You order chicken wings at KFC in the boxes of 6, 9 and 20. What is the largest number of wings that you cannot obtain by buying in any combination of the boxes?

43

You know that you can purchase any number of wings that is divisible by 3 but of course it should not be 3 itself. Thus, you should just try to purchase using the combinations of 9 and 6 boxes till the number is divisible by 3. If that number is not divisible by 3, then you have to use a box with 20. If the number that remains is divisible by 3, you are game. But if it is not divisible by 3, you can use a second box of 20. The number now will necessarily be divisible by 3. Thus the largest number of wings that cannot be bought will come after buying two boxes of 20 and leaving a remainder that is divisible by 3: 3 + 20 + 20 = 43.