https://github.com/bcrist/hand-solitaire
Monte Carlo simulation of the game of Hand Solitaire
https://github.com/bcrist/hand-solitaire
Last synced: 2 months ago
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Monte Carlo simulation of the game of Hand Solitaire
- Host: GitHub
- URL: https://github.com/bcrist/hand-solitaire
- Owner: bcrist
- License: mit
- Created: 2026-01-28T02:19:50.000Z (6 months ago)
- Default Branch: main
- Last Pushed: 2026-01-28T05:29:05.000Z (6 months ago)
- Last Synced: 2026-01-28T21:16:26.531Z (6 months ago)
- Language: Zig
- Size: 42 KB
- Stars: 0
- Watchers: 0
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: readme.md
- License: license
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README
# Hand Solitaire Simulator
Hand Solitaire is single-player game for passing time when all you have is a deck of playing cards. Apart from the cards, all it requires is two hands and a place to stash a discard pile, so it's ideal for road trips, waiting in long lines, etc.
This repo simulates many games and computes a histogram of how many iterations are required to win.
## Setup & Rules
1. Start by removing any jokers and shuffling the deck.
2. Hold the deck in your off hand with all cards face-down.
3. Draw the card on the bottom of the deck and place it face-up on top of the deck, slightly to the side so you can see the card underneath.
4. If there are at least 4 face-up cards and the first and fourth card have the same value (A23456789TJQK) then take those cards and the two cards between them and place them face-up on the discard pile.
5. If there are at least 4 face-up cards and the first and fourth card have the same suit, remove the two cards between them, but leave the suited cards in the hand. Place the two removed cards face-up on the discard pile, then go back to step 4.
6. Return to step 3 until all of the face-down cards have been revealed.
7. Continue drawing face-up cards from the bottom of the hand to see if they trigger any discards at the top of the hand. If you've drawn at least 3 face-up cards in a row without being able to discard, no further discards will be possible.
8. If all cards have been placed in the discard pile, you've won! Otherwise you have completed one "iteration." Shuffle the remaining cards in your hand, then place them on the *bottom* of the discard pile. Flip the discard pile over and go back to step 2.
## Statistics & Notes
* The chance of winning on the first iteration is approximately 1.3%
* The chance of winning in 10 iterations or less is approximately 22.5%
* The chance of winning in 25 iterations or less is approximately 51%
* There is no upper limit to the number of iterations that might be required to win, since the shuffle in step 8 does not guarantee that an "unvisited" permutation is selected, but after simulating several million games, I have only seen one winning game that required over 500 iterations.
* Approdimately 0.06% of all deck permutations are impossible to win (see below).
* The shuffling of remaining cards in step 8 is very important! If you omit this, then approximately half of all starting decks can never win.
It is possible to get stuck in a "loop" where it is impossible to win. The game is entirely deterministic except for the shuffle in step 8, so if only a few cards remain at the end, there are only a few permutations for the next iteration's deck. Usually after the next iteration(s) the set of possible decks will have expanded to include some winning decks, but there are many small sets of decks where all possible permutations for the next iteration are also in the same set, and none of the decks in the set can win. Interestingly, almost all of these "poison" deck sets contain 48 different decks, though I have also seen sets of size 4, 8, 12, and 36.
Some people play a variant of Hand Solitaire without step 7. This step tends to reduce the number of cards remaining in the hand at the end of each iteration, so you might expect that playing without it would decrease the chance of getting stuck in an unwinnable set of decks, but playing without it also significantly decreases the number of winning decks overall, and it turns out that effect is more important. Without this rule, the average number of iterations needed to win is approximately doubled, and the chance of getting an unwinnable deck is approximately 0.13%.
Here is a histogram of 1,000,000 random games:
