AMC 8 · 2013 · #9

Grade 6 patternalgebra

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

sequences-geometricexponentspattern-recognition pattern-recognitionsystematic-enumeration ↑ Prerequisites: exponentsmulti-digit-arithmetic

📏 Medium solution 💡 2 insights

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Problem

The Hulk's jumps double each time: 1, 2, 4, 8, … meters. On which jump does the distance first exceed 1,000 meters?

Pick an answer.

(A)

$9^\text{th}$

(B)

$10^\text{th}$

(C)

$11^\text{th}$

(D)

$12^\text{th}$

(E)

$13^\text{th}$

AMC 8 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Look for a Pattern

The jumps are a clean doubling sequence, so Tool #5 (Look for a Pattern) is the natural fit: spot that the n-th jump is 2ⁿ⁻¹. Tool #2 (Systematic List) then turns the search into bookkeeping — list the powers of 2 in order until one breaks past 1,000. No algebra or logarithms are needed; just keep doubling and stop at the first value over 1,000.

1STEP 1

Each jump doubles the one before, so the distances form a doubling pattern: 1, 2, 4, 8, 16, …

Jump 1{:} 1, Jump 2{:} 2, Jump 3{:} 4, Jump 4{:} 8

2STEP 2

Rewrite each distance as a power of 2, so the n-th jump is 2ⁿ⁻¹.

Jump n = 2ⁿ⁻¹

3STEP 3

List the powers of 2 in order until one clears 1,000: 2⁸ = 256, 2⁹ = 512, then 2¹⁰ = 1024.

2⁷ = 128, 2⁸ = 256, 2⁹ = 512, 2¹⁰ = 1024

4STEP 4

2⁹ = 512 stays under 1,000 but 2¹⁰ = 1024 clears it, so the first winning exponent is 10.

512 ＜ 1000 ＜ 1024 → exponent = 10

5STEP 5

Since jump n has distance 2ⁿ⁻¹, exponent 10 means n − 1 = 10, so n = 11.

n - 1 = 10 → n = 11 → (C)

Answer

11^th

The 11th jump is 2¹⁰ = 1024 m, which is just barely above 1,000 m, and the previous jump 2⁹ = 512 m is just barely below. That "just barely" feel is exactly what a doubling sequence should produce near a threshold, since each step jumps over a 2× gap. The answer (C) sits in the middle of the choices, also consistent with the way AMC distractors flank a correct answer.

💡Key takeaway

This AMC 8 problem only needs Grade 6 "powers of 2" expressions you already know!

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