Play Pig Online

What Is Pig?

Pig is the smallest possible push-your-luck game. One die, two players, and one decision repeated until somebody reaches 100: roll again, or hold? Every roll of 2 through 6 adds to your turn total, and holding moves that total to your score. Roll a 1 and the turn total is gone.

Because there is nothing else to think about, Pig makes the push-your-luck trade-off visible in a way that busier games hide. You can calculate the value of one more roll in your head, then watch the calculation change as the scores change. That is why Pig is a favorite of mathematics teachers, why computer scientists use it to introduce dynamic programming, and why it is the ideal first step before games such as Farkle, Zombie Dice and Ten Thousand, which layer combinations and multiple dice on top of the same core decision.

A Short History of Pig

Pig has no recorded inventor. The first known description in print is in Scarne on Dice (1945) by John Scarne, the American card and dice expert, magician and author who documented most of the gambling and folk dice games of his era. Scarne presented Pig as a simple game already in circulation, so it was clearly being played before the book appeared.

The game’s second life began in classrooms. Its rules can be explained in one sentence, its probabilities involve only a single die, and yet its best strategy is surprisingly hard to pin down, which makes it a perfect teaching example. In 2004 Todd W. Neller and Clifton G. M. Presser of Gettysburg College published “Optimal Play of the Dice Game Pig” in The UMAP Journal (volume 25, issue 1). Using a technique called value iteration, they computed the exact best decision for every possible combination of your score, your opponent’s score and your turn total. The result was not a single number but a jagged, three-dimensional surface, proof that the folk rule “hold at 20” is a good approximation and nothing more. Neller went on to publish teaching material built around Pig, and the game is now a standard assignment in introductory artificial intelligence courses.

The commercial relative most people have met is Pass the Pigs (originally PigMania, 1977), in which two small plastic pigs replace the die and score according to how they land. It shares Pig’s roll-or-hold structure, but the odds are set by the shape of the pigs rather than by six equal faces.

Pig Rules

Equipment

• One six-sided die
• Pencil and paper to keep score

Object

Be the first player to reach 100 points.

How a Turn Works

1. Roll the die.
2. If it shows 2, 3, 4, 5 or 6, add that number to your turn total. Choose to roll again or hold.
3. If it shows 1, your turn ends immediately. Your turn total is lost and nothing is added to your score.
4. When you hold, add the turn total to your score and pass the die.
5. Winning. As soon as a player holds with a score of 100 or more, they win. There is no final turn for the other player in the standard rules.

Scoring Table

How to Score: Worked Examples

• You roll 4, 6, 3 and hold. Turn total 13 goes to your score. Three rolls in a row without a 1 happens about 58% of the time.
• You roll 5, 2, 6, 4 and then 1. The turn total had reached 17; the 1 wipes it and you score zero. The 17 was never yours.
• You are at 92 and roll a 6, then a 3. Turn total 9 makes 101. Hold immediately; there is nothing to gain by rolling.
• You are at 40, your opponent is at 85, and your turn total is 22. The hold-at-20 rule says stop. But your opponent needs only 15 more and will probably win within two turns, so a 22-point hold barely matters. Keep rolling; the optimal policy is far more aggressive here than the simple rule.
• You are at 96 with a turn total of 2. Two more points win the game. The odds of surviving one roll are 5 in 6; the roll is obviously right, and any non-1 face wins.

The Odds Behind Pig

The value of one more roll

Suppose your turn total is t. One more roll has six equally likely outcomes. Five of them add 2, 3, 4, 5 or 6 (an average of 4), and one of them costs you t. The expected change to your turn total is:

(2 + 3 + 4 + 5 + 6) ÷ 6 − t ÷ 6 = (20 − t) ÷ 6

• At t = 15 the roll is worth +0.83 points on average.
• At t = 19 it is worth +0.17.
• At t = 20 it is worth exactly 0.
• At t = 25 it is worth −0.83.

That is the whole argument for hold at 20: below 20, rolling gains on average; above 20, it loses.

Survival probabilities

The chance of avoiding a 1 for k consecutive rolls is (5/6)^k:

Working through every sequence of rolls, a player who always rolls until reaching 20 succeeds on 37.5% of turns and is wiped out on the other 62.5%. Successful turns bank an average of about 21.7 points, so the strategy averages about 8.1 points per turn overall. Holding at 25 succeeds on 30.0% of turns for an average of about 8.0 points per turn: almost the same expected score, but with more variance, which is exactly what you want when you are behind.

How good is hold at 20?

Neller and Presser measured simple rules against the optimal policy in head-to-head play:

• Hold at 20 loses about 8% more often than optimal play would.
• Hold at 25 is better, about 4.2% below optimal.
• “End race or keep pace” (roll for the win once either player is close to 100; otherwise roll until your turn total covers the deficit or reaches 20) comes within 0.9% of optimal.
• With both players playing optimally, the first player wins about 53% of the time, so going first is a real but small edge.

The optimal policy itself is not a rule you can memorize. Its hold target starts near 20 when both scores are low, climbs steeply when you are far behind, and collapses toward “just get to 100” once either player is within reach of the finish.

Pig Strategy

1. Hold at 20 as your baseline

In the early and middle game, when both players are below about 70 and the scores are close, roll until your turn total is 20 or more and then hold. This single rule captures most of what matters and costs you only a few percentage points against perfection.

