Clay Kitchings :: EMAT 6600 ::
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? No (see the chart0
How would you purchase exactly 53 McNuggets? (Yes, see the chart and explanation)
What is the largest number for which it is impossible to purchase
exactly that number of McNuggets?
I used Excel to make a list of possible combinations of McNuggets
one could purchase. It is
important to note that once we obtain a string of six consecutive integers (of
possible nugget purchases), weÕll know we can discontinue the search. In that
case, we could always simply add one more box of 6 nuggets to obtain the next
six consecutive integer values, and so on.
|
Total Nuggets* |
Possible
combinations of nuggets |
||||||||||
|
6 |
6 |
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9 |
9 |
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12 |
6 |
6 |
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15 |
6 |
9 |
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18 |
9 |
9 |
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20 |
20 |
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21 |
6 |
6 |
9 |
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24 |
9 |
9 |
6 |
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26 |
20 |
6 |
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27 |
9 |
9 |
9 |
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29 |
20 |
9 |
|
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30 |
6 |
6 |
9 |
9 |
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32 |
6 |
6 |
20 |
|
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33 |
9 |
9 |
9 |
6 |
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35 |
6 |
9 |
20 |
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36 |
9 |
9 |
9 |
9 |
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38 |
9 |
9 |
20 |
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39 |
6 |
6 |
9 |
9 |
9 |
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40 |
20 |
20 |
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41 |
20 |
6 |
6 |
9 |
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42 |
9 |
9 |
9 |
6 |
9 |
|
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44 |
20 |
9 |
9 |
6 |
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45 |
6 |
9 |
6 |
9 |
6 |
9 |
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46 |
20 |
20 |
6 |
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47 |
20 |
9 |
9 |
9 |
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48 |
6 |
6 |
6 |
6 |
6 |
6 |
6 |
6 |
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49 |
20 |
20 |
9 |
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|
*This
column is the row-sum of the numbers in the cells to the right.
I color-coded strings of consecutive integer values to help
visualize consecutive strings of integers. Notice that from 44-49, we have a string of six consecutive
integers. Starting at these six
integers, 44, 45, 46, 47, 48, 49, we could add six to each of them and obtain
50, 51, 52, 53, 54, 55. Then, we could add another box of 6 to each of the
previous and obtain the next set of integer values from 56 – 61, and on
and on and on.
According to the findings, 43 is the largest possible amount of
exact nuggets one could purchase with choices of 6, 9, and 20 count boxes.
There is an accompanying Excel file that provides
more evidence that integers were not left out in the table above.
Extension:
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 McNuggets?
We
can purchase exact amounts of McNuggets for all integer values strictly above
37. Please see the accompanying Excel
file.