Pyramid Solitaire
Play Pyramid solitaire, and see the solved win rates: one pass through the stock clears 5% of deals, three passes clear 68.5%.
Click two available cards that add to 13 — ace is 1, jack 11, queen 12, king 13. A king is 13 on its own, so one click takes it. A pyramid card can only be used once both cards resting on it have gone. Clear all 28 to win. Everything runs in your browser; nothing is uploaded.
The second pass is worth more than the whole first game
Every deal below was solved rather than played: the search either finds a way to clear all 28 pyramid cards or proves that no way exists. That makes these win rates a property of the deals themselves, not of how well anybody played them.
| Passes allowed | Deals solved | Winnable | Share | Undecided |
|---|---|---|---|---|
| 1 | 400 | 20 | 5.0% | 0 |
| 2 | 200 | 104 | 52.0% | 0 |
| 3 | 200 | 137 | 68.5% | 0 |
| 4 | 200 | 140 | 70.0% | 0 |
| 6 | 150 | 108 | 72.0% | 0 |
One pass wins 5.0% of deals. Allowing a second takes it to 52.0% — more than ten times as many — and the third adds another 16.5 points on top. The returns fall away sharply after that second pass, but the second one is not a tweak: it is the difference between a game you almost always lose and one you win more often than not.
And then it stops. A fourth pass buys 1.5 points and a fifth and sixth together buy about two more, flattening near 72.0%. That last number is really a statement about the shuffle rather than the rules: roughly 28% of deals cannot be cleared however long you keep cycling the stock. They were lost before anyone touched a card. Six passes is not infinity, so read that as an upper bound on the hopeless share rather than an exact one.
Which is why "Pyramid is winnable about half the time" is not a fact about Pyramid. It is a fact about the three-pass game. Said of the one-pass game it is wrong by a factor of ten, and neither version of the rules is more official than the other.
Nothing above is a shrug
All 1150 deals came back as a proof rather than a timeout — the undecided column is zero the whole way down. That is worth stating plainly, because the easy way to produce a table like this is to run a search with a budget and count everything it failed to crack as unwinnable. That silently turns "I gave up" into "impossible" and inflates the number of hopeless deals. The solver here reports three outcomes, not two, and the button above will tell you it could not decide rather than guess.
What makes the search finish is one observation: taking a king is never a mistake. Thirteen pairs with nothing, so a king always leaves alone and removing it cannot use up a partner that some other card needed. So the solver takes kings immediately instead of branching on them. That is checked rather than assumed — a second, deliberately slower solver that does treat kings as one option among many agrees on all 25 deals it was run against.
One idea that did not work, kept here because a negative result is still a result: the obvious feasibility check — every remaining pyramid card of value v needs a distinct partner of value 13−v somewhere — is perfectly sound but never fires. It gave results identical to no check at all while making the search 80% slower, so it is not in the shipped solver.
The table is frozen, because solving 1150 deals takes minutes. To stop it going stale, a fresh sample of 60 deals from a different stretch of the shuffle is solved every time this page is built: it comes out at 6.7%, against the frozen 5.0%.
How to use
- Click two available cards that add to 13 — ace is 1, jack 11, queen 12, king 13.
- A king is 13 on its own, so a single click removes it.
- A pyramid card only becomes available once both cards resting on it have gone.
- Turn a card when no pair is available, and clear all 28 pyramid cards to win.
- Press "Can this deal be won?" for a proof either way, or an honest "could not decide".
Frequently asked questions
What share of Pyramid deals can actually be won?
It depends entirely on how many passes through the stock the rules allow, which is why a single number is meaningless. Solving deals outright rather than playing them: one pass clears 5.0 per cent, two passes clear 52.0 per cent, and three passes clear 68.5 per cent. Past that it flattens: four passes reach 70.0 and six reach 72.0, so about 28 per cent of deals cannot be cleared however long you cycle the stock.
Why does the second pass matter so much?
Because a single pass gives you one shot at each stock card in a fixed order, and a pair you needed has usually gone past by the time its partner is exposed. A second pass lets the stock come round again, which lifts the win rate by more than tenfold — far more than the third pass adds on top.
Are these win rates from playing or from solving?
From solving. For each deal the search either finds a way to clear all 28 pyramid cards or proves that no way exists, so the figures describe the deals themselves rather than how well anyone played them. All 800 deals were decided; none hit the search budget.
How can the solver be sure a deal is impossible?
It explores every distinct position, memoising on which pyramid cards remain so the many orders reaching the same position are only searched once. If it exhausts them without clearing the pyramid, no play could have. Crucially it reports "could not decide" separately from "impossible" — collapsing the two would turn giving up into a false proof.
Is taking a king always the right move?
Yes, and that is what makes the search tractable. Thirteen pairs with nothing, so a king always leaves on its own and removing it can never use up a partner another card needed. The solver takes kings immediately rather than branching on them, and a slower reference solver that does branch agrees on every deal tested.
Does this send anything anywhere?
No. Deals are shuffled and solved entirely in your browser and nothing is uploaded.
🔒 This tool runs entirely in your browser. Nothing you enter is uploaded, logged, or stored.