Are there truly unwinnable Calculation decks?

Question

Calculation is a solitaire variant known for its large skill ceiling; supposedly, the large majority of games can be won by skilled human players, whereas the games to be won by uninformed play are in a slim minority (of these two facts, I can only attest to the latter!).

I imagine that, like pretty much every other solitaire variant, there should be starting configurations of Calculation that are unwinnable even when you know all 48 cards in the deck in advance. Can you provide an example of such a configuration, with a human-verifiable proof that it cannot be won? Or can you prove that every configuration is winnable?

This is to assume no waste pile, and four tableaux.

Rules of Calculation: Suit never matters in this game, only rank. Take out an ace, two, three, and a four, and use them to start four foundation piles, and make the other 48 cards into a shuffled deck. Four tableaux piles start empty. The goal is to build the four sequential foundations:

• A,2,3,4,5,6,7,8,9,10,J,Q,K ("one foundation");
• 2,4,6,8,10,Q,A,3,5,7,9,J,K ("two foundation");
• 3,6,9,Q,2,5,8,J,A,4,7,10,K ("three foundation");
• 4,8,Q,3,7,J,26,10,A,5,9,K ("four foundation").

A legal move consists of either (1) drawing a card from the deck, and placing it on one of the four tableaux, or (2) taking a card from the top of a tableau and moving it to a foundation (but not to another tableau) which is ready to receive it. Play ends when you run out of legal moves, at which point you win if and only if all four foundations have been completed.

Answer 1

trolley813's answer is correct, but I think another configuration would make things clearer. The key thing I take from it is that dealing every card with the same rank one after the other reduces the player's options a lot.

I start dealing the following cards:

3 3 3 4 4 4 5 5 5 5 6 6 6 6 7 7 7 7 ...

One of the 3s will go on the first foundation pile before the game is won, but it can't go there now. As the tableau piles are initially identical, I can say (without loss of generality) that I deal this card onto the first tableau pile.

One of the 4s will go on the first foundation pile. It can't go on the first tableau pile, as this would stop me from playing the 3. I'll deal it onto the second one.

By the same logic, one of the 5s and one of the 6s will have to go on the first foundation pile, and I'll put them on the third and fourth tableau pile respectively.

I now have four 7s, one of which will have to eventually go on the first foundation pile. It can't go there at that point, and it can't go on a tableau pile as they all contain a card which need to go on the first foundation pile before the 7. The game is inevitably lost at this point. ∎

Answer 2

Partial answer (without a proof, only some assumptions):

The configuration (all 48 cards placed in order) like
Q Q Q Q 9 9 9 9 10 10 10 10 3 3 3 5 5 5 5 7 7 7 7 J J J J A A A K K K K 2 2 2 4 4 4 6 6 6 6 8 8 8 8 (or a variation) should be likely unwinnable at all (and if it isn't, I would like to see a solution for it!).

Explanation:

Since all the deuces, fours, sixed and eights are placed in the very end of the deck, you cannot move any cards to the foundation until you draw all the other cards (so you have to arrange all of them on the tableaux), and particularly all the kings and aces. So, at least 2 of the 4 tableaux will be "blocked" with an ace and a king respectively (you cannot place all of them in a single tableau, because aces come before the kings, thus they should be placed first, but all the aces cannot be buried under kings since it would be impossible to place the aces onto the foundations). But the aces are "tricky" since they cannot be used early in the game (an ace can be at least 7th card on a foundation).
The same can be said about the jacks, and (at lesser extent) fives and sevens. Since they come after the "early" cards like threes and queens in the deck, so it would be difficult (if possible at all) to place them on the tableaux without blocking the "early" cards which are more useful.