Klondike Probability: What the Math Says

Klondike is the most-played solitaire variant on the planet, and until recently it was also one of the least formally studied. Everyone had an opinion about how often it was winnable; almost no one had done the math. The research history of Klondike probability is a decade-long project, mostly academic, to replace opinion with data. The headline finding is that a serious fraction of Klondike deals — roughly one in five — cannot be solved no matter how well they are played. That finding sounds obvious once you have it, but before the simulations existed it was disputed, and the specific bound was unknown.

The motivation for running the math is practical. A player who believes Klondike is 95 percent winnable will blame bad luck for every loss. A player who knows Klondike is 80 percent winnable under perfect play will take losses more honestly, studying position features and improving. The numbers also discipline claims. When a casual article writes "most Klondike games are winnable," we can check: yes, most are, but only if the player can approach solver-quality decisions. Under normal human play, the story is different. The research exists to separate those two claims.

This page summarizes what the research actually says, distinguishes between solvability (what a perfect player could do) and observed win rate (what humans actually do), and explains the methodology behind the numbers. We also flag what is still uncertain, because Klondike probability is a living field and the numbers have error bars that popular coverage usually omits.

The canonical Klondike solvability reference is Bjarnason, Fern, and Tadepalli's 2007 paper "Lower Bounding Klondike Solitaire with Monte-Carlo Planning," which ran Monte Carlo simulations to estimate a lower bound on solvability. Their finding, across a large sample of random deals with full observation of face-down cards, was a solvability rate of roughly 82 percent for the draw-one variant. The "lower bound" framing matters: the true solvability could be higher, because Monte Carlo samples do not exhaustively explore every branch. What the paper established is that at least 82 percent of deals are definitely solvable under optimal play with full information.

Later analyses pushed against that bound. Subsequent heuristic solvers, running on larger samples and with stronger search techniques, have reported draw-one solvability figures in the range of 82–91 percent depending on methodology and the fraction of deals declared unsolvable versus undetermined by the solver. The academic consensus remains that the true draw-one solvability rate is in the low eighties to high eighties. No paper has established a tight upper bound; the problem space is large enough that exhaustive proof of unsolvability on a single deal takes significant compute.

It is worth emphasizing that "solvable" in the research sense means a winning sequence exists under the stated rules and assumptions. It does not mean the game is easy, and it does not mean a player could realistically find the sequence. A solvable deal may require fifty moves of careful reordering, stock cycling, and foundation timing. The solvability rate is a theoretical ceiling; how close any player gets is a separate question with a separate answer, which is what the human-win-rate section of this page addresses.

Draw-three solvability is reported slightly lower, typically in the range of 78–82 percent, depending on assumptions about stock cycling and the restrictions modeled. Some implementations allow unlimited redeals; others limit to one or three passes. Every restriction pulls the number down. The honest read is that Draw 3 is a few points harder than Draw 1 for a perfect player and a lot harder for a human player, because the gap between perfect play and human play is wider in Draw 3.

A useful frame for the numbers: roughly one in five Klondike deals is unwinnable even under perfect play in Draw 1. That is the core finding. It does not depend on the player; it is a property of the deal itself. The remaining four in five deals are solvable, but solving them requires the kind of multi-step planning that humans rarely deliver under real-time conditions. The probability of winning under human play is therefore substantially lower than the solvability figure, and that gap is the focus of this page.

The research methodology matters for interpreting the numbers. Monte Carlo planning does not prove unsolvable; it samples moves and estimates success rates. A deal is classified as unsolved if the solver could not find a win within its computational budget. That classification is conservative: some "unsolved" deals would win under deeper search. The published figures are therefore lower bounds in the strict sense, and the true solvability rates may be a few points higher than the headline numbers.

The gap between optimal and human play in Klondike is wider than most players realize. Published solvability ceilings sit in the low eighties, but observed human win rates across the population sit much lower. Casual Draw-1 players land around 15–30 percent. Regular players who have read a strategy guide reach 40–55 percent. Experienced players with deliberate practice reach 60–70 percent. Expert players with long study histories climb into the 70s, and a hypothetical perfect player would sit at the 80–82 percent ceiling.

Draw 3 widens the gap. Casual players win 5–10 percent of Draw 3 games. Regular players reach 10–15 percent. Experienced players hit 15–20. Experts push into the 25–33 percent range. The solvability ceiling for Draw 3 under optimal play is somewhere between 78 and 82 percent depending on assumptions, so expert human play recovers only about a third of the available solvability. The rest is lost to imperfect cycle tracking, suboptimal move ordering, and the kinds of mistakes that are nearly unavoidable under real-time play conditions.

The population-level distribution tells a similar story. In aggregated player-data studies from various digital platforms, the bulk of players sit between 20 and 40 percent win rates in Draw 1 and between 8 and 18 percent in Draw 3. A minority, typically under 10 percent of the active player base, reaches the experienced tier. Expert-level play is rare. The distribution is not a failure of solitaire players; it is a reflection of how much of the game's skill depth goes unused under casual play conditions. Players who treat Klondike as a puzzle rather than a reflex game move through the distribution quickly.

