My friend and I play in a chess tournament

[BMAD]

I was on my junior-high chess team! One time, another seventh grader and I entered an eighth-grade tournament. Every player played every other player once (round-robin); a win counted as one point and a draw was 1/2 point. My friend and I got a total of 8 points, while all the 8th-graders got the same number of points (as each other). How many eighth-graders were in the tourney, and why?

*Note there are two possible answers

[Prime]

1). 7 8th graders in the tournament, 9 people total. Where two 7th graders got 8 points (could be 4 each), and 7 8th graders got 4 points each.

2). 14 8th graders, 16 people total. Each 8th grader got 8 points, the two 7th graders got 8 points between the two of them.

Corollary: none ot the players was Bobby Fischer.

[witzar]

17 players, each game ended in a tie

[vinay.singh84]

# of eigth graders

------------------------
5 (1 pt each)
6 (2 pts each)
10 (5 pts each)
15 (8 pts each)
30 (16 pts each)

[witzar]

12 players. Divide 8-graders into 2 groups of 5 players. You beat each player from group 1, and tie with each player from group 2 and with your friend. This gives you 5 + 2.5 + 0.5 = 8 points. Your friend beats each player from group 2 and ties with other players, also scoring 8 points. Each game between two 8-graders is a tie, so each 8-grader gets 4.5 against other 8-graders and 0.5 total against you and your friend, finishing competition with 5 points.

Edited April 1, 2013 by witzar

[witzar]

32 players. Divide 8-graders into 2 groups of 15 players. You tie with each player from group 1, lose with each player from group 2, and tie with your friend. This gives you 7.5 + 0 + 0.5 = 8 points. Your friend ties with each player from group 2, loses with each player from group 1 and ties with you, also scoring 8 points. Each game between two 8-graders is a tie, so each 8-grader gets 14.5 points against other 8-graders and 1.5 total against you and your friend, finishing competition with 16 points.

[jhawk]

First I read it as the 7th graders got a total of 8 points between them.

The number of games played is (N-1)*(N/2) which is also the number of points awarded. So the 8th graders received a totl of (N-1)*N/2)-8 ponts. These points are then split evenly between N-2 players. Thus,( (N-1)*N/2-8)/(N-2) must be a multiple of 1.0 or 0.5. The only solutions that neet this is for N=9 or N=16

[witzar]

So it looks like I have misunderstood the question assuming that both 7-graders scored 8 points each.

Anyway, "my version" of the puzzle was still interesting to solve.

[jhawk]

If in fact the 7th graders got 8 ponits each, then the possible solutions are 7,8,12,17,or 32

players in the tournament

Edited April 1, 2013 by jhawk

[witzar]

If in fact the 7th graders got 8 ponits each, then the possible solutions are 7,8,12,17,or 32

players in the tournament

There are no solutions with 7 or 8 players. To score 8 points you need to have at least 8 opponents.

[dark_magician_92]

either 7 or 14 eighth graders!!