Pyramid Solitaire

Pyramid: take it apart two cards at a time

Pyramid is solitaire by simple addition. Twenty-eight cards are dealt face up in a pyramid seven rows tall, and the whole game is one rule: any two cards you can reach that add up to thirteen leave the table together. A Queen goes with an Ace, a Jack with a two, a nine with a four — and a King, which is thirteen on its own, goes alone. Nothing is built, there are no foundations, and suits never matter. The other twenty-four cards are the stock, turned one at a time to supply partners, and you win by clearing the pyramid. Every card in it is visible from the first second, which is why a deal thrown away here stings more than almost anywhere else.

How do you set up Pyramid?

Shuffle one 52-card pack and deal a pyramid of seven rows: one card at the top, two below it, three below those, and so on down to seven across the bottom — 28 cards, every one face up. Each row overlaps the row above, so every card except those on the bottom row has two cards lying across its lower half. The remaining 24 cards are the stock, face down. That's the whole layout; there are no foundations to fill and nowhere to move a card except out of the game.

What adds up to 13?

Cards count their face value, with an Ace as one, a Jack as eleven, a Queen as twelve and a King as thirteen. Thirteen splits only six ways, so there are six pairs and one card that needs no partner at all:

Which cards can you play?

Three kinds of card are in play at any moment: any card in the pyramid that is free, the top card of the waste pile, and the card just turned from the stock. A pyramid card is free when nothing lies on it, and because every card above the bottom row is covered by two cards, both of them have to go first. At the deal only the seven cards along the bottom are free. Clear two neighbors there and the card they were both lying on comes free; clear one and the card above stays pinned, however much of it you can see.

The tempting mistake is the one in the picture: a card and the card lying across it that add up to thirteen. Under the 1949 rules, which this board follows, they can't pair — the four has to find a different nine, or the nine a different four. Some sites allow it once the covering card is the last one left, and so did Microsoft's early-1990s version, Tut's Tomb. It is not a small change. With three passes through the stock, allowing it lifts the deals a perfect player can win from 66.2% to 82.8%; with one pass, from 8.8% to 23.6%.

How do the stock and the waste work?

When nothing pairs — or whenever you like — turn the next card of the stock. The card you turn is in play at once. Turn another and the first moves onto the waste pile, where only the top card can be reached; everything beneath it waits until the cards above have been paired away. On the board above, the stock and the turned card sit in the pyramid's upper-left corner, the waste in the upper right, and clicking the stock turns the next card.

The turned card and the top of the waste can pair with each other. Albert Morehead and Geoffrey Mott-Smith put it plainly in 1949 — the top of the waste "may be paired with the next card turned from the stock, or with any released card in the pyramid" — and Microsoft's version plays it that way, but several of the sites that teach Pyramid leave it out and let only the waste top pair. How much it matters depends on the passes. With one, it more than doubles the deals a perfect player can win, from 3.6% to 8.8%, because a stock card you can't use straight away never comes round again. With three, it is worth about one deal in a hundred (66.2% against 65.2%): anything you couldn't pair the first time comes back.

How many times can you go through the deck?

When the stock runs out, the waste is turned over to become a new stock, in the same order as before minus whatever you paired along the way. How many times you may do that is the rule that varies most from book to book. Johnson and Pope's 1928 rules allowed a single pass; George Bonventure's 1930 book allowed one redeal; Morehead and Mott-Smith's Par Pyramid allowed two, which is three passes in all — the same three that Microsoft's version gives you. Some sites let you go round forever. The toggle above the board plays one pass or three.

The second pass is the one that matters. It takes perfect play from 8.8% of deals to 58.6%, because every card you turned and couldn't use comes round again. The third adds another 7.6 points, and going round forever adds only 0.4 more: once you can go through the stock three times, what beats you is the pyramid, not the stock.

Do you have to clear the whole deck, or just the pyramid?

Just the pyramid — and that is the older rule, not a modern softening. The earliest printed versions all stop when the pyramid is gone: Dean Bryden's 1927 Fun With Cards, which may be the game's first appearance; Johnson and Pope's 30 Games of Solitaire in 1928, probably the first book to use the name; and Bonventure's Tower of Twenty-Eight in 1930. It was Helen Coops in 1939 and then Morehead and Mott-Smith who demanded all 52 cards, and because their book became the standard reference, Wikipedia now describes the all-52 game as Pyramid and the original as "Relaxed Pyramid." Microsoft's version counts the pyramid, and so does this board. Whatever is left in the stock and the waste once the last pyramid card goes is simply left.

Can you win every game of Pyramid?

No — about a third of deals can't be cleared by anyone, even with three passes. A solver that knows the order of the stock, which is how every published win rate for Pyramid is measured, clears 66.2% of deals with three passes and 8.8% with one. Those are ceilings rather than targets, since the solver knows what's coming and you don't.

Honest play wins a good deal less. The computer player from the strategy section below, which never sees a card before it's turned, clears 31.5% of deals with three passes — close to the 30% that Solitaired reports from 355,570 games played on its site under nearly the same rules — and 1.2% with one, about one deal in 85. That one-pass figure is why Games Digest called the game next to impossible in 1938. The table shows where the gap between the two goes.

