Another sports (actually tournament) puzzle

Hint and challenge

This puzzle was inspired by one in Martin Gardner's little book, _Aha! Insight_.

The challenge is to see the answer that requires no paper and pencil or difficult figuring.

One question (kind of a dumb one ;-))

I don't really follow sports or playoffs. You said that there are 14 teams with a bye the first round. How do the teams with a bye get into the mix? (I am not clear on who they would play the first time.)

Two answers

1) Irrelevant. At least once you have the Aha! insight.

2) In the first round, 36 contestants would face off to determine the 18 who would joint the 14 with first round byes in the round of 32.

The remaining losers in the first round would form a losers bracket, where four would compete to eliminate two, so the 16 remaining could be combined with the losers of the main bracket's round of 32 to form the losers bracket round of 32.

But you really don't have to follow all that to get the answer.

OK

Not quite sure that I totally understand, but I am going to guess 36.

(32 +4)

No :-(

An easier version

Maybe I over-obfuscated ;-)

What if it's a regular single-elimination tournament? 50 contestants. 14 get a first round bye. How many games are played.

This version is a lot easier to get "the hard way" than the double-elimination example, but has the same "aha!" approach to an easy answer.

WAG

35

Nope

Re: Another sports (actually tournament) puzzle

In order to answer this I think you have to make some assumptions:

1. You mention that the champion from the loser's bracket has to beat the champion twice. When I was wrestling and later coaching wrestling we did not consider these matches as part of the loser's bracket so my answer may be -2 from the answer you got.

2. Even if there are byes the wrestler (and/or team) has to go out and accept the bye and it is considered a match. Therefore, I am not subtracting these matches out.

Given that it becomes pure bracketology (is that a word?). Thirty-two losers will wrestle 16 matches, then 8, then 4, then 2, then 1 for a total of 31 matches. You may have to add 2 to 33 if do not believe in number 1. Also, if do not count byes as matches (number 2) you would have to subtract 14 from this number since all byes will pass to losers bracket and none will win.

So my answers depending on the assumptions you make are 31, 33, 17, or 19. There is probably some mathematical way to come up with this but since I was not a math major.

No :-(

I agree that the contest(s) between the winner of the winners bracket and the winner of the losers' bracket are not part of the losers' bracket.

I was not counting byes as a contest. Put another way, if this were a tennis tournament, how many court times would you have to book?

Your answer fails to recognize the folks who get added to the losers bracket after the initial round.

Answers

Let's take the easier version first:

Round 1: 14 byes, and the other 36 contestants take part in 18 contests

Round 2: 32 participants, 16 contests

Round 3: 8 contests

Round 4: 4 contests

Semifinals: 2 contests

Finals: 1 contest

Total: 49 contests

Original version:

Nah, I'm not going to even try the hard way.

Now for the aha! insight that avoids the above calculation:

In a single elimination tournament, one player loses each contest and is eliminated from the tournament. If you start with 50 players and end up with 1 winner, 49 people had to lose a contest, thus there were 49 contests!

In the original problem, a person leaves the tournament in any of three ways, as the winner (one contestant), as the loser of the final contest between the winners of the two brackets (one contestant), or as one who suffered his second defeat in the losers bracket (48 contestants). So there are 48 contests in the losers bracket.

Re: Answers

Wait a minute, Alex. The first round of matches is not considered part of the loser's bracket or the winners bracket.

Re: Answers

Jim, I'm not sure what your point is.

In the "easier version", I am talking about a single elimination tournament.

In the original version, in a double-elimination tournament, does anyone go home except by losing in the losers bracket, or by winning either of the brackets and playing in the finals?