McNuggets Module
Developed by Zeb Engberg with Jeanine King for the GK-12 project.
May 16, 2014
Overview and interesting questions: McNuggets are a chicken-related food sold by McDonalds. In the final decade of the last century, McNuggets could be purchased in boxes of 6, 9, and
20. As a customer, what possible amounts of McNuggets are you able to order from McDonalds?
For example, it is possible to order 15 McNuggets because it can be obtained from a box of 6 and
a box of 9. On the other hand, it is impossible to order 16 McNuggets.
Given a positive number n, say that n is a McNugget number if it’s possible to order n McNuggets. Said differently, a McNugget number is an integer that can be obtained by adding together
a combination of McNuggets coming from boxes of 6, 9, and 20 McNuggets. Conversely, a nonMcNugget number is an integer which cannot be obtained from any combination of 6, 9, or 20
McNuggets. Here are a few interesting questions:
• What are the McNugget numbers?
• Is there a largest non-McNugget number?
• What is it, and how could you mathematically prove that this is correct?
In the mathematical community, this sort of problem is generally called the Frobenius problem
(see the Mathworld article or Wikipedia entry for more information), and it can be phrased in terms
of postage stamp values, coins, points-values in a game of football, etc. There are many interesting
unsolved problems connected to the Frobenius problem. In particular, how does one calculate the
largest non-McNugget number?
In this lesson, we will explore such questions. Before considering the McNugget problem, we
will start with a question about the possible scores in a game of football.
Learning goals: Through this lesson, students will:
• Calculate the largest non-McNugget number (and the analogous number in other problems).
• Mathematically argue for why their proposed number is in fact the largest non-McNugget
number.
• Create ways to extend the McNugget problem to a more general context.
• Have fun thinking about McNuggets and football.
Background: This lesson assumes no background on the students part except for a willingness
to be creative and a motivation to try.
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Materials: There is a worksheet included at the end of this guide. Manipulatives are also helpful to students, particularly if students have had limited experience with mathematical abstraction.
Manipulatives should be simple: a collection of blocks could be used to represent McNuggets and
three distinct containers could be used to represent boxes of 6, 9, and 20 McNuggets.
Lesson plan:
Football
Start by giving students the following problem. When mathematicians play football, they are
only able to score points through kicking a field goal or scoring a touchdown (always with the extra
point). A field goal is worth 3 points and a touchdown is worth 7 points. What are the possible
scores you might see at the end of a game? What are some examples?
Could the score from a game read 10–6? Sure. A field goal and a touchdown give 10 points,
and two field goals give 6 points.
Could the score read 9–2? Well 9 is possible, since 9 = 3 + 3 + 3. But 2 is not.
Could the score read 8–7? How would you get 8 points?
Now give students a few minutes to ponder the following question: What is the largest number
that you cannot score in this game of football?
Eventually you want them to be able to mathematically prove that the answer is 11. Here is
one way of prove this. Note that
12 = 3 + 3 + 3 + 3
13 = 7 + 3 + 3
14 = 7 + 7.
We can get every number larger than 14 by adding an appropriate multiple of 3 to one of the above
numbers. (This may nicely lead into a discussion of the remainder when a number is divided by
3. This remainder can be easily computed by simply adding together the digits of the number
in question.) On the other hand, you cannot get 11 (which you can see by doing two cases: one
with and one without a touchdown). So 11 is the largest number you cannot score in this game of
football.
You can also ask the following question: Of the remaining numbers between 1 and 10, which
are possible scores and which are not?
Generalizing
Ask students if they can come up with a way to generalize or abstract this problem. This
may lead into a discussion of what it means to abstract a mathematical problem. This would
be an excellent discussion to have. Encourage students to “think like a mathematician”. Is a
mathematician satisfied with playing mathematical football? No!. What other games could they
play?
