Bulls and Cows Solver

Bulls and Cows (or Cows & Bulls, or Pigs and Cows) is a logic game based on deduction in which the player must guess a secret code in as few attempts as possible. The game is typically played by two players with distinct roles: the first chooses a secret code (made of digits, letters, or symbols), and the second one tries to guess it. After each guess, feedback is given in terms of “bulls” and “cows”: bulls indicate correct symbols in the correct position, while cows indicate correct symbols in the wrong position. The game ends when the secret code is fully discovered.

The Bulls and Cows solver can be configured by code length (number of positions) and by the number of possible symbols, depending on the game to solve.

The syntax to use is CCCC A B where CCCC is a guess, A is the number of “bulls” (correct symbols in the correct position), and B is the number of “cows” (correct symbols in the wrong position).

In the classic version, the code has 4 positions and uses digits (0–9), but any combination of letters, digits, or symbols can be used.

For a code of size N, each guess must contain N characters followed by 2 numbers indicating the feedback (bulls and cows). Separate each guess with a line break.

It is also possible to use wildcards (unknown positions indicated by ?, *, - or _) both in guesses and in scores.

According to Donald Knuth, in an equivalent setup (4‑position code with 6 possible symbols), the code can be found in at most 5 steps. The algorithm is:

1 - Generate the set E of all possible codes.

2 - Use 1122 (or an equivalent) as the first guess.

3 - If correct, the game ends. Otherwise, remove from E all codes that are inconsistent with the feedback (bulls and cows).

4 - For each remaining untested guess, evaluate how many possibilities from E would be eliminated for each possible feedback. Score each guess by its worst‑case elimination (the minimum number eliminated). Choose a guess that maximizes this score, preferably one from E. Repeat from step 3.

This guarantees that the code will be found in 5 steps or fewer.

Mathematically, if the player's strategy is known, some combinations can be harder than others. However, with an optimal strategy (like Knuth's), all combinations can be solved in 5 steps or fewer.

If the strategy is unknown, no combination is inherently harder than another.

In practice, some patterns are trickier for humans:

— Repeated symbols are often overlooked.

— Uncommon symbols (high digits, rare letters) are tested later.

— Non-intuitive patterns (e.g., 1221, 5656) are harder to anticipate.

Some commonly cited difficult combinations include: 1221, 2354, 3311, 4524, 5656, 6643.

A well-known variant is the Mastermind game, which uses colors instead of digits or symbols, typically played with a physical board and pegs.

Another variant uses words instead of symbols, greatly increasing the number of possible combinations.

It is also possible to allow empty positions (blanks), if agreed upon by players.

Bulls and Cows is often used as a basis for puzzles or mini-games in software and video games.

Bulls and Cows is an old game with no clearly identified inventor. It was played long before Mastermind, often as a pencil-and-paper number game.