Algorithm to find exactly two sets of numbers that have the same sum
This is called the Partition Problem, and it's NP-complete, meaning that it's very unlikely that any algorithm can solve every instance of the problem quickly. On the bright side, if there are $n$ numbers and their total sum is $t$, there is a fairly simple $O(nt)$-time dynamic programming algorithm (see section "Pseudo-polynomial time algorithm" on that webpage), and this is fast enough to solve many practical instances.