Pyramid Solitaire: the ceiling depends on how many times you see the stock
Pyramid is a game of pairs. Twenty-eight cards form a pyramid, and you clear it by removing two uncovered cards that add up to 13: a 6 with a 7, a Jack with a 2, a Queen with an Ace. Kings count as 13 on their own and leave alone. The other 24 cards form a stock you can turn through for more partners.
On this site you get three passes through that stock. We asked a solver that knows the stock order to play 4,000 deals. It cleared the pyramid in 62.2%, proved 14.0% impossible, and ran out of search time on the remaining 23.9%. So between 62.2% and 86.0% of deals can be won with perfect play.
Then we took the redeals away. With a single pass through the stock, the solver could only prove 7.3% of deals winnable. That comparison is the heart of this page: in Pyramid, the second and third passes are what make the game winnable. You can play it here free, in any browser.
Pyramid's ceiling, with and without redeals
| Result | Three passes (as on this site) | One pass |
|---|---|---|
| Solved with perfect play | 62.2% (95% range 60.6–63.6%) | 7.3% (95% range 6.5–8.1%) |
| Proven impossible | 14.0% | 34.8% |
| Undecided at the search limit | 23.9% | 57.9% |
| Search limit per deal | 1,000,000 positions | 200,000 positions |
| Deals | 4,000 | 4,000 |
The one-pass test had a smaller search limit and left more than half the deals undecided, so its "solved" figure is a floor. Even so, the gap is dramatic. Pyramid is not part of the Blake & Gent (2026) study, so there is no published figure to compare.
What the solver knows that you don't
Our solver knew the order of the stock from the start. That makes its result a ceiling. A greedy computer player that sees only face-up cards, playing all three passes without any look-ahead into the stock, cleared the pyramid in 25.2% of the same 4,000 deals.
That gap, between about a quarter and at least 62%, is room for skill. It mostly comes down to choosing which pairs to take. Two different 7s might each pair with a 6; the right one is the 7 that uncovers something useful.
Features of the deal worth knowing
- A King near the top. In 40.8% of deals, at least one King sits in the top three rows of the pyramid. Kings leave on their own, but one up there only becomes available once the cards overlapping it have been paired away, so it will not help you early in the game.
- A slow start. In 3.6% of deals, there is no King and no pair available at the start, counting the bottom row and the top stock card. You will have to turn stock cards before your first move.
Pyramid rules on this site
- One 52-card deck. Twenty-eight cards form a pyramid of seven rows, all face up.
- A card is available once no card overlaps it from the row below.
- Remove two available cards that total 13. Jacks count 11, Queens 12, Kings 13; Aces count 1.
- Kings are removed on their own.
- The other 24 cards form the stock. The top stock card and the top waste card can both be used in pairs.
- You get three passes through the stock.
- You win when the pyramid is cleared, even if cards remain in the stock or waste.
Getting closer to 62%
Pair from the pyramid first
A pair made of two pyramid cards clears two cards you have to remove anyway. Stock cards can wait for a later pass.
Choose the partner that uncovers most
When two cards could each pair with the same partner, take the one that frees another card.
Watch for "lonely" cards
If all four 6s are gone, the remaining 7s have no partner. Keep a rough count of which ranks are running low.
Treat pass one as a scouting run
You have three passes. On the first, notice where the useful cards sit, so you can plan your pairs for the next two.
Undo is your stock map
The solver's big advantage is knowing the stock. Undo lets you peek ahead and back up, which narrows that gap.
Pyramid on the ceiling ladder
With three passes, Pyramid lands in the middle of the games we measured:
- Golf: 92.7% winnable (knows the stock order, exact).
- Pyramid: 62.2–86.0% winnable (knows the stock order).
- Klondike Turn 3: 66.2–90.5% winnable (sees every card).
- Baker's Solitaire: 31.8% winnable (exact).
Pyramid and Golf make an interesting pair. Both are single-deck games where a solver that knows the stock order does far better than a player who does not. Pyramid's proven-impossible share, 14.0%, is roughly twice Golf's 7.3%, so a Pyramid deal that refuses to open is more often a genuine dead end.
Inside the test, one step at a time
The solitaire-online.us team ran this study in September 2026 to estimate the ceiling for Pyramid on this site, and to see how much the redeals matter.
- Reading the rules from our game code. The 28-card pyramid, pairs totaling 13, Kings removed alone, three passes through a 24-card stock, and the rule that clearing the pyramid wins, all come from this page's game files. We assumed the face-up top card of the stock can be paired directly, as the code suggests; we did not confirm this in a browser.
- Rebuilding the game. We wrote Pyramid as a program with those rules, plus a one-pass version for comparison.
- Dealing the cards. Seeds 1 to 4,000 for both tests, so the same deals were played with three passes and with one.
- Choosing the players. A solver that knows the stock order gives the ceiling. A greedy player that sees only face-up cards gives a floor.
- Running it. Up to 1,000,000 positions per deal with three passes, and 200,000 with one.
- Checking against published research. Pyramid is not covered by Blake & Gent (2026), so there is no outside figure.
- Working out the margin of error. In 19 repeats out of 20, the true share solved with three passes would fall within 60.6–63.6%.
- What the numbers cannot tell you. About a quarter of deals stayed undecided with three passes, so the ceiling is a range. The one-pass figures had a smaller search budget and are floors.
The code behind the ceiling
The study is written in Python 3.11 with nothing but the standard library. Pyramid has its own module, pyramid.py, and runner.py spreads the deals over 19 worker processes on a 20-core machine. Each deal is shuffled from a numbered seed.
The solver is a depth-first search over which pyramid cards remain, which stock cards are used, where you are in the stock and how many passes are left. It remembers positions that already failed so it never repeats them. It also stops early when some pyramid card has no possible partner left outside the cards covering it, which is a sound proof that the deal cannot be won. The three-pass run took about 16 minutes. For the percentage ranges we used the Wilson score interval.
We relied on an AI assistant, Claude by Anthropic, to help write and review this code and to draft the page. None of the figures came from the AI. They all come from running the solver and the greedy player.
Because each shuffle comes from a numbered seed, anyone with the same code gets exactly the same deals and the same verdicts.
By Admin, solitaire-online.us. Study date: September 18, 2026.
Ceiling questions from players
How many Pyramid games can be won?
With three passes and perfect knowledge of the stock, between 62.2% and 86.0% of our 4,000 deals. 14.0% were proven impossible.
Do the redeals really matter that much?
Yes. With one pass, the solver could prove only 7.3% of deals winnable.
Does the game end when the pyramid is gone?
No. On this site you win as soon as the pyramid is cleared.
How often does a simple strategy win?
A greedy player that can't see the stock cleared 25.2% of deals. Planning which pairs to take is what closes the gap.
What's a similar quick game?
Golf Solitaire is another fast, single-deck puzzle, and its ceiling is even higher. See all games on the games page.