One kind of loss is visible before you play a card. Look for a card in the pyramid with all four of its partners among the cards resting on it, directly or through others — a six, say, with every seven somewhere in the triangle of cards below it. Each of those sevens has to leave before the six comes free, and a seven can only leave with a six, but there are only three other sixes. The deal is dead before it starts. That happens in 10.7% of deals (counted over 200,000), the same share Michael Keller found in 100 million. Most losses are quieter: a card whose last reachable partner was spent on something else a few moves earlier.

Strategy that holds up

Every habit below was tested the same way: a computer player that sees only what you see — the pyramid, the waste top and the turned card, never the stock — played the same 20,000 deals with the habit and without it, so the differences are measured on identical cards.

• Take Kings the moment they're free. A King pairs with nothing, so removing it can never cost you a move: it only uncovers the cards beneath it, or clears the waste or the turned card so the next card can be reached. Advice to hold Kings back is common, and there is no position in which it helps.
• Pair inside the pyramid before you spend stock cards. When more than one pair is on offer, preferring the one made of pyramid cards wins 7.5 more deals in a hundred than picking at random (±0.3). Doing the opposite — using the stock's cards first — loses 5.5 (±0.3). The pyramid is what you have to clear; the stock is only a supply of partners, and every one spent on the wrong card is gone.
• Of two pairs, take the one that frees more. If two different fours could go with your nine, take the one whose departure uncovers a card. That is worth another 3.9 deals in a hundred (±0.2) on top of the pyramid-first habit, and together the two habits take a three-pass player from 20.0% of deals to 31.5%.
• Count partners before you use them up. Every card left in the pyramid needs one of the four cards of its partner rank. Before pairing a seven from the stock with a six on the waste, check how many sixes are still in the pyramid and where the other sevens are. A deal is lost the moment the last reachable partner of a pyramid card is used somewhere else — the dead-on-the-deal pattern above, arrived at by your own hand.
• Treat the first pass as information, if you'll use it. With three passes you see every stock card on the first, and what you do on that pass matters more than anything else on this list. A player who remembers every card and plays perfectly afterward wins 58.6% of deals after simply watching the first pass go by, but only 35.4% after grabbing every pair on the way — the greedy pairs spend partners that turn out to be needed later. Taking only pairs inside the pyramid on the first pass costs far less (52.8%); it is the stock cards spent early that do the damage. The catch is memory. Our honest player, which remembers nothing, does much worse sitting the first pass out — 14.9% of deals instead of 31.5% — because for a player who won't use what it saw, a pass without pairs is a pass thrown away.

How is Pyramid scored?

This board simply tells you whether you cleared the pyramid, on which pass, and how many cards were left if you didn't — the number worth keeping a record of is how often you win at each setting. The traditional scoring is Morehead and Mott-Smith's Par Pyramid, which plays three passes and pays for speed: clear all 52 cards on the first pass and score 50; clear the pyramid on the first pass and score 50 less the cards you can't get out of the waste using the remaining passes; clear it on the second or third pass and score 35 or 20, less the waste in the same way; fail, and lose a point for every card left. Par is a net score of zero over six games. Microsoft's version and many others pay points for every pair instead, with bonuses for clearing whole rows that grow toward the apex.

Where does Pyramid come from?

Pyramid is almost certainly American, and younger than most people assume. It belongs to an old family of patiences that discard cards in pairs adding up to a total — David Parlett files it with Baroness, Block Eleven and the rest as simple addition games — but the overlapping 28-card pyramid itself first turns up in the late 1920s: Bryden's "Elizabeth's Solitaire" in 1927, then under its own name in 1928. Games Digest called it next to impossible in 1938. Morehead and Mott-Smith's Complete Book of Solitaire and Patience Games(1949) listed it as "Pyramid (Pile of Twenty-eight)," estimated the chance of winning at one in fifty and the playing time at four minutes, and put it into the standard American repertoire. Michael Keller, whose Solitaire Laboratory has traced the printed history, calls that one-in-fifty "guesswork," and his solver found 6.03% of deals winnable under those exact rules — all 52 cards, one pass.

Computers made it famous. Microsoft's second Entertainment Pack, at the start of the 1990s, shipped a version called Tut's Tomb that dealt the stock three at a time, and Pyramid has been part of Microsoft Solitaire Collection since it launched with Windows 8 in 2012 — which is where most people now learn its rules, and why three passes and a pyramid-only win feel standard.

Pyramid's relatives and variations

Relaxed and strict Pyramid differ only in the win: the pyramid alone (here) or all 52 cards. Par Pyramid is the scored three-pass game above. Overlap pairing — a card with the last card lying across it — is the rule that makes some online versions, and Tut's Tomb, easier. Giza deals the 24 stock cards face up below the pyramid instead of into a stock, so every card is visible and the game becomes a pure puzzle. And the pyramid shape turns up in games that aren't Pyramid at all: TriPeaks and Pyramid Golf lay cards out in peaks but play by Golf's rule, one rank up or down from the waste rather than pairs to thirteen. If that one-way, nothing-is-ever-built feel is what you like here, try Golf Solitaire next; if you want a solitaire where cards go back onto the table, the original is Klondike.