Pyramid Solitaire: how the three passes change everything

By Admin · Research by the free-solitaire.nz team · Figures from our September 2026 simulation study

Pyramid Solitaire has one of the simplest ideas in card games: remove pairs of cards that add up to 13, and clear the pyramid. It also has a reputation for being nearly impossible to win. So we tested it.

We ran a solver on 4,000 random deals using the rules on this site, where you get three passes through the stock. With perfect play, at least 62.2% of deals can be won — and with a single pass, fewer than one in ten could be proven winnable. The redeals are what make our Pyramid a fair game.

Pyramid at a glance

Because some deals were left undecided, the true share of winnable deals lies somewhere between 62.2% and 86.0%. Either way, most Pyramid deals on this site can be cleared.

How to play Pyramid Solitaire

The layout

• 28 cards dealt face up in a pyramid of seven rows, each row overlapping the one above.
• The remaining 24 cards form the stock. The top stock card is face up, and cards are turned onto a waste pile.
• On this site you can go through the stock three times (two redeals).

The goal

Clear every card from the pyramid. On this version, the game is won as soon as the pyramid is empty, even if cards remain in the stock or waste.

How cards are removed

• A card is available when no card overlaps it from the row below.
• Remove two available cards that add up to 13. Jacks count 11, Queens 12, Kings 13, Aces 1. So Queen + Ace, Jack + 2, 10 + 3, 9 + 4, 8 + 5 and 7 + 6.
• A King is worth 13 on its own and is removed by itself.
• The top stock card and the top waste card can be paired with pyramid cards or with each other.

Why the redeals matter so much

With one pass, every stock card gets exactly one chance. If its partner is still buried when it goes by, it is gone. With three passes, the stock becomes a resource you can come back to: you can uncover a card in the pyramid, then pick up its partner on the next time through.

That is why our single-pass run could prove only 7.3% of deals winnable, while the three-pass run proved 62.2%. (The single-pass run also had a smaller search budget and left 58% of deals undecided, so its true figure is higher than 7.3% — but the gap is still dramatic.)

Strategy that follows from the numbers

1. Find the buried Kings first

In 40.8% of our deals, at least one King sat in the top three rows of the pyramid. Kings clear themselves, but only once uncovered, and a King near the top blocks everything above it from being freed. Plan a path to it early.

2. Look for partners in the same “shadow”

A card can never be paired with a card that covers it. So if a 9 sits higher in the pyramid and a 4 lies in the rows underneath it, that 4 cannot be its partner — the 9 needs one of the other 4s, from elsewhere in the pyramid or the stock. Our solver used this rule to prove deals impossible: when a pyramid card has no possible partner left outside the cards covering it, the game is lost.

3. Clear the pyramid, not the stock

On this site you win when the pyramid is empty. Pairs made purely from stock and waste cards help only when they uncover something useful. Prefer pairs that free pyramid cards.

4. Use your first pass to learn

With three passes, your first run through the stock is partly reconnaissance. Note which cards are coming, make the safe plays, and save the tricky decisions for the second pass. Unlimited Undo on this site makes this even easier.

5. Don't panic at a slow opening

Only 3.6% of our deals offered no King or pair at all on the opening layout (bottom row plus the top stock card). Most deals give you something to do straight away.

Common mistakes we see

• Pairing the easy cards first. A pair from the bottom row is tempting, but if it uses up the only partner for a card higher in the pyramid, you may have just lost the game. Check what each pair does to the cards above it.
• Burning the stock too fast. Three passes sounds generous, but each card you flip past without using is only coming back on the next pass. Pause before each turn.
• Forgetting the waste pile. The top waste card is still in play. It can pair with the stock card or a pyramid card — easy to overlook.
• Not tracking the four of a kind. There are exactly four cards of each rank. If three 9s are gone and the last 4 is buried under the pyramid, the fourth 9 needs that 4 — plan for it early.

A simple counting habit helps: each partner rank (Ace and Queen, 2 and Jack, and so on) forms a family of eight cards that must pair off four times. When one family is running short in the stock, the matching pyramid cards need to be freed while partners are still around.

