I saw the technique of using a mathematical system called the "Four minute clock". Basically, the clock has 4 numbers, 0, 1, 2 and 3, and only one hand. We can make a summation table which will look like the image attached. According to the text, if there is symmetry (reflection) across the main diagonal, the system is commutative. The problem came up when I attempted a 5 minute clock. Same clock, except this time we have 5 numbers: 0, 1, 2, 3, and 4. Filling out the table, we see that there is NO evident symmetry along the main diagonal, yet the commutative property still applies to this system. Namely: a+(b+c) = (a+b)+c The system does not pass the commutativity test, yet it still works? Am I missing something?