2. Raise the target when you are behind

Expected points per turn barely change between a target of 20 and a target of 25, but the higher target produces bigger swings. When you are trailing by 30 or more, swings are your friend. Move your target to 25 or even 30 and accept more wipeouts in exchange for the occasional 35-point turn that gets you back in the game.

3. Lower the target when you are ahead

The mirror image: with a comfortable lead, you want low variance. Holding at 15 or so keeps your average up while giving your opponent fewer turns to catch up on a lucky streak. Do not overdo it; holding at 10 wastes too much expected value.

4. End the race when either player is close

Once your opponent is at 80 or more, treat every turn as a race to 100: their next hold may end the game, so a modest 20-point bank is almost worthless. Roll until you either win or crash. The same applies when you are within 20 of the finish: roll for the win rather than holding at 20 and giving your opponent another turn.

5. Never hold below 100 when a hold cannot win but a few more points can

If you are at 85 with a turn total of 12, holding gives you 97 and guarantees your opponent another turn. Rolling once more wins outright five times out of six. Take the roll.

6. Do not “cover” a loss with a bigger risk

After a wipeout it is tempting to roll to 30 or 40 on the next turn to make up the lost points. The die has no memory. Set your target by the scoreboard, not by what happened last turn.

7. Think in turns, not points, near the end

The question late in the game is “how many turns does each of us need?” If your opponent needs one more decent turn and you need two, a safe hold that leaves you still needing two turns does nothing; a risky roll that could reduce you to one turn does.

Common Mistakes

• Holding at small totals early. Stopping at 8 or 10 “to bank something” gives away expected value every turn; the roll is worth +1.7 points at t = 10.
• Rolling on at 30+ with a lead. The roll is worth −1.7 at t = 30 and the wipeout would cost a whole turn’s work.
• Holding on 99. It happens more often than you would think; check the sum before you stop.
• Believing a 1 is “due.” Each roll is an independent 1-in-6 chance.
• Playing the same target regardless of score. The right target depends on both scores; a flat rule is a starting point, not a plan.

Variants and House Rules

• Two-Dice Pig. Roll two dice. A single 1 ends the turn and loses the turn total; double 1s lose your entire score as well. Doubles other than 1s are often worth double their pips.
• Big Pig. Two dice; double 1s add 25 to the turn total instead of penalizing you, and other doubles score twice their pips. Two-dice versions have very different break-even points (the chance of at least one 1 on two dice is 11 in 36, about 30.6%), so the “hold at 20” rule does not carry over.
• Skunk. A classroom version for large groups. Everyone stands; the leader rolls repeatedly; anyone still standing when a 1 comes up loses the round’s points. Sitting down is the equivalent of holding.
• Hog. Each player chooses how many dice to roll and throws them once. Any 1 scores zero for the turn; otherwise the total of all the dice is banked. The decision moves from “how many rolls” to “how many dice.”
• Piglet. Played with a coin: heads adds 1, tails ends the turn, and the target is much lower. Useful as a teaching toy.
• Pass the Pigs. The commercial game with plastic pigs; positions such as “razorback,” “trotter” and “snouter” score, a plain side-down landing ends the turn, and a “pig out” loses the turn total.
• Playing to 50 or 200. Shorter or longer races shift the optimal targets slightly but leave the basic hold-at-20 logic intact.

If you like Pig, the natural next steps on Dice.now are Farkle, which multiplies the roll-or-bank decision across six dice and a scoring table; Zombie Dice, where “roll a 1” becomes “take three shotgun blasts”; and Bunco, a party game with no decisions at all if you just want to roll. For the same one-die feel with a puzzle attached, try Shut the Box.

Glossary

• Turn total: Points accumulated in the current turn; not safe until you hold.
• Hold: Stop rolling and add the turn total to your score.
• Pig (verb): To roll a 1 and lose the turn total. Sometimes called “pigging out.”
• Hold at n: A strategy that rolls until the turn total reaches n, then holds.
• Break-even point: The turn total (20 in single-die Pig) at which one more roll has zero expected value.
• Optimal policy: The best decision for every combination of scores and turn total, computed by Neller and Presser in 2004.
• Jeopardy game: The general name for dice games in which points accumulate during a turn and can be lost by a bad roll.

Play Pig Online on Dice.now

The game at the top of this page is classic single-die Pig against the computer, first to 100 points. You always roll first. Press Roll to throw the die; a 2 through 6 is added to your turn total and a 1 (shown in red) ends your turn and clears it. Press Hold to bank the turn total; the button shows exactly how much you would bank. The status bar tracks both scores and your turn total, the scoreboard shows how many points each player still needs, and the game log records every roll and every hold by both players.

The computer plays in the open, one roll at a time. Its strategy is deliberately simple and close to what a good human does: it holds at 20 or more, raises its target to 25 when it is 30 or more points behind you, and holds immediately whenever the turn total is enough to win. It never uses the full optimal policy, so a player who applies the strategy tips above, especially the end-race rule, can beat it more often than not.

Rolls use your browser’s cryptographic random number generator, so every face is equally likely. Your best single turn and your win-loss record are stored on your device only; nothing is sent to our servers. Roll the die.