Klondike Solitaire

For decades, Klondike — the green-felt card game everyone just calls "Solitaire" — shipped with Windows and was dealt billions of times. The rules feel simple: build the four suits up from Ace to King, moving cards between seven columns and flipping the face-down cards as you free them.

And yet there is a question about this game that nobody on Earth can answer: out of all possible deals, what fraction can be won if you play perfectly? Card players nicknamed it the solitaire problem, and the best estimates only say "somewhere around 80–90%."

That a game this popular, this old, and this small still hides an unsolved mathematical question is not an accident. The culprit is the very thing that makes Klondike fun — the cards you cannot see.

Here is a Klondike tableau. The columns each end in a face-up card; everything behind it is face-down — you cannot see it until you uncover it. Click a face-up card to "play" it; the hidden card beneath flips over.

<p class="hint">{{hint}}</p>
<div id="board" class="board"></div>
<div class="status" id="status">{{click_to_play}}</div>
<div class="btns">
  <button id="reveal" type="button">{{btn_reveal}}</button>
  <button id="deal" type="button" class="ghost">{{btn_new_deal}}</button>
</div>

* { box-sizing: border-box; }
body { font-family: system-ui, sans-serif; color: #222; margin: 0; }
.hint { font-size: .9rem; color: #444; margin: 0 0 .7rem; line-height: 1.45; }
.board { display: grid; grid-template-columns: repeat(7, 1fr); gap: 6px; margin: .4rem 0;
         background: #0d6b3f; padding: 10px; border-radius: 10px; }
.col { display: flex; flex-direction: column; gap: 3px; min-height: 120px; }
.card { height: 30px; border-radius: 5px; display: flex; align-items: center;
        justify-content: center; font: 700 14px ui-monospace, monospace; user-select: none; }
.down { background: #1d4ed8; background-image: repeating-linear-gradient(45deg,#1d4ed8,#1d4ed8 4px,#2563eb 4px,#2563eb 8px);
        border: 1px solid #1e40af; }
.up { background: #fff; border: 1px solid #cbd5e1; }
.up.red { color: #dc2626; }
.up.black { color: #111; }
.up.playable { cursor: pointer; box-shadow: 0 0 0 2px #fde047; }
.up.playable:hover { box-shadow: 0 0 0 2px #facc15; }
.empty { height: 30px; border: 1px dashed rgba(255,255,255,.4); border-radius: 5px; }
.status { font-size: 1rem; font-weight: 600; margin: .5rem 0; min-height: 1.4em; }
.status.ok { color: #0a7d33; }
.status.bad { color: #c92f3c; }
.btns { display: flex; gap: .5rem; flex-wrap: wrap; }
button { font: 600 14px system-ui, sans-serif; padding: .45rem .9rem; border: 1px solid #0d6b3f;
         background: #0d6b3f; color: #fff; border-radius: 8px; cursor: pointer; }
button.ghost { background: #fff; color: #0d6b3f; }

// Code not found

Notice the trap. Every choice you make changes which hidden cards you'll reach — and you are deciding without knowing what they are. Press Reveal everything to see how a single buried card can have made the deal impossible from the very first move. A perfect player would need to reason about every arrangement consistent with what's hidden, and there are astronomically many.

How hard is Klondike, really? It splits into two very different questions.

• With every card face-up (perfect information), a single deal becomes a search problem: try moves, backtrack, find a winning line if one exists. It's expensive but doable — in 2019, Charlie Blake and Ian Gent built a solver, Solvitaire, that decided millions of deals this way.
• The real game hides cards (imperfect information), so a perfect strategy can't assume any one arrangement. This is the version that matters, and the exact winnable fraction is a famous OPEN problem — what enthusiasts call the solitaire problem.
• The estimates only bracket it. Simulations and solvers put the perfect-information win rate near 82%, and human-style play far lower. The true value for optimal play under hidden cards has never been pinned down.
• Why it stays open: hidden information forces you to reason over a gigantic space of possible card arrangements at once, which is exactly the kind of thing that pushes a problem from "hard" toward "intractable" — the same wall behind P vs NP.

That is the punchline: the most-played computer game in history conceals a question mathematicians still cannot close. Its difficulty isn't bad shuffling — it's missing information made playable.

"Decide well when you can't see the whole state" is one of the deepest shapes a real problem can take, and Klondike is its friendly face:

• Games of imperfect information: poker and bridge bots wrestle with the same hidden-card reasoning that makes Klondike unsolved.
• Planning under uncertainty: a robot or a self-driving car acts on partial sensor data, never the full world — formally a partially observable decision problem.
• Medicine and diagnosis: a doctor chooses tests and treatments without seeing the hidden "true state" of the body, weighing what might be there.
• Teaching the limits of search: because everyone has played it, Klondike is one of the clearest ways to feel why hidden information makes optimal play so hard.

Understand why Klondike resists a clean answer and you've met decision-making under uncertainty — the same difficulty that shadows Minesweeper and countless real planning problems.

Klondike hides a beautiful secret in plain sight: a game played billions of times still guards an unanswered question. Reveal every card and a computer can grind out whether a deal is winnable; keep the cards hidden — as the real game does — and the exact odds of perfect play remain unknown to this day.

So the next time a deal strands you with no good move, take comfort. You haven't necessarily played badly. You may have run into the solitaire problem — a small, friendly puzzle that has quietly outlasted everyone who tried to solve it, much like the questions behind P vs NP.