What is the largest number of ping pong balls that you can not purchase?
By getting two boxes of 6, you have 12 ping pong balls. But you can not get 13 ping pong balls since no combination of 6, 15, and 20 adds up to 13.
So, in other words, what is the greatest number of
ping pong balls that can not be made from 6, 15, and 20?
Beginning with 50, every number following can be made up of combinations of 6, 15, and 20.
| # of ping pong balls | # of packs of 6 | # of packs of 15 | # of packs of 20 |
| 50 = | 0 | 2 | 1 |
| 51 = | 1 | 3 | 0 |
| 52 = | 2 | 0 | 2 |
| 53 = | 3 | 1 | 1 |
| 54 = | 9 | 0 | 0 |
| 55 = | 0 | 1 | 2 |
You can get all numbers greater than 55 by adding 6 or multiples of 6 to the numbers above. So this shows that all numbers above 49 can be made from combinations of 6, 15, and 20. All that we had to show was the first instance where six numbers in a row could be made (since 6 was the smallest box).
I wrote a computer program in C++ to solve this problem. The following output shows the number of ping pong balls which can be made from packages of 6, 15, and 20:
6, 12, 15, 18, 20, 21, 24, 26, 27, 30, 32, 33, 35, 36,
38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 50, and all numbers
after it.
| 1. Walt Arrison | Philadelphia, Pennsylvania |
| 2. David Powell | Winchester, Virginia |
| 3. Keith Mealy | Cincinnati, Ohio |
| 4. Chip Schweikarth | Winchester, Virginia |
| 5. George Gaither | Winchester, Virginia |
| 6. Bob Hearn | Winchester, Virginia |
| 7. Richard Johnson | La Jolla, California |
| 8. Kirstine Wynn | Winchester, Virginia |
| 9. Erin McGinnis | Winchester, Virginia |
| 10. Bill Hall | Wellington, Florida |