Going through a desk that belonged to my Grandad, I found an old bill but unfortunately the first and last digits of the total had faded and were no longer legible. All that can still be read is
72 turkeys £_67.9_
where the underscores represent the unreadable digits.
What was the price of each turkey?
I found this nice puzzle on Cut-The-Knot, that treasure store of problems. It was original a question in a Stanford examination. You can find my solution further down the page.
Solution
We can ignore the decimal point for the moment and just try to deduce the missing digits, let's call them X and Y, from the fact that X679Y must be divisible by 72, the number of turkeys. 72 is even, so Y must be an even digit; 0, 2, 4, 6, or 8.
Moreover 72 is a multiple of 9, and a number is only divisible by 9 if the sum of its digits is divisible by 9. The digit sum of the known digits is 24 and for each possible value of Y we can then work out what the corresponding value of X must be for the digit sum to meet this condition.
The possible combinations are
| Y | X | Resulting number |
| 0 | 5 | 56790 |
| 2 | 3 | 36792 |
| 4 | 1 | 16794 |
| 6 | 8 | 86796 |
| 8 | 6 | 66798 |
We also know that 72 is divisible by 8 and so the number we want must be divisible by 8 also. So now we test the candidate numbers in turn and only 36792 passes the test.
Finally 36792/72 = 511, so the price per turkey back then must have been £5.11