Pyramid Solitaire

Twenty-eight in a pyramid

How to play Pyramid Solitaire

Twenty-eight cards are dealt face up in seven rows, one card at the apex and seven along the base, each row overlapping the one above. The remaining twenty-four go face down to the stock.

Every card has a numeric value — ace is 1, jack 11, queen 12, king 13 — and the whole game is the number thirteen. Any two available cards that add to exactly thirteen come off together. Kings are already thirteen, so they leave on their own.

A card is only available when nothing covers it. At the start that means the seven cards along the base, and nothing else. Draw from the stock one card at a time; the top waste card can pair with the pyramid, and when it goes, the card beneath it is back in play. The deal is won when all twenty-eight pyramid cards are gone. What is left in the stock or waste does not matter.

The only arithmetic

Which cards pair with which

There are only seven combinations to learn, and after a couple of deals you stop counting and start seeing them.

Pairing rules

Which moves are legal

Nothing that leaves the pyramid ever comes back, and nothing moves between pyramid positions. Every move is a removal.

Clearing technique

Pyramid Solitaire strategy

We tested the usual advice by playing the same 100,000 deals several ways, none of which could see the order of the stock. The winner: before each draw, find the sequence of removals that clears the most pyramid cards, and make its first move. When two pairs compete for the same card, the right one is the pair that opens the way to more removals, and that is only visible if you look past the first move.

Take every king the moment it is free. A king needs no partner, and removing one only ever uncovers cards, so there is never a reason to wait. Using the waste card does not cost you stock, either: when it goes, the card underneath becomes the new top.

Count the partners you still need. There are four of every rank, so an ace in trouble has at most four queens to rescue it, and a queen the ace is covering, directly or through other cards, cannot help, because the ace is in its way. The Hint button plays the planning strategy above.

Three ways to run out

Where Pyramid deals get lost

Clearing the pair that opens nothing

When two pairs are available, they are almost never worth the same. Removing a pair from the far left of row seven uncovers one card; removing one that sits under a shared parent uncovers a card that unlocks two more. Always ask what the removal exposes, not just whether it is legal.

Drawing past a card you still need

Each draw covers the waste card. It comes back only when every card drawn on top of it has been removed, or, with passes left, on the next time through. On the last pass, look at what the waste card could pair with in the next move or two before you bury it.

Losing track of the partners

There are four of every rank. If three queens have gone and an ace sits high in the pyramid, one queen is all that ace has left — and if the ace is covering that queen, the deal cannot be won. Counting the partners of the cards you are stuck on tells you early whether to press on or start again.

Measured on this page's game

Is Pyramid Solitaire winnable?

With three passes, usually; with one, almost never. We played 100,000 shuffled deals through the same game code that runs at the top of this page, in both modes, and gave every deal to a solver that knew the order of the stock.

With one pass, only 3.9% of deals can be cleared at all — about 1 in 26. That is a little above the 3.4% Solitaire Laboratory reports for the same rules from Mark Masten's million-deal solver run, and his overlap figure sits a similar amount below ours. For the face-up stock card, where solitairewinrates.com has measured 100 million deals, our solver agrees (9.3% against 9.0%), and every win it claims is replayed through this page's engine. Careful play without seeing the stock clears 0.8%.

Three passes turn it into a different game. Knowing the order of the stock, 65.7% of deals can be cleared, and careful play clears 36%. On 30% of all deals the cards were there to win and careful play still lost, so there is plenty of room to play better than our strategy does.

A lost deal can end anywhere. Of the careful three-pass games that did not clear the pyramid, half finished with 7 or fewer cards left, and 15% of them with just one or two. When a deal is lost, the result screen asks the solver whether it could have been cleared, so you can tell a hopeless deal from a missed chance.

Rules that change the odds

Pyramid's rules, measured side by side

Pyramid is printed with several small rule differences, and each one moves the odds a long way. We ran the same solver on 20,000 of our deals under each ruleset.

Published figures: Solitaire Laboratory, from Mark Masten's solver over a million deals, for one pass with and without overlap discards; solitairewinrates.com, over 100 million deals, for the face-up stock card. Every figure here counts a deal as won when the pyramid is cleared.

The one- and three-pass rows are the two modes of the game above; the others show how far a single rule can move the odds.

Other names

Par Pyramid, Giza, and Pyramid's history

Solitaire Laboratory traces the modern game to Dean Bryden's Fun With Cards of 1927, and notes that Johnson and Pope's 30 Games of Solitaire, a year later, may be the first book to call it Pyramid. The three-pass mode here matches the variant known as Par Pyramid, in which the waste is turned over and dealt twice more.

Other relatives keep the thirteen rule and change something else: Relaxed Pyramid, Tut's Tomb, Apophis and Triangle among them. The best known is Giza, which deals every card face up and has no stock at all, so it plays as a puzzle rather than a gamble.

Other pairing games

Pyramid against the other clearing games

TriPeaks is the closest relative and the better game for a quick session: the same two-card blocking, but building a run one rank at a time means a single good card can clear half a peak. Pyramid asks for exact partners, which is stricter and slower.