Pyramid Solitaire: pairs that make 13, and odds better than you'd guess
Pyramid is the solitaire you can learn in thirty seconds. Cards are laid out in a triangle; you remove any two uncovered cards whose ranks add up to 13, and Kings go on their own. Empty the triangle and the game is yours.
Simple to learn doesn't mean simple to win. We had a simple "no peeking" computer player, one that sees only the face-up cards, play 4,000 games. It cleared the pyramid in 25.2% of them — about one in four. A full search that knew the order of the stock proved at least 62.2% of the same deals winnable. The distance between those two numbers is what this page is about.
Pyramid in numbers
| Measurement | Result | Deals | Player |
|---|---|---|---|
| Wins for a player who sees only face-up cards | 25.2% (95% range 23.9–26.5%) | 4,000 | Greedy bot, 3 passes |
| Deals proven winnable with the stock order known | 62.2% (range 60.6–63.6%) | 4,000 | Our solver, 3 passes |
| Deals proven impossible | 14.0% | 4,000 | Same solver |
| Deals not settled within the search limit | 23.9% | 4,000 | Same solver |
| Deals with no King or pair available at the very start | 3.6% | 4,000 | Reading the deal |
| Deals with at least one King in the top three rows | 40.8% | 4,000 | Reading the deal |
So with perfect knowledge, somewhere between 62% and 86% of deals can be cleared, and at least 14% can't. A casual player clears about a quarter. There's plenty of room between the two.
Pyramid rules as coded here
- The pyramid: 28 cards in seven rows, from one card at the top to seven at the bottom, all face up. Once no card overlaps it, a card is free to play.
- The stock: the other 24 cards. They're turned over singly onto a waste pile.
- Three passes. When the stock runs out you can turn the waste over and go through again — twice, for three passes in total.
- Pairs: remove two available cards whose ranks add to 13. Ace counts 1, Jack 11, Queen 12. So Ace + Queen, 2 + Jack, 3 + 10, 4 + 9, 5 + 8, 6 + 7.
- Kings count 13 on their own and are removed alone.
- Stock and waste cards can pair with pyramid cards or with each other.
- You win when the pyramid is cleared, even if cards are left in the stock or waste.
Three passes vs. one: what we found
We ran the solver a second time allowing only a single pass through the stock. With just one pass, it could prove only 7.3% of deals winnable (and couldn't settle 58% within a smaller search limit, so that figure is a lower bound). With three passes, at least 62.2%. Don't treat your second and third passes as leftovers; they're where most wins are made.
Beating the one-in-four baseline
Look for the buried Kings early
In 40.8% of our deals, at least one King sat in the top three rows. Those Kings can only leave once the cards below them are gone, so they decide which parts of the pyramid you need to clear first.
Don't always take the first pair
If two 6s are available and one 7, which 6 you use matters. Take the one that uncovers more of the pyramid.
Watch the stock on your first pass
You'll see all 24 stock cards once before your first redeal. Notice where the partners you need are. On the next pass, you'll know whether to hold a pyramid card for a stock card coming up.
Work evenly across the bottom row
Clearing one side of the base while the other stays full can leave the top of the pyramid locked. Spread your pairs.
Use Undo to compare choices
When two pairs are available and you're not sure which to take, try one, go through part of the stock, and undo if it goes badly. Our greedy bot couldn't do that.
Where Pyramid games slip away
- Pairing a card with a stock card when its partner in the pyramid would have uncovered more.
- Taking the first pair you see without checking alternatives.
- Rushing through the first pass without remembering the stock order.
- Giving up when the opening looks bad. Only 3.6% of deals start with nothing to do at all.
Play Pyramid here
Free to play in any browser, on desktop, tablet or phone — nothing to download, and no sign-up or email. The cards are large and easy to read, with a zoom control. You get Hint, unlimited Undo, Restart and Auto Play, and you can drag or tap to pair cards. Wins, time and moves are kept for the current game, and you can mute the sound, go fullscreen or change table colors. No gambling, and nothing to buy.
If you enjoy fast, card-matching games, try Golf next. For a longer classic, there's Klondike, and for a puzzle with every card visible, FreeCell.
The research behind these numbers, step by step
Pyramid was tested in September 2026 by the play-solitaire-free-online.us team at Solitairen OOD, following the rules coded on this site.
- Rules from our game code. A 28-card pyramid and a 24-card stock with two redeals, giving three passes; pairs adding to 13 are removed and Kings go alone; the game is won when the pyramid is clear, even if stock or waste cards remain. One assumption: the face-up top card of the stock can be paired directly, as well as the top of the waste. We took that from the code but didn't confirm it in a browser.
- Game rebuilt in code. We programmed Pyramid with three passes, and separately with one pass for comparison.
- Numbered deals. Seeds 1 to 4,000, our own deal numbers.
- Two players. A greedy bot that sees only face-up cards (a floor), and a solver that knows the stock order and searches up to 1,000,000 positions per deal (a ceiling no player can reach, because nobody knows the stock order).
- Running it. Both players on all 4,000 deals with three passes; the solver again with one pass, using a 200,000-position limit.
- Published comparison. Pyramid isn't among the games in Blake & Gent (2026), and we found no published figure for this exact version, so there's nothing to check ours against.
- Margin of error. The "95% range" is where the true share almost certainly lies. For proven wins with three passes: 60.6% to 63.6%.
- Limits. 23.9% of deals were undecided, so the honest answer is a range (62.2% to 86.0%), not one number. The bot is basic. And if the stock-card assumption is wrong, the figures would change.
The code behind the test
Pyramid is coded in pyramid.py, Python 3.11 with the standard library alone. 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 redeals are left. It remembers positions it has already failed from. It also cuts off a search early when some pyramid card has no possible partner left outside the cards covering it, since that position can never be won. The greedy bot shares the same game model but sees only face-up cards. runner.py spread the 4,000 deals over 19 worker processes on a 20-core machine, and the three-pass run took about 16 minutes. Percentages use Wilson score intervals.
We used Claude, Anthropic's AI assistant, to help write and review the Pyramid code and to draft this page. No figure came from the AI; all of them came from running the simulations, which we reviewed before publishing.
By Admin, play-solitaire-free-online.us (Solitairen OOD). Research dated September 18, 2026.
Questions new players ask
Which pairs add up to 13?
Ace + Queen, 2 + Jack, 3 + 10, 4 + 9, 5 + 8 and 6 + 7. A King is 13 by itself.
What share of Pyramid deals are winnable?
With the stock order known, at least 62.2% in our test and at most about 86%. At least 14.0% were proven impossible.
How many passes through the stock do I get?
Three times in total on this site. Those extra passes make a big difference: with one pass, our solver could prove only 7.3% of deals winnable.
Must the stock be empty to win?
No. Here you win as soon as the pyramid is cleared.
Is Pyramid mostly luck?
Less than it looks. A basic player won about one in four; a full search found wins in far more of the same deals. Choices matter.