two part analysis question

[D datonman]

A city is hosting a table tennis tournament for its residents. Each team has exactly two players, and each team is on exactly one team. In each round, each team plays exactly one other team and either wins or loses. The winning team advances to the next round and the losing team is eliminated. No team or player drops out except by losing a game. The tournament is in progress, and exactly 512 players participated in the first round.

From the available options, select a number of tournament rounds and a number of teams such that after the specified number of rounds there will be the specified number of teams remaining in the tournament. Make only two selections, one in each column.

[DavidG@VeritasPrep]

A city is hosting a table tennis tournament for its residents. Each team has exactly two players, and each team is on exactly one team. In each round, each team plays exactly one other team and either wins or loses. The winning team advances to the next round and the losing team is eliminated. No team or player drops out except by losing a game. The tournament is in progress, and exactly 512 players participated in the first round.

From the available options, select a number of tournament rounds and a number of teams such that after the specified number of rounds there will be the specified number of teams remaining in the tournament. Make only two selections, one in each column.

We know that if there are two players per team, and there are 512 players, there are 512/2 = 256 teams. After each round, half the teams are eliminated. So after one round, there will be 128 teams left. After two rounds, there will be 64 teams left. After 3 ---> 32 left. After 4 ---> 16 left. After 5 ---> 8 left. After 6 ---> 4 left. After 7 ----> 2 left. After 8 ---> 1 left, and this team won the tournament.

Of the above options, the only possibility that will work is that after 4 rounds, there will be 16 teams remaining.

[BTG Expert Team]

What we see: a Two-Part Analysis question. 512 players, 2 per team, losers drop out each round. Pick a number of rounds and a number of teams left after those rounds, from the choices 2, 4, 8, 16, 32 in each column. (The stem's "each team is on exactly one team" means each player is on one team.)

Step 1. Count teams, not players.
What we see: 512 players, exactly 2 per team.
What it means: \(512 \div 2 = 256\) teams start round 1.

Step 2. Write the rule for each round.
What we see: every team plays one other team, and the loser is out.
What it means: each round cuts the number of teams in half. After \(r\) rounds, \(256 \div 2^r\) teams remain: 128, 64, 32, 16, 8, 4, 2, 1 after rounds 1 through 8. After 8 rounds there is one winner and the tournament is over.

Step 3. Match it to the choices.
What we see: rounds can be 2, 4, 8, 16 or 32.
What it means: 2 rounds leaves 64 teams (not a choice). 4 rounds leaves 16 teams (a choice). 8 rounds leaves 1 team (not a choice). 16 or 32 rounds can't happen.
What we do and why: only one pair works, so we pick that round count and that team count. Check from the other side: 32 teams needs 3 rounds, 8 needs 5, 4 needs 6, 2 needs 7, and none of those is a rounds choice.

The trap: halving 512 players instead of 256 teams. That gives 32 left after 4 rounds and 2 left after 8 rounds, and both pairs sit in the table, so the wrong method looks like it works.
Takeaway: in elimination events, count the units that actually play (teams), then halve once per round: \(\text{start} \div 2^r\).

Answer: Rounds 4, Teams 16