# Liar's dice

Liar's dice is a class of dice games for two or more players requiring the ability to deceive and to detect an opponent's deception.

In "single hand" liar's dice games, each player has a set of dice, all players roll once, and the bids relate to the dice each player can see (their hand) plus all the concealed dice (the other players' hands). In "common hand" games, there is one set of dice which is passed from player to player. The bids relate to the dice as they are in front of the bidder after selected dice have been re-rolled.

The genre has its roots in South America, with games there being known as Dudo, Cachito, Perudo or Dadinho; other names include "pirate's dice," "deception dice" and "diception." The drinking game version is sometimes called Mexicali or Mexican in the United States; the latter term may be a corruption of Mäxchen ("Little Max"), the name by which a similar game, Mia, is known in Germany, while Liar's dice is known in Germany as Bluff. It is known by various names in Asia.

## Single hand

Five six-sided dice are used per player, with dice cups used for concealment.

Five dice are used per player with dice cups used for concealment.

Each round, each player rolls a "hand" of dice under their cup and looks at their hand while keeping it concealed from the other players. The first player begins bidding, announcing any face value and the minimum number of dice that the player believes are showing that value, under all of the cups in the game. Ones are often wild, always counting as the face of the current bid.

Turns rotate among the players in a clockwise order. Each player has two choices during their turn: to make a higher bid, or challenge the previous bid—typically with a call of "liar". Raising the bid means either increasing the quantity, or the face value, or both, according to the specific bidding rules used. There are many variants of allowed and disallowed bids; common bidding variants, given a previous bid of an arbitrary quantity and face value, include:

• the player may bid a higher quantity of any particular face, or the same quantity of a higher face (allowing a player to "re-assert" a face value they believe prevalent if another player increased the face value on their bid);
• the player may bid a higher quantity of the same face, or any particular quantity of a higher face (allowing a player to "reset" the quantity);
• the player may bid a higher quantity of the same face or the same quantity of a higher face (the most restrictive; a reduction in either face value or quantity is never allowed).

If the current player challenges the previous bid, all dice are revealed. If the bid is valid (at least as many of the face value and any wild aces are showing as were bid), the bidder wins. Otherwise, the challenger wins. The player who loses a round loses one of their dice. The last player to still retain a die (or dice) is the winner. The loser of the last round starts the bidding on the next round. If the loser of the last round was eliminated, the next player starts the new round.

### Variants

• Instead of the current player being the only one who can raise the bet, challenge (or "call up") the previously-made bid, any player may raise or challenge a bid at any time. The first challenge made ends the round, and the challenger closest to the current bidder in the direction of play has priority if multiple players challenge at the same time.
• If played with the above variant, the player who made the last bid may count aloud from 1 to 10. If he reaches 10 with no one challenging or increasing the bid, the round ends with that player earning back a die. A player may have more than 5 dice that way, and any player who reaches 10 dice that way wins the game.
• With the above-mentioned variants, some players may stay quiet and win easily. To avoid that, the following rule may be added: Each time a player loses a challenge, he loses a die normally, but the two players sitting to their left and right lose a die as well (unless one of them was the player to win the challenge).
• Another solution to the above-mentioned variants is to force all players to choose a side: Each player holds a two-sided item (preferred a coin or a card), and decides which side means 'true', and which means 'lie'. When a player challenges, all players must join the challenge, placing their items on the table on either 'true' or 'lie', hidden beneath their hands. Once all players have joined, the items are revealed and the table is divided into players who support either side of the challenge. Every player on the losing side loses a die at the end of the challenge.
• With some bidding systems, a player may elect to choose one or more dice of matching value from under their cup, place them outside the cup in view of the other players, re-roll the remaining dice, and make a new bid of any quantity of that face value.
• When a player has no two dice with the same face, he may choose to pass once in a game round. If he does so, the bid will not be raised. The next player can raise the bid using standard rules, or call the bluff. By doing so, he challenges the claim of the passing player having no two dice with the same face. This is commonly used in multi-round games where dice are removed from the game, as it helps players with few dice left to gain more information about the other dice without risk.
• Instead of raising or challenging, the player can claim that the current bid is exactly correct ("Spot On"). If the number is higher or lower, the player loses to the previous bidder, but if they are correct, they win. A "spot-on" claim typically has a lower chance of being correct than a challenge, so a correct "spot on" call sometimes has a greater reward, such as the player regaining a previously lost die or all other players losing a die.

### Elements of strategy

As with any game of chance, probability is highly important. The key element is the "expected quantity": the quantity of any face value that has the highest probability of being present. For six-sided dice, the expected quantity is one-sixth the number of dice in play, rounded down.[citation needed] When wilds are used, the expected quantity is doubled as players can expect as many aces, on average, as any other value. Because each rolled die is independent of all others, any combination of values is possible, but the "expected quantity" has a greater than 50% chance of being correct, and the highest probability of being exactly correct. For example, when 15 dice are in play and wilds are used, the expected quantity is 5. The chances of a bid of 5 being correct are about 59.5%; in contrast, the chances of a bid of 8 being correct are only about 8.8%.

