Chicken McNuggets can be purchased in quantities of 6, 9, and 20 pieces. You can buy exactly 15 pieces by purchasing a 6 and a 9, but you can't buy exactly 10 McNuggets. What is the largest number of McNuggets that can NOT be purchased?

Hint 1

We need a systematic way to figure out which quantities we can purchase, so let's start out by writing down the numbers 1, 2, 3, .... We'll hope the number is not too big, maybe not bigger than 50, so write down a table

 1  2  3  4  5  6  7  8  9  10
11 12 13 ...

down through 50. Then underline each number that can be purchased. You can use this method to keep from getting lost: First underline 6, 9, and 20, because you know those can be purchased. Then move up the table to each underlined item (keeping one finger on it so you don't lose your place), and separately add 6, 9, and 20 to that item and underline the answer. So you would start by putting your finger on 6, then underlining 6 + 6 = 12, 6 + 9 = 15, and 6 + 20 = 26. Then move your finger up to the next underlined item (that's 9) and again add 6, 9, and 15. Keep going until your finger runs off the end of the table.

Hint 2

Here's what the table should look like:

 1  2  3  4  5  6  7  8  9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50

From looking at the table we would guess that any number past 43 can be purchased, but how can we be sure? For example, can we tell from this table that 51 can be purchased?

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