These games based on the bar breaking or pile splitting are not very challenging as such. However, they furnish an edifying experience besides giving a chance for a knowledgeable person to show off if he/she is the only one who knows the secret. Here is a couple of examples.
Two players take turns breaking a bar. The last to break a piece wins the game.
It's a great way to learn of odd and even numbers. Any one privy to the secret would know what is preferable: to start the game or to be a second player - depending as whether the total number of squares is even or odd.
Marbles, checkers, or stones are arranged in several piles. A move consists in selecting a pile and splitting it into two. The player to split the last pile is the winner. (Explanation: it clearly does not matter how many piles one starts with. Imagine starting with a single pile and then making a few moves "that do not count.")