We have also observed consistency effects. Players who improve their Draw 1 win rate from 25 to 50 percent typically do not see proportional gains in Draw 3, and vice versa. The skills overlap but do not transfer one for one. Draw 1 teaches sequencing and foundation discipline; Draw 3 teaches memory and cycle planning. A player who masters one mode and neglects the other can sit at different skill levels between them. That is why "your Klondike win rate" is meaningless without saying which mode.

The gap has practical implications. When a beginner asks "is Klondike winnable?", the truthful answer depends on which ceiling we are citing. The solvability ceiling is roughly 80 percent in Draw 1, so in principle most deals are winnable; but the human ceiling under realistic play is lower. Under casual play, only around a third of Draw 1 deals will be won. Under expert play, nearly three in four are won. That range defines what "winnable" means in practice.

One of the tactical questions that simulation can answer is when recycling the stock is expected-value positive. In Draw 1 with unlimited passes, cycling is essentially free: every pass reveals the same 24 cards, and the player chooses which to play on which pass. The EV of cycling there is positive until no new legal moves become available, at which point the game is locked.

In Draw 3, cycling has a real cost and a clear benefit. The cost is tempo and attention. The benefit is the cycle shift: every played card changes which cards are accessible on the next pass, so cycling is how we surface cards that were trapped. Simulation data suggests that the first three or four passes of the stock produce most of the accessible-card changes, and that diminishing returns set in after roughly the fifth pass. Players who cycle more than six or seven times without progress are usually in a locked position whether they recognize it or not.

The expected value of a specific cycle is computable when the stock composition is known. If we need a particular ace, and we know its position in the stock, we can compute how many passes are required to bring it to the accessible position and whether those passes conflict with other card-retrieval goals. Simulation work formalizes this as a small dynamic program: each potential play is weighted by the number of cards it would unlock on the next pass, and the player chooses the play with the highest expected unlock. A heuristic solver using this rule plays Draw 3 significantly better than one that plays greedy-best-tableau-move, which is a useful piece of evidence that stock-cycle planning is the dominant Draw 3 skill.

Pass-limited Draw 3 changes the math. When the rules allow only three passes through the stock, each pass becomes strategically scarce. A pass spent without a stock play is nearly wasted; a pass that produces two or three productive plays is well spent. Under three-pass rules, the solvability ceiling drops a few points because some deals that would have been solvable under unlimited passes run out of passes in the restricted mode.

Not all Klondike deals are equally hard. Some opening configurations are significantly harder to solve than others, even before any play begins. Simulation work on starting-position features identifies a few properties that predict difficulty: the number of face-down Kings in deep columns, the distribution of aces across tableau versus stock, and the density of same-suit pairs in the top face-up row.

These features have been studied piecewise rather than in a unified model. We have rough estimates of how much each factor shifts solvability, but no single formula that combines them into a deal-specific prediction.

Face-down Kings are expensive because Kings can only land on empty columns, and empty columns require clearing a column first. A deal with all four Kings buried deep in the tableau has a harder time producing empty columns on demand, because the same Kings will need to come out before they can serve as column anchors. Deals with Kings already face-up on the tableau or accessible in the stock are significantly easier.

Aces matter similarly. An ace buried under several face-down cards takes longer to reach, delaying the foundation start and often cascading into lost tempo. Aces already visible on the tableau or reachable in the stock's top group reach the foundation within the first few moves. The best opening configuration is one where all four aces are accessible within the first pass of the stock.

A third feature that matters is the suit distribution in the top face-up row. A row that contains all four suits offers more flexibility for building alternating sequences than a row where three suits share the same color. Red-red-red starts force reliance on stock draws for any black card, delaying tableau building. Mixed suit starts allow immediate alternating-color joins from move one, which accelerates column emptying. The effect size is modest — a few percentage points in solvability — but it is consistent across simulation samples.

Face-down density is the broader signal. A deal with heavy face-down blocking in multiple columns has more work to do before sequences can form. Heuristic solvers use these position features to prioritize early work and to flag deals that are likely to be unsolvable, saving compute for the marginal cases.

Every solvability number circulating on the web has an error bar, and most presentations omit it. A claim that Klondike is "82 percent solvable" means nothing without a sample size and a confidence interval. A run of 1,000 simulated deals gives a different margin of error than a run of 1,000,000. Published research usually reports 95 percent confidence intervals, and those intervals are wider than readers expect.

Confidence intervals are easy to state and harder to internalize. A 95 percent interval means that if we ran the experiment many times, 95 percent of the intervals we produced would contain the true parameter. It does not mean there is a 95 percent probability the true value is in any specific interval. That subtlety matters when headlines turn research findings into single numbers. An "82 percent solvable" claim is a point estimate, and the point estimate sits inside an interval that includes a range of possible true values. Good popular coverage reports the interval. Most does not.

