AMC 8 · 2018 · #19

Grade 4 countingpattern

Archetype Branching State Enumeration └ Sub-archetype Rule-Governed Process Simulation

paritypattern-recognitionsystematic-enumerationcombinations-basic pattern-recognitioncaseworktree-enumeration ↑ Prerequisites: paritycombinations-basic

📏 Medium solution 💡 3 insights 📊 Diagram

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Problem

A sign pyramid has 4 rows. The bottom row has 4 cells; each cell holds either a + or a -. The rule for filling every cell above the bottom is: write + if the two cells directly below it match, write - if they differ. Among all ways to fill the 4 bottom cells, how many produce a + at the very top?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2018 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Make a Systematic List

There are only 2⁴ = 16 possible bottom rows, which is small enough to list completely (Tool #2). To stay organized and to spot why the answer must be exactly half of 16, we first solve the same puzzle for a tiny pyramid with a 2-cell bottom row (Tool #9 Easier Problem) and a 3-cell bottom row, then look for the pattern (Tool #5). The pattern says: in a sign pyramid, flipping the leftmost bottom cell always flips the top sign, so exactly half of all bottom fillings give a +. That same reasoning, verified by listing, gives the answer for 4 cells without any algebra.

1STEP 1

Read the rule off the diagram: a cell is + exactly when its two cells below match, and - when they differ.

cell above = + if children match, - if children differ

2STEP 2

Warm up on a 2-cell bottom: of its 2² = 4 rows, the top is + in exactly 2 of them — half.

top=+ count for n=2 : 2 out of 4 =

\frac{1}{2}

3STEP 3

Now the 3-cell bottom: among its 2³ = 8 rows, the top is + in exactly 4 of them — again half.

top=+ count for n=3 : 4 out of 8 =

\frac{1}{2}

4STEP 4

See why: flipping the leftmost bottom cell flips every cell above it up to the top, so rows pair off half-and-half.

#{top=+} = #{top=-} =

\frac{1}{2}

· 2ⁿ = 2^ n-1

5STEP 5

Apply it to 4 cells: half of 2⁴ = 16 rows win, so the count is 16 ÷ 2 = 8 — choice (C).

#{top=+} =

\frac{1}{2}

· 2⁴ =

\frac{1}{2}

· 16 = 8 → (C)

Answer

The total number of bottom fillings is 2⁴ = 16, so any answer above 16 would be impossible — choice (E) is the maximum reasonable bound. Both very small (2) and very large (16) answers would be suspicious because the pyramid rule is symmetric in + and - (swapping every sign in the bottom flips every cell, and the top of an even-height pyramid behaves predictably), suggesting the count should be exactly half. Half of 16 is 8, which is choice (C). The explicit list of 8 bottom rows in the last step confirms the count and matches the answer.

💡Key takeaway

This AMC 8 problem only needs Grade 4 "follow a rule and look for the pattern" thinking you already know!

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