AMC 8 · 2006 · #22

Grade 6 arithmetic

Archetype Small-Domain Constrained Search └ Sub-archetype Factor-Pair Enumeration

pattern-recognitionoptimization-countingdigit-constraintsconvert-to-algebra pattern-recognitionconvert-to-algebracasework ↑ Prerequisites: multi-digit-arithmeticpattern-recognition

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

Three different one-digit positive integers A, B, C sit in the bottom row of a small pyramid. Each cell above is the sum of the two cells it sits on top of, all the way to a single top cell. Over all valid placements, find the difference between the largest and smallest possible top numbers.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Solve an Easier Related Problem

The pyramid rule turns three bottom numbers into one top number, but the relationship is not obvious by inspection. Tool #9 (Easier Related Problem) says: plug in a concrete small triple, walk the pyramid up, and see what the top cell really is in terms of A, B, C. That single experiment reveals that the middle cell is counted twice, so the top equals A + 2B + C. Once that formula is in hand, Tool #6 (Guess and Check) picks the bottom cells smartly: to maximize, put the largest digit 9 in the middle (it counts twice) and the next two largest 8, 7 on the outside; to minimize, mirror the idea with 1, 2, 3.

1STEP 1

Try one concrete bottom row: A = 1, B = 2, C = 3 fills upward to a top of 8.

Bottom: 1, 2, 3 → Second row: 1+2 = 3, 2+3 = 5 → Top: 3+5 = 8

2STEP 2

Reading the top as a bottom-cell sum gives Top = A + 2B + C — the middle B is counted twice.

Top = (A+B) + (B+C) = A + 2B + C

3STEP 3

To maximize, the doubled middle takes the biggest digit: B = 9 with 8, 7 outside gives top 33.

Max top = 8 + 2 · 9 + 7 = 8 + 18 + 7 = 33

4STEP 4

To minimize, the middle takes the smallest digit: B = 1 with 2, 3 outside gives top 7.

Min top = 2 + 2 · 1 + 3 = 2 + 2 + 3 = 7

5STEP 5

Subtracting the two extremes gives the requested difference: 33 - 7 = 26.

33 - 7 = 26 → (D)

Answer

Sanity check both extremes against the choices. The largest possible top is at most 9 + 2 · 9 + 9 = 36 if repeats were allowed, so 33 is in range. The smallest is at least 1 + 2 · 1 + 1 = 4 if repeats were allowed, so 7 is in range. Their difference 26 matches answer (D), and 26 is the only choice that fits both bounds. Spot-checking another swap also confirms the formula: bottom 2, 9, 8 gives second row 11, 17, top 28, matching 2 + 2 · 9 + 8 = 28.

💡Key takeaway

Build one small pyramid by hand and the rule jumps out: the top cell is A + 2B + C, so the middle digit counts twice. Put 9 in the middle for the max and 1 in the middle for the min, and the difference is 33 - 7 = 26.

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