For a simple proportion estimate, the 95 percent confidence interval scales roughly with the inverse square root of the sample size. A sample of 10,000 deals with an observed 82 percent solvability produces a confidence interval of roughly plus or minus 0.8 percentage points. A sample of 1,000 widens that to plus or minus 2.4 points. A sample of 100 widens it further, to plus or minus 7.5 points. Small-sample claims should always be read with that margin in mind.

Sample size matters a second way: the deal generator. A study that samples from a uniform random distribution of 52-card permutations is measuring the underlying game. A study that samples from only Microsoft-numbered deals is measuring that specific subset — which may have systematic properties that a uniform sample does not. Most published research uses uniform random samples, which is the right default for game-wide claims. Claims about specific deal numbering schemes require their own samples and their own confidence intervals.

The published Klondike literature uses samples large enough to produce tight confidence intervals on solvability, but the methodology caveats remain. Every heuristic solver makes tradeoffs about search depth, time budget, and unsolved-deal treatment, and each tradeoff shifts the reported number slightly. A range of 78–85 percent across serious studies is not a failure to converge; it is the reality of running different analyses on different samples with different solvers.

Running our own Klondike simulations is straightforward in principle. We generate uniform random permutations of a 52-card deck, deal them into the 28-card tableau plus 24-card stock configuration, and run a solver. The solver either finds a winning move sequence and returns "solvable," finds no win within its budget and returns "undetermined," or proves no win exists and returns "unsolvable." We aggregate across a large sample and report solvability as the fraction of deals that returned "solvable."

Deal generation matters for reproducibility. A named, seeded random deck lets different researchers run the same deals and compare solver results directly. Without a shared seeding convention, every paper generates its own private deal set, which makes cross-study comparison suggestive rather than decisive. A small open benchmark of labeled Klondike deals would resolve much of that friction.

The solver design is the interesting part. Heuristic solvers use domain knowledge — ace priority, King placement, face-down reveals — to prioritize moves, running a depth-limited search with pruning. Exhaustive solvers explore every reachable move sequence, guaranteed to find a win if one exists but prohibitively slow for some deals. Most published Klondike research uses heuristic solvers with a fixed compute budget, which is why results are framed as lower bounds.

Time budget is the operational constraint. A deal that is solvable in principle may be unsolvable within a five-second compute budget per deal. Research papers usually disclose their per-deal budgets and the fraction of deals the solver could not finish. That fraction is an important caveat because it sets an upper bound on solvability: a run that budgets five seconds per deal and fails to solve ten percent of deals within that budget cannot claim a solvability rate above ninety percent on that sample. Longer budgets push the number up; shorter budgets push it down. When we compare two papers' numbers, we compare their budgets first.

Face-down treatment matters. Under full-observation simulation, the solver sees every card in the tableau including the face-down ones. Under partial-observation simulation, the solver only sees what a human player would see. Full-observation solvers report higher solvability because they can plan around cards a human would not yet know about. Partial-observation solvers report numbers closer to the human ceiling. Both are valid research targets; they answer different questions.

The Klondike research field has unresolved questions worth naming. First, the true solvability ceiling is still a range rather than a single number. Published bounds vary from the low eighties to the high eighties depending on methodology, and no exhaustive-search result has closed the gap. Second, the partial- observation ceiling, which is closer to what human play can achieve, is less well characterized than the full-observation ceiling. Third, the interaction between redeal limits and solvability is underexplored; most research assumes unlimited passes, but real digital implementations vary.

There are also methodological gaps. The community has no shared benchmark suite of Klondike deals the way chess has tactical-test suites. Different solvers are evaluated on different samples, which makes direct comparison hard. A common test set of, say, 10,000 labeled deals would advance the field. We would like to see someone publish one.

A fourth open area is the relationship between stock position and solvability. We have qualitative evidence that deep-buried aces and kings matter, but we lack a formal model connecting stock composition to solvability probability. A characterization of, for instance, the conditional solvability of a deal where the ace of hearts sits at stock position 23 would let us tell players which stock orders tend to favor Draw 3 play and which tend to block it. That sort of result would be useful to strategy writing and to teaching.

Fifth, we have limited data on how human win rates interact with deal difficulty. It is plausible that strong human players win a larger fraction of easy-to-solve deals than weak players do, but it is also plausible that the gap closes on hard deals where nobody plays well. We would like to see a study that labels deals by their solver difficulty and cross-tabulates with observed human performance. That study would tell us, in quantitative terms, which parts of the skill curve are learnable and which are mostly luck-bound.

Finally, the relationship between opening-position features and solvability is still heuristic rather than formalized. We know that buried Kings and aces make deals harder, but we do not have a single predictive model that takes a starting position and returns a solvability probability. Such a model would let players identify likely-unsolvable deals early and concede efficiently — useful in Vegas scoring especially.