AMC 10 · 2012 · #14

Grade 7 countingprobability

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

interval-arithmeticdigit-sum work-backwards ↑ Prerequisites: interval-arithmetic

📏 Long solution 💡 4 insights

Problem

Two players alternate a doubling and an adding move while the number stays under a cap. Add the digits of the smallest winning start.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

How to solve

Strategy Work Backwards

The winning condition is stated at the end of the game ("be the last under 1000"), while the question asks for the start value N — a classic Tool #11 (Work Backwards) setup. Working from the finish, Bernardo wins exactly when he hands Silvia a number so large that her +50 pushes it to 1000 or beyond; that pins his winning move to the window 950–999. Tool #4 (Introduce a Variable) turns the alternating doubling-and-adding into tidy expressions in N, so Bernardo's outputs are just 2N, 4N+100, 8N+300, 16N+700. Tool #14 (Extreme Principle) then hunts for the smallest N hitting that window, which happens on the last of those outputs. Tool #6 (Guess and Check) confirms the winner by replaying the actual game for N=16 and for N=15.

1STEP 1

Translate the win into a number window

Winning is landing in one narrow window.

B ＜ 1000 and B + 50 ≥ 1000 ⟹ 950 ≤ B ≤ 999

2STEP 2

Name the start and track the moves

Every move stays a linear expression.

N → 2N → 2N+50 → 4N+100 → 4N+150 → 8N+300 → 8N+350 → 16N+700

3STEP 3

List Bernardo's four numbers

One player produces exactly four numbers.

Bernardo's outputs: 2N, 4N+100, 8N+300, 16N+700

4STEP 4

Solve for the smallest winning start

The inequality gives a smallest start of 16.

950 ≤ 16N+700 ≤ 999 → 250 ≤ 16N ≤ 299 → 15.625 ≤ N ≤ 18.6875 → N_min=16

5STEP 5

Replay the game to check

Replaying the game confirms it.

N=16: 32,82,164,214,428,478,956; 956+50=1006 ≥ 1000 (B wins) | N=15: 990 (S wins)

6STEP 6

Add the digits

Its digits add to 7, choice (A).

N = 16 → 1 + 6 = 7 → (A)

Answer

The window logic and the algebra agree: N=16 gives Bernardo's final number 956, which lands in 950 ≤ B ≤ 999 so Silvia's +50 dies at 1006, and the direct replay confirms all earlier numbers stay legal. The neighbor N=15 produces 940, one short of the window, and Silvia wins — proof that 16 is the true minimum, not just a value that happens to work. Its digit sum 1+6=7 is exactly choice (A), and 7 is the smallest offered digit sum, matching the fact that a small smallest-N should have a small digit total.

💡Key takeaway

Work backwards from the finish: Bernardo wins when his number lands in 950–999, and turning the doubling-and-adding into one expression 16N+700 shows the smallest start is 16, whose digits add to 7.

Translate the win into a number window
Name the start and track the moves
List Bernardo's four numbers
Solve for the smallest winning start
Replay the game to check
Add the digits