Winning Conditons for Solitaire

Despite using Linux, I play solitaire often. I have reasoned through something approximating mathematics that any game with two face down cards can be won.

Let's assume one card is face down. If the card is an ace, then it is possible to move the card on top of it off since all the kings are face up. If the card is a king, then all the aces are available, so the cards can be put away until the king is unburied. For any card in between, both the aces and the kings are available, so the card can be unburied by a combination of moving cards up and moving stacks around. So, any game with one buried card can be won.

Now, assume we have two face down cards. If any two cards are buried, there are three possibilities: two cards of the same suit, two cards of the same colour, two cards of a different colour. If the cards are of a different colour, then the two complementary cards must be face up. Therefore, half of the deck is exposed, so the cards can be shuffled until half of the deck is put away, freeing any buried cards. This argument also applies if two cards of the same suit are buried. If the cards are of different suits, then considerably more shuffling is needed, but they can be exposed.

This no longer holds with three face down cards. It's harder to explain, but easy to show. If king covers a king of the same colour covers ace of the first suit.

The problem is, once you know you can always win with only two cards buried, it becomes demotivating to finish the game. Thank goodness for the finishing animation.