I have a bag of balls. Each ball has a number on it that can be from 1 to 99. Not all those numbers are used and there are fewer than 99 balls. No number may be used more than once. I have selected the numbers so that if you pick any three balls at random the sum of the numbers on them will be divisible by six.

I can’t have 99 balls in the bag because some of the combinations will not be divisible by 6. For example, 1 + 2 + 4 = 7. So if I include 1 and 2, I cannot include 4. Or, for example, 5 + 23 + 30 = 58. Since 58 is not divisible by 6 I cannot have all three of those numbers in the bag. I have to pick some combination of balls so that the sum of any three will be divisible by 6.

What is the largest number of balls I can have in the bag?

Try your luck, then check our solution.

Thanks to Lawrence Lee for suggesting this puzzle.