The rules downloadable above don't contain info on how to hold a Kenté tournament. Here's how it's done.
Let's say you have fifteen people in your tournament. Begin by dividing the players at random into five groups of three. Assign Black, White, and Green to the players at random. Each group plays a game. If there are any boards with a tie for first place, the game is played again with the same three players (colors again assigned at random) until there is a winner. Once the ties have been broken, we have five winners advancing.
But five does not divide evenly by three. To solve this problem, take the second-place finisher who had the highest score in the first round and promote that player to the second round along with the first place finishers. (In the event that two or more second place finishers tie for highest score, take the one whose opponenents' combined scores are the lowest).
We now have six players in the second round. Two games are played, with the two winners and the highest-scoring second-place finisher promoted to the third round.
Note that it will sometimes be necessary to take two second place finishers to the next round to arrive at a number of players divisable by three.
Once the third round is played, you might think you are done. You have a winner. But that is not the case. This first winner is just one of three who will play in the final round.
All the players except this first winner play again through a series of rounds to determine the second winner. A third winner is determined in the same way. At that point, the three winners play a final round to determine the overall winner.
What this means is that every player in the tournament has three chances to make it to the final round.
But you may have noticed a problem with this. When we run the tournament the second time, we only have fourteen players instead of fifteen. This is similar to what would happen if you started out with a number that wasn't divisable by three (say, sixteen).
In this case, you hold an elimination round to get to a number divisable by three. For example if you have fourteen players, you need to get to twelve, so you hold a round where only the two lowest scorers are eliminated.
Similarly, if you started with sixteen players, you would hold an elimination round to eliminate one player and get to fifteen.