AMC 8 · 2004 · #5

Grade 3 counting

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

logical-deductionpattern-recognitioncomplementary-counting identify-subproblemscomplementary-counting ↑ Prerequisites: multi-digit-arithmetic

📏 Short solution 💡 2 insights

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Problem

Sixteen teams play a single-elimination tournament: every game eliminates the losing team, and the tournament ends when one champion remains. How many games are played in total?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Find an Invariant

Tool #11 (Find an Invariant) is the cleanest path here. Across the whole tournament, the relation "each game eliminates exactly one team" never changes — that one-loser-per-game rule is the invariant. Since every team except the champion must be eliminated, the number of games equals the number of eliminated teams: 16 - 1 = 15. Tool #9 (Solve an Easier Problem) lets us first try a small bracket (like 4 teams) to see why the rule "games = teams - 1" must always hold, before applying it to 16 teams.

1STEP 1

Warm up small: 4 teams need 2 + 1 = 3 games, and 3 = 4 - 1.

4 teams → 2 + 1 = 3 games = 4 - 1

2STEP 2

Every game knocks out exactly one team, so games = eliminated teams.

games played so far = teams eliminated so far

3STEP 3

All 16 teams but the champion lose, so eliminated teams = 16 - 1 = 15.

eliminated teams = 16 - 1 = 15

4STEP 4

By the invariant, total games = eliminated teams = 15 → (D).

total games = eliminated teams = 15 → (D)

Answer

Double-check by adding up the rounds: with 16 teams, Round 1 has 16 ÷ 2 = 8 games, Round 2 has 8 ÷ 2 = 4 games, Round 3 (semifinals) has 4 ÷ 2 = 2 games, Round 4 (final) has 1 game. Total: 8 + 4 + 2 + 1 = 15 games. That matches choice (D). The smaller choices (A) 4, (B) 7, (C) 8 count only some rounds (for example, 8 is just Round 1), so they undershoot. Choice (E) 16 would mean even the champion lost a game, which contradicts "the champion never loses."

💡Key takeaway

Every game in a single-elimination tournament knocks out exactly one team. So the number of games equals the number of teams that have to be eliminated — which is everyone except the champion. For 16 teams, that's 16 - 1 = 15 games, no bracket drawing required.

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