Perfect play versus real play

Our solver knows the order of the stock, which no player does. Our greedy bot does not: it sees only the face-up cards and grabs the first good pair, and it won 25.2% of 4,000 deals with three passes. Careful players who plan ahead and use Undo will land somewhere between the bot's 25% and the solver's 62%-plus.

Playing Pyramid on free-solitaire.nz

Pyramid here is free and needs no download, no sign-up and no email. It runs in any modern browser on phone, tablet and computer.

• Large, clear cards, with a zoom control for the whole pyramid.
• Hint, unlimited Undo, Restart and Auto Play.
• Tap or drag and drop to pair cards.
• Wins, time and moves tracked for your current game.
• Sound off, fullscreen mode and a choice of table colours.

No gambling, nothing to buy, and no clock ticking — ideal for a quiet evening in Christchurch or a coffee break anywhere.

How we did the research, step by step

The free-solitaire.nz team ran this study in September 2026. These are the steps we took, in order.

1. Read the rules from our own game. From this site's game code (“Pyramid Easy”): a 28-card pyramid and a 24-card stock with three passes. Two uncovered cards adding to 13 are removed together, and a King is removed on its own. The game is won when the pyramid is cleared, even if stock cards remain.
2. Wrote down one assumption. We treated the face-up top card of the stock as available to pair directly, alongside the top of the waste pile. We read this from the game code but did not test it card by card in a browser. If the site behaves differently, the figures would shift.
3. Rebuilt the game in code. We wrote a program that plays Pyramid by those rules.
4. Generated numbered deals. 4,000 deals, each from a fixed seed, so every figure can be reproduced.
5. Ran a solver that knows the stock order. A depth-first search that remembers positions it has already seen, allowed up to 1,000,000 positions per deal. It also stops early on a position it can prove is hopeless.
6. Sorted every deal into three groups. Solved: 62.2% (95% range 60.6–63.6%). Proven impossible: 14.0%. Undecided when the budget ran out: 23.9%. So between 62.2% and 86.0% of deals are winnable.
7. Measured what the redeals are worth. With a single pass and a 200,000-position budget, only 7.3% of deals were proven winnable, with 58% left undecided.
8. Ran a player that only sees what you see. A greedy bot that sees only face-up cards and never undoes won 25.2% of the same 4,000 deals.

We are not aware of a published winnability figure for this exact version of Pyramid, so there was no outside result to compare against. Ranges are 95% confidence intervals.

The technical side

Pyramid runs in pyramid.py, written in Python 3.11 with only the standard library. A shared runner.py uses Python's multiprocessing to play 19 deals at once, and each deal is shuffled from a numbered seed.

• The solver is a depth-first search that knows the stock order. It tracks which pyramid cards are gone, which stock cards are used, where you are in the stock and how many redeals remain, and it remembers every position that failed so it never retries one. It also cuts a line short as soon as some pyramid card has no possible partner left, which is a sound test for a hopeless position. Each deal gets 1,000,000 positions; the single-pass comparison used 200,000.
• The greedy bot sees only face-up cards.
• Scale. The 4,000 deals took about 16 minutes.
• The 95% range is a Wilson score interval.

The impossibility test only removes positions that genuinely cannot be won, so a deal the solver rules out is truly lost, not merely skipped. We used an AI assistant (Claude, by Anthropic) to help write and review the code and draft this page. The AI did not produce any figure; every number comes from running the code, and the team checked the results before publishing.

Frequently asked questions

Is Pyramid Solitaire really that hard?

With a single pass through the stock, yes. With the three passes on this site, most deals can be won with good play — our solver proved at least 62.2%.

Do I have to clear the stock too?

No. On this version you win as soon as the pyramid is empty.

What adds up to 13?

Queen + Ace, Jack + 2, 10 + 3, 9 + 4, 8 + 5 and 7 + 6. Kings are 13 on their own.

How long does a game take?

Usually five to ten minutes.

Do I need an account?

No. Nothing to install and no sign-up.

What should I try next?

For another quick clearing game, try Golf Solitaire. For something longer, try Spider Solitaire or classic Klondike. See every game on our games page and read more research on the blog.