## World Cup

30

January

A set of soccer games is to be organized in a round robin (e.g. every team plays a match against each other one time). If 45 games are played, how many teams participated?

”

If 10 teams participated, then the first team plays games against the other 9 teams. The second team has already played against the first team so it has to play games against only the other 8. In this way the second-last team has to play against only one team, and the last team has already played against the others. Therefore the total number of games is 9 + 8 + …….. + 2 + 1 = 45.

If 10 teams participated, then the first team plays games against the other 9 teams. The second team has already played against the first team so it has to play games against only the other 8. In this way the second-last team has to play against only one team, and the last team has already played against the others. Therefore the total number of games is 9 + 8 + …….. + 2 + 1 = 45.

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“EVERY team plays a match against EACH OTHER one time.”

Since there’s no mention of teams being eliminated (until the answer, that is) that means that every team plays every other team one time, so there is no whole number answer to this question.

The formula for this answer can also be expressed as N*(N-1)/2.

If N*(N-1)/2 = 45

multiply both sides by 2

N*(N-1) = 90

Some simple guessing will get you to N=10 fairly quickly.

I first misread this as having 45 teams, and needing to solve the for the number of matches. I got to 990 using the formula, and was surprised to see the answer of 10. But then who would have a 45-team round robin? That’s ridiculous.

10

Andrew

not Andrewl

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n*(n-1)/2=45

n^2-n-90=0

n=10

Ah, a nice easy one.

Just wanted to say, I really enjoy these puzzles; I very much appreciate the time and effort you have spent in creating this wonderful site filled with puzzles. Truly a place to “tickle” my brain.