Pyramid Solitaire: where are the Kings, and why it matters
Pyramid is a matching game. Twenty-eight cards form a triangle of seven rows, and you remove pairs that add up to 13 — a 6 with a 7, a Queen with an Ace, a Jack with a 2 — while Kings, worth 13 on their own, go by themselves. Only cards that aren't covered by others can be played.
That makes the position of the Kings a big part of reading a Pyramid deal. We dealt 4,000 games under this site's rules. In 40.8% of them at least one King sat in the top three rows of the pyramid, where it's buried under almost everything. In 3.6%, there was neither a King nor a pair available on the very first move.
How winnable are the deals overall? A solver that knew the stock order proved 62.2% of deals winnable and 14.0% impossible, and couldn't settle the remaining 23.9% within its limit. A greedy player that couldn't see the stock won 25.2%.
A Pyramid deal, by the numbers
| What we looked at | Result | Deals | How we got it |
|---|---|---|---|
| At least one King in the top three rows | 40.8% (95% range 39.3–42.3%) | 4,000 | Inspecting the deal |
| No King or pair available at the start | 3.6% (3.0–4.2%) | 4,000 | Inspecting the bottom row and the stock's top card |
| Proven winnable, three passes | 62.2% (60.6–63.6%) | 4,000 | Solver that knows the stock order, 1,000,000 positions |
| Proven impossible, three passes | 14.0% | 4,000 | Same solver |
| Undecided, three passes | 23.9% | 4,000 | Same solver |
| Proven winnable with only one pass | 7.3% | 4,000 | Same solver, 200,000 positions |
| Greedy player's wins, three passes | 25.2% | 4,000 | Sees only face-up cards |
Put together, between 62.2% and 86.0% of deals can be won with three passes if you knew the stock order.
How Pyramid is played on klondike.nz
Building the triangle
Twenty-eight cards in seven rows, from one card at the top to seven at the bottom, each row overlapping the one above. The other 24 cards form the stock, with the top card face up.
Winning
Clear the pyramid. On this site the game is won when the pyramid is empty; cards left in the stock or waste don't count against you.
Moves
- Remove two uncovered cards that add up to 13. Aces count 1, Jacks 11, Queens 12.
- Remove a King on its own.
- The face-up top card of the stock and the top card of the waste can be paired with pyramid cards or with each other.
- Turn stock cards to the waste. You get three passes through the stock.
Reading a Pyramid deal
Find the high Kings
A King in the top three rows (as in 40.8% of our deals) can't be removed until most of the pyramid below it is gone. It's not a problem in itself, but it tells you which part of the pyramid must be cleared completely.
Look for "orphan" cards
Each card needs a partner: there are four cards of every rank, so a 6 has four possible 7s. If every 7 still in play is sitting on top of that 6 in the pyramid, the 6 can never be removed. Our solver used exactly this test to prove deals impossible.
A frozen start isn't fatal
Only 3.6% of deals begin with no move at all. When it happens, you start from the stock.
Use all three passes
With a single pass through the stock, our solver could prove only 7.3% of deals winnable. With three passes, 62.2%. The extra passes are the whole game: hold a card back when its partner will come round again.
Don't pair the first thing you see
Our greedy player, which couldn't see what the stock held, won 25.2%. If a 9 in the pyramid and a 9 in the stock both fit your 4, choose the one that uncovers more.
A first-minute checklist for Pyramid
All 28 pyramid cards are visible from the start, so you can plan before you touch the stock. This routine is built on the figures above.
- Find the Kings. Kings on the bottom row can go at once. Kings in the top three rows — 40.8% of deals have at least one — tell you which part of the pyramid must be cleared completely.
- List the bottom-row pairs. These are your immediate moves. Also check the face-up stock card against them.
- Check for trapped cards. For any card deep in the pyramid, look at where its possible partners are. If every one still in play sits on top of it, it can never be removed.
- Plan for three passes. If a card's partner is further down the stock, you'll see it again. Don't force a weak pair now.
- Choose pairs that uncover. When two different cards would both pair with the same card, take the one whose removal exposes more of the pyramid.
- Don't worry about the stock at the end. On this site you win once the pyramid is empty.
Three ways players misread a Pyramid deal
- "No move at the start means a dud." It happens in only 3.6% of deals, and even then the stock gives you moves.
- "Take every pair as soon as it appears." The stock comes round three times. Holding a card back for a better partner is often the winning play.
- "A King near the top is a lost game." It happens in 40.8% of deals. It means more of the pyramid must be cleared, not that the game can't be won.
The deals that really can't be won are usually ones where a card has no reachable partner — which is worth checking early.
Play Pyramid on klondike.nz
Free in any modern browser, on desktop, tablet or phone. No download, no sign-up, no email. Large cards with zoom; Hint, unlimited Undo, Restart and Auto Play; tap or drag; wins, time and moves tracked for the current game; sound, fullscreen and colour options. A tidy little puzzle for a morning coffee.
Where these numbers come from
The klondike.nz team ran this study in September 2026.
- Rules from our code. A 28-card pyramid, 24-card stock, pairs to 13, Kings alone, three passes, and a win when the pyramid is cleared. We assumed the face-up top stock card can be paired directly; we took that from the game code but didn't confirm it in a browser.
- A program version. Rebuilt to those rules.
- Numbered deals. Seeds 1 to 4,000.
- Looking at each deal. We checked where the Kings were and whether any move was available at the start.
- Two players. A solver that knows the stock order (1,000,000 positions per deal for three passes, 200,000 for one) and a greedy player that doesn't.
- Comparison. The published study we compare against doesn't include Pyramid, so there's nothing to check against.
- Error bars. 60.6–63.6% is the 95% range around 62.2%.
- Limits. The solver knows the stock order, so it's a ceiling; 23.9% of deals were left undecided.
Under the bonnet
The simulation is written in Python 3.11 with nothing beyond the standard library, in pyramid.py. A shared runner spreads the deals over 19 worker processes using multiprocessing; the three-pass run took about 16 minutes. The solver is a depth-first search that remembers positions it has already ruled out and cuts off any position where some pyramid card has no possible partner left — a sound test for impossibility. Wilson score 95% intervals give the ranges. We used Claude, an AI assistant by Anthropic, to help write and review the code and draft this page; the figures come only from running the code.
Questions players ask
Do leftover stock cards stop me winning?
No. On this site you win when the pyramid is empty.
How many times can I go through the stock?
Three passes in total.
Is a King near the top a bad sign?
It means more work, not a lost game. It happens in 40.8% of deals.
How many Pyramid deals can be won?
At least 62.2% with perfect knowledge of the stock, and up to 86.0%.
What else is quick to play?
Golf Solitaire. Or go back to Klondike.
By Admin, klondike.nz. Study date: 18 September 2026.