Unfortunately, even if we have an ultra fast computer, which can test one complete combination in 20 picoseconds (1/50 microseconds), the complete test will run for approximately the present lifetime of the universe. N! gets too big too fast.
There are less simple minded ways to write this program, and they will run faster. But no method now known escapes the buildup of complexity illustrated by N!. Furthermore, Stephen Cook in 1970 showed that there is a class of problems, all equivalent to each other in a precise sense of "complexity", which stand or fall together. Our second tiling problem is in this class, along with over 1000 other, apparently unrelated, problems. None have been shown to escape intractibility, but if only one does then so do all the rest. Computer scientists currently believe that this class of problems will remain intractible. Harel's book also discusses these issues.