There is a classic problem of finding the only number that occurs an odd number of times in a list. The solution is to xor everything and the result is the requested number.
The key properties used here are the involution of xor (i.e. a number's inverse is the same number) and the closure over integer range of the variable.
Now, are there any other operations that could be used for this? I could only find 1/x which, unfortunately, cannot be used due to precision erros.