A school chess club is divided into three groups; an Elite group which meets on Monday, a Tournament group, which meets on Wednesday and a Recreational group, which meets on Friday. The club runs group competitions four times a year, where every player in a group plays every other player in that group twice, once playing as white and once as black.

After the first group competition is over, three players, one from the Recreational group and two from the Elite group, move into the Tournament group. After this, exactly the same number of games are played in total in the second group competition as in the first one.

After the second group competition is over, three players, two from the Recreational group and one from the Elite group, move into the Tournament group. In the third group competition, exactly the same number of games are played in total as were played in each of the first two group competitions.

The Chess coach decides that, after the third group competition, she wants the Tournament and Recreational groups to be the same size, and the Elite group to be half the size of the Tournament and Recreational groups. She finds that she can do this by moving some players from the Recreational to the Tournament group, and the same number of players from the Tournament to the Elite group.

How many players are there in the Chess Club?