In Anderson et. al 2010, "Cognitive and metacognitive activity in mathematical problem solving: prefrontal and parietal patterns", the experimenters taught people how to solve a novel system of arithmetic problems, which they termed 'pyramid problems'.
The basic idea is that a pyramid expression, such as $m\$n$, represents a pyramid with a base of $m$ blocks that is $n$ levels high. Each level of the pyramid is one block smaller than the one below it. The value of the expression is the number of blocks in the pyramid, and the goal of the task was to learn to evaluate these expression. For example, if the expression is $4\$2$, then it represents a pyramid with four blocks as a base, and with a second level composed of 3 blocks, for a total of 7 blocks.
Participants in this study successfully learned how to solve problems of this sort, and also generalized such expressions to handle negative arguments.
I'm interested in finding out if there are other such arithmetic systems. It seems like the kind of thing recreational mathematicians would be interested in. Has anyone heard of similar games?