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?

# 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.

# 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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©1998 by Luke Rapley