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Chicken Nuggets Are a Problem

There's a famous fast-food restaurant you can go to, where you can order chicken nuggets. They come in boxes of various sizes. you can only buy them in a box of 6, a box of 9, or a box of 20. So if you're really hungry you can buy 20, if you're moderately hungry you can buy 9, and if there's more than one of you, maybe you buy 20 and divide them up.

Using these order sizes, you can order, for example, 32 pieces of chicken if you wanted. You's order a box of 20 and two boxes of 6.

Here's the question: What is the largest number of chicken pieces that you can not order? For example, if you wanted 37 of them, could you get 37? No. Is there a larger number of chicken nuggets that you cannot get? And if there is, what number is it?


Source: Puzzler on NPR's Car Talk, May 16, 2005

Hint: Build a table...You can buy 6, but not 7 or 8. You can buy 9, but not 10 or 11, etc...Look for patterns!


Solution Commentary: For commentary, I suggest you read Tom and Ray's discussion of a solution.

Side Note: Can you generalize this problem to any three natutal numbers x, y, and z?

And, if you have access to MAA FOCUS (November 2005, p. 22-23), please read Dan Kalman's "outside-the-box" solution, which is creative but quite suspect.