Pyramid Solitaire: grabbing pairs vs choosing them
Pyramid looks like a game of spotting: find two cards that add up to 13 and take them away. A 6 with a 7, a Jack (11) with a 2, a Queen (12) with an Ace, and a King (13) on its own. Twenty-eight cards sit face up in a triangle seven rows tall, and 24 more wait in the stock. Clear the triangle and you win.
The real skill is not spotting pairs but choosing between them. A greedy program that takes whatever pair it sees, looking only at face-up cards, won 25.2% of 4,000 deals (95% range 23.9–26.5%). A solver that plans every removal, and also knows the stock order, proved 62.2% of the same deals winnable (60.6–63.6%). More than half of the deals it won were deals the greedy approach threw away.
At a glance: what planning adds in Pyramid
| Player and rule | Result | 95% range | Deals |
|---|---|---|---|
| Greedy program, 3 passes (this site), sees only face-up cards | 25.2% won | 23.9–26.5% | 4,000 |
| Solver, 3 passes, knows the stock | 62.2% proven winnable | 60.6–63.6% | 4,000 |
| Solver, 3 passes | 14.0% proven impossible | 12.9–15.1% | 4,000 |
| Solver, 3 passes | 23.9% undecided | 22.6–25.2% | 4,000 |
| Solver, 1 pass only (comparison) | 7.3% proven winnable | 6.5–8.1% | 4,000 |
The solver ran out of search budget on 23.9% of deals, so the true winnable share with three passes is somewhere from 62.2% to 86.0%. The greedy result is a floor and the solver result is a ceiling for a player who cannot see the stock. No published study we use covers Pyramid, so there is no outside figure to check against.
The passes are planning time
This site gives you three passes through the stock. We also tested a single pass: the solver could prove only 7.3% of deals winnable and 34.8% impossible (33.4–36.3%), with 57.9% undecided. That test had a smaller search budget (200,000 positions per deal instead of 1,000,000), so the comparison is not exact. The direction is still plain: extra passes are what let you learn the stock and come back for the partners you need. A player who uses the first pass to plan gets far more from them than one who just keeps pairing.
Pyramid rules on spidersolitaireonline.us
- Triangle: 28 face-up cards in seven rows. A card is free when nothing in the row below covers it.
- Stock: 24 cards, flipped one at a time onto a waste pile, with three passes in all.
- Pairs: remove two free cards that total 13. Ace is 1, Jack 11, Queen 12.
- Kings: worth 13, removed alone.
- Stock and waste: the top waste card can pair with a triangle card. From our reading of the game code, the face-up top stock card can pair too.
- Win: clear the triangle. Cards left in the stock or waste do not count.
Thinking-ahead habits for Pyramid
When two cards could make the pair, pick the one that opens more
Two 7s may both match your 6. The greedy program takes whichever it meets first. A planner takes the one that frees more cards in the row above. Over thousands of deals, choices like this are much of the gap between 25.2% and 62.2%.
Plan around high Kings
In 40.8% of deals (39.3–42.3%) at least one King sat in the top three rows. It can only go once everything under it is gone, so it tells you which side of the triangle to clear first.
A dead start is rare
Only 3.6% of deals (3.0–4.2%) opened with no King and no pair available from the bottom row and stock. Nearly every deal gives you a way in; the question is which way.
Don't spend a stock card you will want later
If a stock card is the only partner for a buried triangle card, leaving it for the next pass can be worth more than using it now.
Let Undo stand in for foresight
You cannot see the stock like the solver does, but unlimited Undo lets you try a line, see what it uncovers and take it back.
Playing Pyramid here
Pyramid is free in any modern browser on computer, tablet or phone, with no download, account or email address. Large cards with zoom, Hint, unlimited Undo, Restart, Auto Play, and drag-and-drop or tap are all included. Wins, time and moves are tracked; sound can be switched off; fullscreen and table colors are available.
How we measured it
The spidersolitaireonline.us team ran these tests in September 2026 on this site's own Pyramid rules.
- Read the rules from our game code: 28-card triangle, 24-card stock, three passes, pairs to 13, Kings alone. We assumed from the code, without testing in a browser, that the face-up stock card can be paired directly.
- Rebuilt the game in Python, with a one-pass setting for comparison.
- Dealt deals 1 to 4,000 from fixed seeds and used the same deals for every test.
- Ran two players. A greedy program that sees only face-up cards, and a solver that knows the stock order and searches for a clearing line (1,000,000 positions per deal for three passes, 200,000 for one pass).
- Computed 95% ranges with the Wilson score method.
- Named the limits. The solver knows more than any player; a quarter of deals were left undecided; and if the stock-card assumption is wrong, the figures would move.
The code is Python 3.11, standard library only. The solver remembers positions that already failed and drops a line as soon as some triangle card has no partner left anywhere. Claude, an AI assistant from Anthropic, helped us write and review the code and draft this page. It produced none of the figures: all of them come from running the code, and the team checked them before publishing.
Pyramid strategy questions
Is Pyramid mostly luck?
Less than it looks. The same deals went from 25.2% won with greedy play to 62.2% proven winnable with full planning.
Are some deals impossible?
Yes. 14.0% were proven impossible even with the stock order known.
How many passes do I get?
Three on this site.
What is a similar quick game?
Golf has an even wider gap between greedy and planned play. For a deeper planning game, try Spider or FreeCell, or browse the games page.
By Admin, spidersolitaireonline.us. Research run September 2026.