Alex Y
That does it. Congrats.

Thanks for playing.

My solution was less elegant: Paint a true statement, such as 2+2=4, on three of the doors, 2+2=5 on three doors, and a statement of unknown truth on the other three, and ask if the statement on the door to freedom was true.
But each of my proposals for statements of unknown truth had some potential issues. One was an unproven mathematical conjecture, such as an infinite number of twin primes (but what if, unknown to the suitor, a proof has just been published?) My favorite is that Spain will win the first World Cup to be held after 2050.

Another solution similar to yours is to tell the king that you are thinking of a set of doors that includes the three leftmost doors, but none of the three rightmost. Is the door to freedom in the set of doors I am thinking of? (Note that the doors in the middle don't have to be included or excluded at random -- just by a selection process that the King does not know.)