TriPeaks Odds & Streak Math

The board in numbers

Every TriPeaks deal starts from the same arithmetic. Twenty-eight cards form the three peaks — rows of 3, 6, 9, and 10 from the apexes down — one card seeds the foundation, and the last 23 cards of the deck become your stock. To win, all 28 peak cards must reach the foundation, but you only have 23 stock draws to restart stalled sequences. That deficit is the entire tension of the game: five of your foundation cards, at minimum, must each carry a chain of two or more plays. In practice winning boards usually feature a couple of long chains rather than many short ones, which is why the probability of extending a run matters more than the probability of any single play.

The base row of ten is fully exposed on move one, and each of the 18 covered cards is pinned by exactly two cards from the row beneath it. As the board empties, the number of exposed cards swings between roughly four and ten, and every exposed card is an independent chance to continue your current run. More exposed cards means better odds on every foundation flip — a fact that quietly rewards spreading your removals across all three peaks instead of tunneling into one.

How likely is any card to be playable?

Ignore suits — TriPeaks does — and think purely in ranks. Against a given foundation card, a random unseen card is playable if it sits one rank up or one rank down. With King-Ace wrapping enabled (our Easy setting), every rank has exactly two neighbors, so 8 of the other 51 cards match: about a 15.7% chance per unseen card. Without wrapping (Medium and Hard), Aces and Kings each have only one neighbor. Averaged over a random foundation rank, the expected number of matches drops to 96/13 ≈ 7.4 of 51 cards, or roughly 14.5% per card.

A percentage point sounds trivial, but it is not the real cost. The real cost is where the missing links sit. Without wrapping, the rank ladder is a line segment with two cliff edges; with wrapping it is a circle. Chains die at cliff edges. Eight of the 52 cards in the deck are Kings and Aces, and in a no-wrap game every one of them is a potential chain terminator — either a wall you cannot play onto from one side, or a dead end once played.

Expected run lengths

A run continues as long as at least one exposed card matches the card you just played. If E cards are exposed and each unseen card matches with probability p, the chance of at least one continuation is roughly 1 − (1 − p)^E. With eight cards exposed and wrapping on (p ≈ 0.157), that is about a 75% continuation chance per step, which implies an average run of roughly four cards from a favorable start. With four cards exposed late in a no-wrap game, continuation falls near 47% and the expected run shrinks toward two.

These are back-of-the-envelope figures — real boards have correlated ranks, and skilled players choose the branch that keeps a run alive rather than playing the first legal card — but the direction of the math is what matters. Runs get shorter as the board empties, so the cheap chains come early and the expensive singles come late. Budget your 23 draws accordingly: an early draw spent while plays still exist is the most expensive mistake in TriPeaks.

Robert Hogue, who invented TriPeaks in 1989, ran a statistical analysis of the original game and found that over 90% of deals are solvable in principle. You will not clear 90% — nobody does — but the gap between the theoretical ceiling and a typical player's clear rate is exactly the space that look-ahead, chain selection, and draw discipline compete for. On our fully open Easy and Medium boards, perfect information puts that ceiling within reach of careful planning; on Hard, the 18 face-down covered cards make perfect play impossible by design, and clear rates fall accordingly.

Triangular streak scoring, and why it warps strategy

Scored versions of TriPeaks — including the original — traditionally pay 1 point for the first card of a streak, 2 for the second, 3 for the third, and so on. A streak of N cards is therefore worth N(N+1)/2 points, the classic triangular number. The consequence is dramatic: doubling a streak roughly quadruples its value.

Under a bonus system like this, optimal play stops being "clear the most cards" and becomes "concentrate your clears." It can be correct to decline a lone playable card, draw from the stock, and hunt for a longer chain instead — the marginal card of a 12-streak is worth twelve times the opening card of a new one. It also changes fork decisions: at a choice between two legal plays, the score-maximizer always traces both branches and takes the longer chain, even when the shorter branch unlocks a more useful covered card.

Our version keeps score the way most solitaire purists prefer — wins, losses, and best times rather than points — so here the streak table reads as an efficiency chart instead: cards cleared per stock draw. The strategic lesson survives the translation. Long chains are how you beat the 28-versus-23 deficit, and a play that extends a live run is almost always better than one that strands you. When win-rate and chain-length goals conflict — a chain-breaking play that frees a peak apex, say — take the win. See the strategy guide for how to spot those exceptions.

What Easy wrapping does to clear rates

Turning on King-Ace wrapping looks like a small rules toggle and plays like a different game. Mechanically it adds two links to the rank circle, taking the average per-card playability from about 14.5% to 15.7%. Strategically it does three bigger things. First, it deletes dead ends: a sequence like Queen-King-Ace-2 becomes one continuous four-card chain that no-wrap rules would cut in half. Second, it makes every rank equal — you no longer route plans around the eight Kings and Aces in the deck. Third, it compounds with the continuation math above: raising the per-step continuation chance a few points multiplies through every step of every chain, so long runs become disproportionately more common.

Stack the toggles and the three difficulties separate cleanly. Easy gives you wrapping plus a fully face-up board — the highest ceiling and the most solvable deals. Medium keeps the open board but restores the cliff edges at King and Ace, cutting chains shorter and making stock discipline the deciding skill. Hard removes both crutches: no wrapping, and the 18 covered cards start face-down, so you are estimating rather than calculating. Track your own clear rate per difficulty in the stats panel — the spread between your Easy and Hard percentages is a decent measure of how much of your game is planning versus pattern luck. The full rules cover each setting's exact behavior.

Odds cheat sheet