means we need 14 digit cards with a 6. Hence we require 5 extra digit cards per board, so 10 extra are required overall.