Making 5-tuples out of their respective 4-tuples
Go over all $\binom{36}{5}$ possible five-number combinations. For each one, calculate the five four-number combinations, and check whether all of them are unused.

In order to check whether a given four-number combination is unused, you can use a table. This will allow each such test to proceed very quickly. You can either use an array of size $36^4$ using the trivial encoding, or a slightly more sophisticated one of size $\binom{36}{4}$.