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# Probability - Think About It!

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In a single-elimination tournament, each game is between two players. Only the winner of each game advances to the next round. In a particular such tournament there are 256 players. How many individual games must be played to determine the champion?

Dec 9, 2021

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The "Think About It" name usually hints at an elusive trick to the question. There doesn't appear to be a trick anywhere; here's my best attempt:

If we arrange the players into two-player groups, we'll cut the players in half. 128 games have been played.

Here is the player to game progression:

(256,0)

(128,128)

(64,128 + 64)

(32, 128 + 64 + 32)

(16, 128 + 64 + 32 + 16)

(8, 128 + 64 + 32 + 16 + 8)

(4, 128 + 64 + 32 + 16 + 8 + 4)

(2, 128 + 64 + 32 + 16 + 8 + 4 + 2)

(1, 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1)

Fun fact: sum of powers of 2 starting from 2^0 up to 2^n = 2^n-1

Proof: our answer is 255.

Oh man... now I get the trick (think about it).

Dec 9, 2021
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thank you soo much!!!

AlgebraGuru  Dec 9, 2021
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Tinfoilhat:   Good logic       Give yourself a point

AlgebraGuru:   You should give Tinfoilhat a point too.

Melody  Dec 9, 2021
edited by Melody  Dec 9, 2021
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k, do i just like it?

AlgebraGuru  Dec 9, 2021
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yes, you click on the heart.  :)

Melody  Dec 9, 2021
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Fun fact: sum of powers of 2 starting from 2^0 up to 2^n = 2^n-1

It’s probably not fun if it’s not a fact.... But it’s definitely NOT a fact.

The formula for the sum of powers of (2) is $$2^{(n+1)}-1 \text { where (n) is an integer}$$

... now I get the trick (think about it).

OK, I’ve thought about it...

Your formula (2^n – 1) gives the correct number of games played when (n=8), but how does the problem solver find the (8)? ...And after finding it, how does it help in solving the problem?

This is a binary question. There are 256 Players. Each game eliminates one player. How many players must be eliminated until only one remains? Answer: 255. Because 256 – 255 = 1

For this kind of question, the answer is always one less than the number of players.

After thinking about it, I’m nominating your solution for a Fields MeTal. Category: Rube Goldburg solution methods (Overly complex solutions for simple questions).

You’re not likely to win.  This is the best one https://web2.0calc.com/questions/help-algebra_116 so far....

GA

--. .-

GingerAle  Dec 14, 2021
edited by Guest  Dec 14, 2021