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If students get stuck on the idea of abstracting, ask them “what did we start with?” (we started
with 3 and 7). The numbers 3 and 7 are two specific numbers. How can you generalize this? What
if you started with other numbers?
Here is the most general version (unlikely that students will get here).
Given positive numbers a1 , a2 , . . . , an , determine which numbers can be written in the form
k1 a1 + k2 a2 + . . . kn an ,
where the ki are nonnegative integers. Is there are largest number not of this form?
It should be noted that if the greatest common divisor of the ai ’s is not 1, then there will be
infinitely many numbers not of the above form. For example, if we take a1 = 4 and a2 = 6, then
any combination of a1 and a2 will be even, so there is no largest number not of this form, since no
odd number is of this form. Similarly, if gcd(a1 , . . . , ak ) = d, then any combination of the ai ’s is a
multiple of d. If d 6= 1, there are infinitely many numbers not divisible by d.
Creating their own example
Ask students to partner up. Each group should pick two distinct single digit numbers (perhaps
they should be coprime [two numbers a and b are coprime if gcd(a, b) = 1] depending on what
students got out of the discussion on the greatest common divisor). Call these numbers A and B.
To avoid a trivial case, mandate that A, B > 1. The case of A = 2 and B = 3 is easy as well and
should be avoided. Don’t let them pick A = 3 and B = 7, since we already looked at this earlier.
What points are you able to score in a game in which you can earn either A or B points? Is
there a largest number of points that you can’t score? Find it.
Give students five or ten minutes to work on this task. After they’ve finished, they could share
their findings. You could even draw a chart on the board to record the results. Something like the
following chart would be nice.
A, B
3, 7
2, 3
3, 5
4, 6
largest number you cannot score
11
1
7
no odd number can be scored
McNuggets
Now we get to what we’ve been waiting for. Explain to students the set up of the McNugget
problem, which goes like this: McNuggets can be purchased in boxes of 6, 9, and 20. A McNugget
number is the total number of McNuggets coming from a combination of boxes. For example, 15
is a McNugget number because it can be obtained from a box of 6 and a box of 9. On the other
hand, 16 is a non-McNugget number.
Ask students to find which of the numbers up to 50 are McNugget numbers. To do this, give
them the attached worksheet (the final page of this pdf), which they can work on individually
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or in small groups. It may be nice to give the students manipulatives, but this isn’t absolutely
necessary. After they finish the worksheet, you should discuss their findings. What is the largest
non-McNugget number up to 50? [answer: 43.] What about the numbers bigger than 50? Can you
prove that 43 is indeed the largest non-McNugget nubmer? [answer: Yes. Since the numbers 44
through 49 are all obtainable, by adding an appropriate multiple of 6, one would be able to obtain
all numbers larger than 49.]
If time permits, here are a few other follow up questions:
• Happy meals contain a box of 4 McNuggets. With this new size box, what is the largest
non-McNugget number? [answer: 11.] Prove it.
• McDonalds has decided to replace happy meals with healthy meals. These contain a box of 3
McNuggets. How does this change things? [answer: In this case, the largest non-McNugget
number would decrease from 43, since 43 = 20 + 20 + 3.]
• If you were designing a new box of McNuggets, what new box of McNuggets would you
introduce into their menu? Why?
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Name:
Figure out which of the first 50 numbers are McNugget numbers. If a number is a McNugget
number, write down how to obtain it from boxes of McNuggets.
1 = not possible
3 = not possible
5=
7=
9=
11 =
13 =
15 = 6 + 9
17 =
19 =
21 = 6 + 6 + 9
23 =
25 =
27 =
29 =
31 =
33 =
35 =
37 =
39 =
41 =
43 =
45 =
47 =
49 =
2 = not possible
4=
6=
8=
10 =
12 =
14 =
16 = not possible
18 =
20 =
22 =
24 =
26 = 20 + 6
28 =
30 =
32 =
34 =
36 =
38 =
40 =
42 =
44 =
46 =
48 =
50 =