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?
| 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 |
| 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.
| 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 |
| 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