Midnight (game)

Midnight (or 1-4-24) is a dice game played with 6 dice.

One player rolls at a time. He rolls 6 dice, and keeps at least one. Any that the player doesn't keep are rerolled. This procedure is then repeated until there are no more dice to roll. Once kept, dice cannot be rerolled. Players must have kept a 1 and a 4, or they are disqualified. If they have a 1 and 4, the other dice are totaled to give the player's score. The maximum score is therefore 24. The procedure is repeated for the remaining players. The player with the highest 4 dice total wins.

If two or more players tie for the highest total, any money bet is added to the next game.

Dice shown in black are those kept from the previous roll.

The player scores 20 (6+3+5+6).

A variant version, called 2-4-24, in which the player must keep a 2 and a 4 to score, rather than a 1 and a 4, is sometimes played.

It is possible to calculate the probability of scoring, if that is the player's sole objective. This would, for instance, be the case if the player was the last to throw and the other players had not scored.

The strategy is to keep a 1 or a 4 when they are first thrown and otherwise to keep just one dice, as required by the rules. Playing this strategy will mean that the player will score unless they fail to throw a 1 or a 4 in 21 (=6+5+4+3+2+1) throws of the dice. Since 21 is the maximum number of throws possible, this strategy must maximise the chance of scoring. The probability of scoring is 1-(2*(5/6)^n-(4/6)^n) where n is the number of dice thrown, in this case 21. This gives the chance of scoring as 95.7%, the maximum possible.

The formula can also be used to calculate the probability of having a 1 and a 4 after each throw, when using this strategy.

So the player has a 41.8% chance of throwing a 1 and a 4 on the first throw of the dice and a 74.2% change of throwing a 1 and a 4 after the second throw of the dice.

The formula can be used to calculate the maximum probability of scoring when the player has less than 6 dice.

So the player has a 87.2% chance of scoring even when they only have 5 dice left and have not kept a 1 or a 4 on the first throw. They have a 93.5% chance of scoring at this stage if they kept a 1 or 4 on the first throw. When a player hasn't kept either a 1 or a 4 and has only 2 dice left, the change of getting a 1 and 4 with these 2 dice is 13.9%.

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Wikipedia