Following Pólya's "guess and test" method, we seek to discover 3-period folding numbers analogous to the exhaustive set of 2-period folding numbers discovered by Hilton and Pedersen in 1981. Most of the rows and columns of the 2-period folding numbers are reported in the Online Encyclopedia of Integer Sequences (OEIS) with various other mathematical interpretations. We provide a table of 3-period folding numbers, but it is not exhaustive, as we demonstrate by showing other sets of 3-period folding numbers that are not in the table. We close the paper with an algorithm for finding more sets of 3-period folding numbers and a conjecture about how many such sets exist.