## PROBLEM: Chicken McNuggets

McDonalds sells Chicken McNuggets in boxes of 6, 9, or 20. Obviously one could purchase exactly 15 McNuggets by buying a box of 6 and a box of 9.

Could you purchase exactly 17 McNuggets?

How would you purchase exactly 53 McNuggets?

What is the largest number for which it is impossible to purchase exactly that number of McNuggets?

What if the McNuggets were available in boxes of 7, 11, and 17? What is the largest number for which it is impossible to purchase exactly that number of McNuggests?

## 6, 9 or 20 McNuggetts

 # of McNuggetts Possible Combinations # of McNuggetts Possible Combinations 1 None 16 None 2 None 17 None 3 None 18 9,9 or 6,6,6 4 None 19 None 5 None 20 20 6 6 21 6,6,9 7 None 22 None 8 None 23 None 9 9 24 6,9,9 or 6,6,6,6 10 None 25 None 11 None 26 6,20 12 6,6 27 9,9,9 13 None 28 None 14 None 29 9,20 15 6,9 30 6,6,9,9 or 6,6,6,6,6
 # of McNuggetts Possible Combinations # of McNuggetts Possible Combinations 31 None 46 6,20,20 32 6,6,20 47 9,9,9,20 33 6,9,9,9 48 6,6,9,9,9,9 34 None 49 9,20,20 35 6,9,20 50 6,6,9,9,20 36 9,9,9,9 or 6,6,6,6,6,6 51 6,9,9,9,9,9 37 None 52 6,6,20,20 38 9,9,20 53 6,9,9,9,20 39 6,6,9,9,9 40 20,20 41 6,6,9,20 42 6,9,9,9,9 43 None 44 6,9,9,20 45 9,9,9,9,9

The data in the above tables are a list of the number of McNuggetts one would like to purchase and then the combination or combinations of 6, 9 and/or 20 McNuggetts packages one could use to purchase that number. If the number of McNuggetts cannot be purchased by only using a 6, 9, or 20 pack of McNuggetts, then in the possible combinations column the word 'none' appears. What we were trying to determine was what would be the largest quantity of McNuggetts one could not purchase by combining only 6, 9 and/or 20 packs of McNuggetts?

From these tables, one can see that 43 McNuggetts is the largest number of McNuggetts that cannot be purchased using only a 6, 9 or 20 pack of McNuggetts.

## 7, 11 or 17 McNuggetts

 # of McNuggetts Possible Combinations # of McNuggetts Possible Combinations 1 None 16 None 2 None 17 17 3 None 18 7,11 4 None 19 None 5 None 20 None 6 None 21 7,7,7 7 7 22 None 8 None 23 None 9 None 24 7,17 10 None 25 7,7,11 11 11 26 None 12 None 27 None 13 None 28 7,7,7,7 or 11,17 14 7,7 29 None 15 None 30 None
 # of McNuggetts Possible Combinations Similarly to the 6, 9 and 20 packs of McNuggetts discussed above, what if one wanted to determine the largest number of McNuggetts that could not be purchased evenly using a 7, 11 or 17 pack of McNuggetts? The tables once again illustrate the number of McNuggetts and possible combinations of 7, 11 and/or 17 McNuggetts in order to be able to purchase the given number of Mcnuggetts. From these tables, one can see that 37 is the largest number of McNuggetts one cannot purchase by only using 7, 11 and/or 17 packs of McNuggetts. 31 7,7,17 32 7,7,7,11 33 11,11,11 34 17,17 35 7,11,17 36 7,7,11,11 37 None 38 7,7,7,17 39 11,11,17 40 7,11,11,11 41 7,17,17 42 7,7,11,17 43 7,7,7,11,11 44 11,11,11,11 45 11,17,17

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