Brain Teasers
Chess Matches
In my class there are 60 students. We decided to hold a chess tournament.
In the first phase, every one plays 6 matches each, each match with a different player.
What is the minimum number of people that will have to back out if all participants play 6 matches ONLY? Also how many matches will be played in the first phase?
In the first phase, every one plays 6 matches each, each match with a different player.
What is the minimum number of people that will have to back out if all participants play 6 matches ONLY? Also how many matches will be played in the first phase?
Answer
The number of players that need to back out is 4. Number of matches is 168.For 6 people to play each other, a minimum of 7 people are required. So the total number of players is a multiple of 7. The nearest is therefore 56 (7x8).
Number, of matches = 7 students per group x 6 matches per student / 2 students per match x 8 groups = 7 x 6 / 2 x 8 = 168
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Comments
You made an assumption that the 7 people in each group were limited to playing each other. The fact is that each person can play 6 games with anyone, as long as they haven't play them before. The tournament could advance the people with the best records after playing 6 games. No one would have to sit out.
Actually nobody needs to back out. The reason is because, as I will demonstrate below, it is possible to have a group of eight people play six matches each. You could then break the 60 classmates into 4 groups of 8 (32 people) and 4 groups of 7 (28 people).
Clearly it is possible for a group of seven people to play six matches each. It is also possible for a group of eight people to play 6 matches each. Here is how: Let person number 1 play persons numbered 2,3,4,5,7,8. Let number 2 play 1,3,4,6,7,8. Let number 3 play 1,2,5,6,7,8. Let 4 play 1,2,5,6,7,8. Let number 5 play 1,3,4,6,7,8. Let number 6 play 2,3,4,5,7,8. Let number 7 play 1,2,3,4,5,6. Let number 8 play 1,2,3,4,5,6. Therefore no one needs to back out.
Clearly it is possible for a group of seven people to play six matches each. It is also possible for a group of eight people to play 6 matches each. Here is how: Let person number 1 play persons numbered 2,3,4,5,7,8. Let number 2 play 1,3,4,6,7,8. Let number 3 play 1,2,5,6,7,8. Let 4 play 1,2,5,6,7,8. Let number 5 play 1,3,4,6,7,8. Let number 6 play 2,3,4,5,7,8. Let number 7 play 1,2,3,4,5,6. Let number 8 play 1,2,3,4,5,6. Therefore no one needs to back out.
Heres another way. Break the players into 8 pools of 7 as suggested and leave 4 out. But in 6 of the pools, don't play the last 2 matches. This leaves 4 players in 6 of the pools with one game to go each. They play against the 4 who were left out of the pools. Thus all players in the pools have 6 matches and so do the 4 left out of the pools. Under this scheme there are 180 matches (6x60/2).
It is actually even easier. 60 players X 6 matches = 360 player-matches. 360 player-matches / 2 players per match = 180 matches.
This set up is even simpler than any mentioned. Arrange 30 boards in a ring. The thirty students inside the ring stay seated, while the 30 students outside the ring find their next game at the board to their right.
OK, so we all agree this teaser is flawed...why is it still here?
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