It is fairly obvious if you think about it and if you are a mathematician, but I was surprised by this theorem.

Theorem: Roll any number of dice all of which can have any (finite) number of sides except at least one die which is the standard six sided die. The probability that the sum is divisible by three is exactly 1/3. Also, the probability that the sum is odd is exactly 1/2.

Corollary: Roll any number of six sided dice. The probability that the sum is divisible by 3 is 1/3. The probability that the sum is even is exactly 1/2.