The members of a chess club took part in a round robin competition in which each player plays with other once. All members scored the same number of points, except four juniors whose total score ere 17.5. How many members were there in the club? Assume that for each win a player scores 1 point, 1/2 for a draw, and zero for losing.

Explanation

Let the number of members be n. Total number of points is `""^(n)C_(2)`.
Therefore, `""^(n)C_(2)-17.5=(n-4)x` (where x is the number of point scored by each player)
`implies n(n-1)-35=2(n-4)x`
or `2x=(n(n-1)-35)/(n-4)` (where x takes the values 0.5, 1, 1.5, etc.)
`=(n^(2)-n-35)/(n-4)`
`=(n(n-4)+3(n-4)-23)/(n-4)`
`=(n+3)-(23)/(n-4)`
`implies (23)/(n-4)` must be an integer
`implies n=27`