Mike Callinan's Write Up on the McNugget problem

Goals/Objectives:  We want to solve the following problem:

McDonald's 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. Using only combinations of boxes of 6, 9, and/or 20 McNuggets,
a. Could you purchase exactly 17 McNuggets?

b. How would you purchase exactly 53 McNuggets?

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

d. Let's say you could only buy the McNuggets in boxes of 7, 11, and 17.
What is the largest number for which it is impossible to purchase exactly
that number of McNuggets?

Conjecture/Hypothesis/Prediction
Peggy Reigle and I decided to calculate all combinations of 6, 9, and 20 up to the LCM of 180 and see answer the questions posed above.

Procedures/Methods:
1.  Peggy Reigle and I created an Excel spreadsheet.
2.  We set up the first worksheet to show combinations of 6 and 9.
3.  We then created another worksheet to add the third combination as if there were 0 boxes of 20.
4.  We created a third, fourth, and fifth worksheet with 1 box of 20, 2 boxes of 20 and 3 boxes of 20.
5.  Finally, we created a worksheet with all possible numbers, leaving a blank when we could not get a combination of the 3 variables.

Conclusion (analysis using technology and understanding why it works)
a. Could you purchase exactly 17 McNuggets?  no

b. How would you purchase exactly 53 McNuggets?  yes (1-20, 1-9, 4-6 or 1-20, 3-9, 1-6)

c. What is the largest number for which it is impossible to purchase
exactly that number of McNuggets?  43 (by process of elimination up to 180)

d. Let's say you could only buy the McNuggets in boxes of 7, 11, and 17.
What is the largest number for which it is impossible to purchase exactly
that number of McNuggets? to be continued
Comments and reactions, what did I learn
Working with 3-dimensional combinations is difficult with a 2-D spreadsheet.

Extensions (what other questions can I ask)
To be continued

Here's my link to the McNuggets Excel worksheet