Cribbage

A deal has four stages and they always come in the same order. You are dealt six cards and throw two of them face down into the crib, a fifth hand that belongs to whoever is dealing. A starter card is then cut and turned; it is not played, but it counts as a fifth card in every hand at the end. Next comes the play, where the two of you lay cards alternately onto a running count that may not pass thirty-one, pegging for fifteens, pairs and runs as you go. Finally comes the show, where the non-dealer counts their four cards plus the starter, then the dealer, then the dealer counts the crib. The deal alternates, so the crib comes round to you every other hand.

The board here is a real 121-hole board with two pegs a side. The front peg is your score and the one behind it is where you stood before your last count, which is exactly how a wooden board works and why you can see at a glance what the last hand was worth. Reaching 121 ends the game immediately — mid-count, mid-play, on the cut, wherever it happens.

Two scoring systems run side by side and they are not the same. During the play you score for what the cards on the table add up to; at the show you score for every combination inside your own five cards. Aces are always low and every court card counts ten, but a jack and a queen are still not a pair.

Twenty-nine. Hold the three fives that do not match the jack you are holding, and cut the fifth card as the five of that jack's suit. Eight combinations make fifteen — four pairings of a five with the jack, and the four ways of picking three fives — for 16 points, the six pairs among the four fives are another 12, and the jack scores 1 for his nobs. That is 29, and nothing beats it.

There are 12,994,800 ways to hold four cards and cut a fifth, and exactly four of them score 29 — one in 3,248,700. Deal a hand a minute, eight hours a day, and you would still be waiting the best part of two decades. Twenty-eight is far commoner at 76 combinations, nineteen times as many: four fives with any ten-count card except the jack the starter matches.

Four totals below the maximum cannot be reached at all: 19, 25, 26 and 27. That is why a hand worth nothing is called a nineteen — it is the score you can never really have. It happens more than you would like: 1,009,008 of those 12,994,800 combinations count zero, about one hand in thirteen. The average over all of them is 4.77 points, which is the number every lay-away decision is measured against.

There are fifteen ways to lay two cards away from six, and choosing between them is the single biggest decision in the game. The right question is never "which four are worth most?" but "which four are worth most once the starter lands, and what am I handing over?" — because the crib is yours on your deal and your opponent's on theirs. A seven-point keep that throws a 5-6 into their crib is usually worse than a six-point keep that throws a king and a four.

The opponent here does that arithmetic properly. For each of the fifteen lay-aways it counts the four you would keep against all 46 cards that could still be cut, exactly, and adds the value the two thrown cards bring to the crib — with the sign flipped when the crib is not its own. Counting the crib exactly would mean 683,100 five-card counts every time it lays away, so that half is sampled 240 ways per candidate from a generator seeded off the deal, which keeps it both fast and repeatable: the same deal always produces the same throw. That sampling is not free, and the honest figure is worth printing — it lands within about 0.18 of a point of the exact crib value, which changes the card it throws roughly one lay-away in ten and costs it about a tenth of a point a game.

It works. Measured over 3,000 deals with both sides laying away for value, the four cards kept average 8.0 points against the 4.77 a random four-plus-starter is worth, and the crib comes in at 4.77 as well. Against an opponent that throws and pegs at random, the same brain won 397 of 400 games, and four out of five of those wins were a skunk. Press Hint and it runs on your own six cards and tells you what it would throw and why.

Yes, and it is larger than beginners expect. The dealer owns the crib — a fifth hand nobody else gets to score — and never has to lead into the count. Counting last is the one thing the deal costs you: the non-dealer's compensation is the lead in the play and the first count at the show, and that order is what decides a photo finish at 120. In 1,600 games the computer played against itself here, the player who dealt the first hand won 889 of them — 55.6%, five and a half games in a hundred better than an even split. A game runs 8.9 deals on average, so each side turns only four or five cribs in the whole game. One crib's worth of advantage really is the entire edge.

The practical consequence is that the two seats want different lay-aways. On your own deal, throw cards that work together — a 5, two cards five apart, a pair, anything that builds fifteens. On your opponent's deal, break up your throw: a king and an ace go together about as badly as any two cards in the deck, because a king caps runs at one end and the ace is too small to make a fifteen with a ten-count.

• The crib flush. Four cards of a suit in the crib score nothing. The flush must run to all five, starter included. Plenty of implementations pay 4 there and quietly inflate every dealer's score.
• Thirty-one and the go. A card that brings the count to exactly 31 scores 2 and ends the series. It does not also collect the point for the last card — the 2 replaces it.
• Runs in the play read backwards. They come off the tail of the cards laid since the last reset, in any order, and an earlier card does not break them. Lay 3, 4, 3, 5 and that last five pays a run of three.
• You must play if you can. Holding a card back to save it for a better moment is not legal. If anything in your hand fits under thirty-one, it goes down.
• Double runs multiply, they do not extend. 4-5-5-6 is two runs of three and a pair — eight points, not a run of four.
• His nobs is a hand card. The starter being a jack is his heels, worth 2, and only to the dealer. The 1 for nobs needs a jack you are actually holding, matching the starter's suit.