Math Stinger # 137

Math Stinger # 137 solution
At Dan's Donuts, doughnuts are sold by the box. Dan is obsessive-compulsive,
and he sell boxes with exactly 6, 9, or 20 doughnuts, no more and no less.
This means there are a few orders that Dan can't fill, for example an order of
5 doughnuts, or 8 or 10. Find the largest order of doughnuts that can not be filled.

The answer is 43.

Notice that combinations of 6 and 9 produce multiples of 3 starting at 6.
Thus we have 6, 9, 12, 15..... 30, 33, 36, 39, 42, 45, 48....

With 20 in the mix we have additional multiples of 3 starting at 26
Thus we have 20, 26, 29, 32, 35, 38, 41, 44, 47, 50....

With 40 in the mix we have additional multiples of 3 starting at 46.
Thus we have 40, 46, 49.... This fills the last gap.

This leaves 43 as the largest number of donuts that cannot be filled.

Correct solutions were sent by Travis Prather, Leslie Epperson, Leon Eskenazi, and John Bradshaw.