Pyramid Solitaire: how three passes change the odds
Pyramid is a game of pairs. Twenty-eight cards form a triangle seven rows deep, each row overlapping the one above. You remove two exposed cards whose values add up to 13 (an Ace counts as 1, a Jack 11, a Queen 12), and a King, worth 13 on its own, is removed by itself. The other 24 cards form a stock that you can go through three times. Clear the pyramid and you've won.
Wherever we can, we compare our results on solitaire-free.uk with the 2026 study by Blake and Gent at the University of St Andrews. Pyramid is not in that study, and we found no other published winnability figure for the version played here. So this page sets out our own measurements of this site's Pyramid, including one rule we had to assume.
The main finding: with knowledge of the stock order, our solver proved 62.2% of 4,000 deals winnable, proved 14.0% impossible and could not decide 23.9%. So between 62.2% and 86.0% of deals can be won with perfect play and three passes.
Our results in full
| What we measured | Three passes (this site) | One pass only |
|---|---|---|
| Proven winnable, stock order known | 62.2% (95% range 60.6–63.6%) | 7.3% |
| Proven impossible | 14.0% | 34.8% |
| Undecided when the search limit was reached | 23.9% | 57.9% |
| Won by a simple player who sees only face-up cards | 25.2% (4,000 deals) | — |
| Published figure | None found for this version of Pyramid | |
The one-pass column used a smaller search budget, so more deals were left undecided and its figures are less precise. Even so, the contrast is striking. With one pass, we could prove only about one deal in fourteen winnable; with three, at least three in five. On this site the redeals are not a luxury. They are what makes Pyramid winnable.
Pyramid's rules on solitaire-free.uk
- Twenty-eight cards in seven rows: one at the top, seven at the bottom. A pyramid card becomes free to use when no card is lying across it.
- Remove two available cards totalling 13. Aces count 1, number cards their value, Jacks 11, Queens 12.
- A King (13) is removed on its own.
- The remaining 24 cards form the stock. The top stock card is face up, and it can be paired with an available pyramid card, as can the top card of the waste pile.
- You can go through the stock three times in all.
- The game is won when the pyramid is cleared. Cards left in the stock or waste don't matter.
Playing Pyramid with the data in mind
Plan for the second and third pass
With only one pass, very few deals can be proven winnable. With three, most can. So the first pass is a chance to learn where cards sit in the stock. You don't need to pair everything straight away.
Start by finding the Kings
In 40.8% of our deals, at least one King was sitting in the top three rows of the pyramid. Kings come off alone, but one buried high up can only be freed once the cards below it are cleared. Plan your route to it early.
Choose which card to pair carefully
When a card has more than one possible partner, prefer the pair that uncovers more of the pyramid. A card from the stock or waste costs you nothing on the table; a pyramid card may be the key to the row above.
A slow opening is normal
In 3.6% of our deals there was no King and no pair available at the start, counting the bottom row and the top stock card. That just means turning stock cards until one appears.
Use Undo to think like the solver
The solver knows the order of the stock. You don't, but after one pass you'll know a good deal of it, and Undo lets you try a line and step back. A simple player that never planned ahead won 25.2%; planning pushes you towards the solver's range.
Pyramid on solitaire-free.uk
Pyramid is free, with no downloads, no account and no email address, in any current browser on a computer, tablet or phone. Cards are large and clear, with zoom; you have Hint, unlimited Undo, Restart and Auto Play; move by dragging or tapping; and the game shows wins, time and moves for the current deal. You can also mute the sound, go fullscreen or pick a different table colour.
For another quick foundation game, try Golf. For a slower puzzle, see Alaska.
Mistakes that cost Pyramid games
- Pairing from the pyramid when the stock would do. If a stock or waste card can partner an exposed pyramid card, using it saves a pyramid card that may be needed for a harder pair higher up.
- Rushing the first pass. With three passes, the first is partly for learning the stock. Taking every pair at once can leave you with unmatched cards later.
- Leaving the top rows till last. The top of the pyramid is where games get stuck. Keep the partners those cards will need.
- Forgetting Kings need no partner. Remove a King the moment it is exposed; it never helps to leave one in place.
How we ran the research, step by step
The solitaire-free.uk team did this study in September 2026. With no published figure for this version, we have tried to make the method as clear as possible, including the one rule we could not confirm.
- Rules from our code. A 28-card pyramid, a 24-card stock with three passes, pairs totalling 13, Kings removed alone, and a win once the pyramid is clear. We assumed that the face-up top card of the stock can be paired directly; this is our reading of the game code, not something we tested in the browser.
- A program version. We rebuilt Pyramid in Python with a solver and a simple player.
- Numbered deals. 4,000 deals, each shuffled from a numbered seed so they can be dealt again, used for both the three-pass and the one-pass runs.
- Two players. The solver knows the order of the stock, which no player does, so it gives a ceiling. It could search up to 1,000,000 positions per deal for three passes, and 200,000 for one pass. The simple player sees only face-up cards and plays greedily.
- Running it. 4,000 deals with three passes, 4,000 with one pass, and 4,000 for the simple player.
- Published comparison. None found, including in the St Andrews study.
- The margin of error. A 95% range is where the true value very probably lies. For the proven 62.2%, it is 60.6% to 63.6%. Because of undecided deals, the full answer is 62.2% to 86.0%.
- What the numbers can't tell you. Undecided deals are unknown, not lost. The simple player is a floor. And if the site in fact does not let you pair the top stock card directly, our figures would be too high.
Behind the numbers: the code
The study used Python 3.11 with nothing beyond the standard library. Pyramid has its own module, pyramid.py. The solver is a depth-first search over which pyramid cards are gone, which stock cards are used, where you are in the stock and how many passes remain. It remembers positions already proven to fail, and it abandons a position early if some pyramid card has no partner left anywhere it could reach, which is a sound test that the deal cannot be won from there. A shared runner spread the 4,000 three-pass deals across 19 worker processes on a 20-core machine in about sixteen minutes. Ranges use the Wilson score interval.
Claude, an AI assistant from Anthropic, helped write and check the solver and draft this page. The AI did not produce the results. They come from running the code, and the team reviewed them before this page was published.
Frequently asked questions
Can most Pyramid deals be cleared?
With three passes and knowledge of the stock order, between 62.2% and 86.0% in our study of 4,000 deals. We know of no published figure for this version.
Why do the redeals matter so much?
Pairs often need a card that is buried in the stock at the wrong moment. Extra passes give you another chance to meet it. With one pass we could prove only 7.3% of deals winnable.
What does a King count as?
Thirteen, so it is removed on its own without a partner.
Must the stock be emptied to win?
No. On this site the game is won as soon as the pyramid itself is cleared.
Which other games are fast?
Golf is similar in pace. The full list is on the games page, and our blog has more on the research.
By Admin, solitaire-free.uk. Study date: 18 September 2026.