However, this was not enough to solve the problem.  Each test took around 40 seconds to 200 seconds, and with 200 tests I wasn't going to wait for that long for the code to finish.

Using the idea above, I reconstructed the whole program into a recursive version. What it does it finds every possible value that can be calculated from a set of numbers, then I run it on every subset of the 6 numbers we are given and test if our goal value is inside. 

In the above code the usage of a mask made it very easy to test if our possible combinations of numbers were disjoint, however in this recursive manner it was not so easy. Instead of a mask, I simply checked if 2 subsets intersected, and if they did, if those numbers were repeated (since otherwise simply checking for empty intersection would eliminate sets which contained 2 of the same number).

I again add my code for the solution of problem, with minor comments. In the end it took 8 seconds to run which is reasonable enough for me!