I can only think of some refinements of depth-first. For example, if all sevens are in the black hole and there are cards left out of the black hole that are both above and below seven, then you can stop the search.

You should be able to do a brute force search to solve this. Your search would be aided by estimating the probability distribution over the remaining (unseen) cards on the table. Specifically, you want to estimate the probability that the next card turned up after putting a face card into the Hole is one higher or lower than either of the other two face cards. You could then bias your search toward maximising this probability?

As an alternative, just build a play book by back constructing it from possible end-game scenarios.

Cheers,

Timkin