The Chicken Nugget Conundrum

Sep 08, 2017

This puzzler was sent in by a fellow named Louis Gee. Here it is:

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 you divide them up.

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

Here's the question:

What is the largest number of chicken pieces that you cannot order? For example, if you wanted, say 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?


You can clearly buy six, you can clearly buy nine, you can obviously buy 12, 15 we've established, 18, 20, 21 you can keep going. Now, if you can buy 15, of course you can buy 30, 45, and you can buy 90.  And if you can buy 18 you can buy 36 and 72. And if you can buy 20, you can buy all the multiples of 100: 1,000, 10,000, 100,000, a million, etcetera.

I kept working upwards, and there were some holes. I couldn't buy 31 for example. I couldn't buy 37, I couldn't buy 43, but then a strange thing happened.

I found out that I could buy 44, 45, 46, 47.

Now if I could buy 46, I could buy 92. Once I got to 43, I realized that was the largest number that I couldn't buy.

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