Thursday, December 13, 2012

Ready, Go, SET!

Riddles and puzzles are a good way to challenge yourself and increase your logic skills.  They also give valuable practice in solving the story problems that seem to haunt so many math students.

Here’s a brain teaser that can be solved with simple math.  (As is the case with many math puzzles, there is more than one way to solve the problem.)  If you think you’re up to it, try to answer the question before reading the solutions and explanations at the bottom of this page.

There are 340 players in a SET* tournament.  Each game in the tournament is played with 4 people.  For each game, there is one winner who moves on to the next round.  Rounds will continue until there is just one winner left – the champion.  In any round, if the number of players is not a multiple of 4, then some players will be chosen randomly to advance automatically so that the number of people playing in that round is divisible by 4.  (This will ensure that each game that is played will have 4 players.)  How many total games will be played in the tournament?

{Scroll down for solutions}


*I love card games.  I also love activities that help people learn and/or increase their brain power.  SET is a card game that is simple to learn, fun for a wide range of players (recommended for players 6 years old and above), and helps build brain power.  As a teacher and card game enthusiast, I give SET my full recommendation.  (And now SET-Junior is available for younger players.)



Method 1 (slow)

To solve the problem, many people get out pencil and paper and begin calculating the results, round by round.  For each round, they keep track of the number of games, and then add these up at the end to find the answer to the question.  This method can be quite time consuming.  One such solution is shown:

Method 2 (fast)

Of course, there is another way to solve this problem – a way that takes much less time.  In the previous solution, the focus was on the WINNERS from each round.  (The number of players in rounds 2 – 5 was determined by figuring out how many winners there were in the previous round and adding this amount to the number of people who were lucky enough to advance automatically.)  A much quicker solution involves focusing on LOSERS:

We know each game that is played will have 4 players.  This means that each game is guaranteed to have 3 losers.  The tournament begins with 340 players, and 339 of those players will be a loser at some point.  All we need to do is divide 339 by 3 to find our answer: There will be 113 games.

When looking at a tough problem, we often hurt ourselves by focusing on the wrong part of the problem.  Sometimes it is helpful to take a step back, reexamine the problem, and figure out if our focus is in the right place.

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