Question

If 32 teams participate in a standard single elimination tournament, how many games must be played to decide a champion?

Answer 1

Answer: 31 games

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Explanation:

• 32/2 = 16 games will happen in round 1. Afterward, 16 teams are left.
• 16/2 = 8 games will happen in round 2. Afterward, 8 teams are left.
• 8/2 = 4 games happen in round 3.
• 4/2 = 2 games in round 4.
• 2/2 = 1 game as the final championship.

Count the number of times you divide by two and we have five occurrences of this. So there five rounds overall.

To get the total number of games played, we add up the quotients

16+8+4+2+1 = 31

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Or as a shortcut we can simply subtract off 1 since 1+2+4+...+2^n = 2^(n+1)-1

We can write that rule as

$\displaystyle \sum_{k=1}^{n}2^k = 2^{n+1}-1$

which is equivalent to

$\displaystyle \sum_{k=1}^{n}2^{k-1} = 2^n-1$