However, a high bid is not necessarily incorrect, because bids incorporate information the player knows. A player who holds several dice of a single value (for instance, four out of the five dice in their hand are threes) may make a bid, with fifteen dice on the table, of "six threes". To an outside observer who sees none of the dice, this has an extremely low probability of being correct (even with wilds), however since the player knows the value of five of those dice, the player is actually betting that there are two additional threes among the ten unknown dice. This is far more likely to be true (about 40%).

Each bid gives others at the table information. Players, through subsequent bids, reveal the players' confidence in the quantity of each face value rolled. A player with two or three of a certain face value under his or her own cup may make a bid favoring that face value. Players can thus use these bids to build a mental picture of the unknown values, which either strengthens or weakens their confidence in a bid they are considering. Others may consider a bid as evidence it is true, and if their own dice support the same conclusion, may increase the bid on that face value, or if their dice refute it may bid on a different face, or challenge the previous bid.

Conversely, bids can also be bluffs. Bluffs in liar's dice can be split into two main categories: early bluffs and late bluffs. An early bluff is likely to be correct by simple probability (depending on the number of players), but other players may believe the bidder made that bid because his or her dice supported it. Thus, the bluff is false information that can lead to incorrect higher bids being made on that face value. Players will thus attempt to trick other players into overbidding by use of early bluffs to inflate a particular face value. A late bluff, on the other hand, is usually less voluntary; the player is often unwilling to challenge a bid, but as a higher bid is even more likely to be incorrect it is even less appealing. A late bluff is thus a critical part of the game; convincing bluffs, as well as reliable detection of bluffs, allow the player to avoid being challenged on an incorrect bid.

Playing Liar's dice involves interpersonal skills similar to other bluffing games such as poker. Being able to reliably detect bluffs through giveaways, or "tells", and analyzing a player's bidding history for patterns that can indicate the likelihood of a bluff, are important skills here just as in poker.

### Dice odds

For a given number of unknown dice n, the probability that exactly a certain quantity q of any face value are showing, P(q), is

${\displaystyle \ P(q)=C(n,q)\cdot (1/6)^{q}\cdot (5/6)^{n-q}}$

Where C(n,q) is the number of unique subsets of q dice out of the set of n unknown dice. In other words, the number of dice with any particular face value follows the binomial distribution ${\displaystyle B(n,{\tfrac {1}{6}})}$.

For the same n, the probability P'(q) that at least q dice are showing a given face is the sum of P(x) for all x such that q ≤ x ≤ n, or

${\displaystyle \ P'(q)=\sum _{x=q}^{n}C(n,x)\cdot (1/6)^{x}\cdot (5/6)^{n-x}}$

These equations can be used to calculate and chart the probability of exactly q and at least q for any or multiple n. For most purposes, it is sufficient to know the following facts of dice probability:

• The expected quantity of any face value among a number of unknown dice is one-sixth the total unknown dice.
• A bid of the expected quantity (or twice the expected value when playing with wilds), rounded down, has a greater than 50% chance of being correct and the highest chance of being exactly correct.[1]

## Common hand

A set of poker dice being rolled behind a screen, played as in the "individual" hand version of liar's dice.

The "Common hand" version is for two players. The first caller is determined at random. Both players then roll their dice at the same time, and examine their hands. Hands are called in style similar to poker, and the game may be played with poker dice:

• Five of a kind: e.g., 44444
• Four of a kind: e.g., 22225
• High straight: 23456
• Full house: e.g., 66111
• Three of a kind: e.g., 44432
• Low straight: 12345
• Two pair: e.g., 22551
• Pair: e.g., 66532
• Runt: e.g., 13456

One player calls their hand. The other player may either call a higher-ranking hand, call the bluff, or re-roll some or all of their dice.[clarification needed] When a bluff is called, the accused bluffer reveals their dice and the winner is determined.[2]

## Drinking game version

The first player rolls two dice under a cup and claims a roll. Most claims are scored by reading the higher die as the 10s place and the lower as the 1s, e.g., a roll of 1 and 4 is read as "41". Doubles are higher than "65", and what would be the lowest roll 2-1, is a "Mexican" and higher than 6-6.

Special rolls:

• 3-1 Social (everyone drinks, cancel all previous rolls, roll again to open)
• 3-2 Reverse (change direction and previous player drinks one sip, cancel all previous rolls, roll again to open)
• 2-1 Mexican (if the cup is lifted revealing a Mexican, the incorrect challenger drinks twice, if the player does not challenge, the player must still drink, since nothing is higher than Mexican)

The next player may do one of two things. If he believes the roller, he simply takes the dice (without looking at the result), rolls, and claims a higher scoring roll. If he does not believe the roller, the cup is lifted, revealing the roller's hand. Either the bluffer or incorrect challenger must drink.