AMC 10 · 2016 · #14

Grade 6 counting

Archetype Small-Domain Constrained Search └ Sub-archetype Sum-Target Search

linear-diophantineparity systematic-enumeration ↑ Prerequisites: linear-diophantine

📏 Medium solution 💡 2 insights

Problem

Count how many ways 2016 can be written as a sum of 2s and 3s when order does not matter. Each way is a count of twos and a count of threes whose total is 2016.

Pick an answer.

(A)

236

(B)

336

(C)

337

(D)

403

(E)

672

AMC 10 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Introduce a Variable

A "way" is just a pair of counts, so tool #4 (Introduce a Variable) names them: a twos and b threes with 2a+3b=2016. Solving that for a shows a is a whole number only when 2016-3b is even, which pins down which values of b are allowed. Then tool #2 (Make a Systematic List) sweeps b in order through its allowed values and counts them; each allowed b gives exactly one way, so counting the b values answers the question.

1STEP 1

Name the two counts

Let a be the number of twos and b the number of threes, so the total condition becomes 2a + 3b = 2016.

2a + 3b = 2016, a ≥ 0, b ≥ 0 whole numbers

2STEP 2

Solve for the number of twos

Choose b first; the leftover splits into twos, giving a = $\frac{2016 - 3b}{2}$ , which must be a whole number ≥ 0.

2a = 2016 - 3b → a = (2016 - 3b)/2

3STEP 3

Find which b are allowed

a is a whole number only when 2016 - 3b is even; since 2016 is even, that forces b to be even.

2016 - 3b even ⇔ 3b even ⇔ b even

4STEP 4

List the even b and count

Even b run 0, 2, 4, …, 672, and there are $\frac{672}{2}$ + 1 = 337 of them, one way each — choice (C).

b ∈ {0, 2, 4, …, 672} → 672/2 + 1 = 337 → (C)

Answer

337

Check the endpoints: b=0 gives a=1008 (all twos), and b=672 gives 3·672=2016 so a=0 (all threes); both are listed in the problem's examples, so the smallest and largest allowed b are right. The count 337 is just over half of 672, which makes sense because exactly the even b work, and it matches one of the choices.

💡Key takeaway

Let b be the number of threes; only even b from 0 to 672 work, and there are 337 of them.

Name the two counts
Solve for the number of twos
Find which b are allowed
List the even b and count

Name the two counts

Solve for the number of twos

Find which b are allowed

List the even b and count

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