Answer to March 19, 2001 Problem

The Ping Pong Ball Problem

 
At the Winchester Sports Store, Ping Pong Balls come packaged in boxes of 6, 15, and 20.

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?

 

Solution to Problem:

The answer is 49 Ping Pong Balls.

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 combnations 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.


Correctly solved by:

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