Six people participated in a checker tournament. each participant played exactly three games with each of the other participants. how many games were played in all?

The 1st player plays 3 games with each of the other 5 players.
the 2nd player plays 3 games with 4 others, and so on.
Games played are 15, 12, 9, 6, 3 = 45 games played in total.

Another way to think of this is that each player plays 15 games each, which is 90 games.  But each game requires 2 players.  Divide 90 by 2 = 45.