This activity can be acquired from Vatzek, the archery master in Ledetchko. Speak with him to learn the rules of the game, and then bet an amount to gamble in the competition. Whatever amount you bet will be matched by the other two villagers.
Once you have placed your bet, you must shoot as many coloured arrows as possible into the logs moving down river. If you have a decent skill level in archery and are familiar with adjusting your aim to a moving target, this should not be a problem. If you find that you need some advice, see Hunting for some basic tips on using the bow, and where to expect trajectory of the shot.
Although the other villagers are slow to the draw, the activity will end when the first log reaches Vatzek, who can be found downstream. You should ensure that you have the most arrows in logs by that time. Once he calls the end, you can speak with him to receive your reward and complete the activity.
Note: You can play Chumps on the River as often as you like. It is a useful way to train your archery skills and earn some extra Groschen in the process.