Chicken Nuggets

Chicken nuggets come in packets of 6,9, or 20. What is the largest number of nuggets that you cannot buy when combining various packets?
Source: NCTM Mathematics Teacher 2006

SOLUTION
Consider the numbers 36,38, and 40
36=9+9+9+9
38=9+9+20
40=20+20
All even numbers greater than or equal to 42 can be formed by adding 6s to these 3 numbers. For example,
42=36+6
44=38+6
46=40+6
48=36+6+6
50=38+6+6
52=40+6+6
54=36+6+6+6
56=38+6+6+6
58=40+6+6+6
\cdots
Similarly, all odd numbers greater than or equal to 45 can be formed by adding 9s. For example,
45=36+9
47=38+9
49=40+9
51=42+9
53=44+9
55=46+9
57=48+9
59=50+9
\cdots
Some of the numbers that cannot be formed from 6,9, or 20 because they are either not multiple of 3 or 20 subtracted from them are not multiple of 3
10,11,13,14,16,17,19,22,23,25,28,31,34,37, and 43
43 is not a multiple of 3
43-20=23 is not a multiple of 3
23-20=3 is too small
43 is the largest number of nuggets that you cannot buy when combining various packets.

Answer: 43

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About mvtrinh

Retired high school math teacher.
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