Bulls and Cows

Bulls and Cows (also known as Cows and Bulls or Pigs and Bulls or Bulls and Cleots) is an old code-breaking mind or paper and pencil game for two or more players, predating the similar commercially marketed board game Mastermind.

It is a game with numbers or words that may date back a century or more. It is played by two opponents.



The numerical version

The numerical version of the game is usually played with 4 digits, but can also be played with 3 or any other number of digits.

On a sheet of paper, the players each write a 4-digit secret number. The digits must be all different. Then, in turn, the players try to guess their opponent's number who gives the number of matches. If the matching digits are in their right positions, they are "bulls", if in different positions, they are "cows". Example:

    Secret number: 4271
    Opponent's try: 1234
    Answer: 1 bull and 2 cows. (The bull is "2", the cows are "4" and "1".)

The first one to reveal the other's secret number wins the game. As the first player has a logical advantage, the game can be balanced over multiple games by alternating the right to go first, or over a single game by granting the second player an equal number of guesses, possibly resulting in a tie.

The game may also be played by two teams of 2–3 players, with the team members discussing their strategy before selecting a move.

A computer program moo, written in 1970 by J. M. Grochow at MIT in the PL/I computer language for the Multics operating system, was amongst the first Bulls and Cows computer implementations, inspired by a similar program written by Frank King in 1968 and running on the Cambridge University mainframe. Because the game has simple rules, while it is difficult and entertaining, there are many computer variants; it is often included in telephones and PDAs.

It is proven that any number could be solved within seven turns. Minimal average game length is 26274/5040=5.2131 turns.