Mastermind (board game)

Mastermind or Master Mind is a code-breaking game for two players. The modern game with pegs was invented in 1970 by Mordecai Meirowitz, an Israeli postmaster and telecommunications expert. It resembles an earlier pencil and paper game called Bulls and Cows that may date back a century.
Gameplay and rules.
The game is played using:
The two players decide in advance how many games they will play, which must be an even number. One player becomes the "codemaker", the other the "codebreaker". The codemaker chooses a pattern of four code pegs. Duplicates and blanks are allowed depending on player choice, so the player could even choose four code pegs of the same color or four blanks. In the instance that blanks are not elected to be a part of the game, the codebreaker may not use blanks in order to establish the final code. The chosen pattern is placed in the four holes covered by the shield, visible to the codemaker but not to the codebreaker.
The codebreaker tries to guess the pattern, in both order and color, within eight to twelve turns. Each guess is made by placing a row of code pegs on the decoding board. Once placed, the codemaker provides feedback by placing from zero to four key pegs in the small holes of the row with the guess. A colored or black key peg is placed for each code peg from the guess which is correct in both color and position. A white key peg indicates the existence of a correct color code peg placed in the wrong position.
If there are duplicate colours in the guess, they cannot all be awarded a key peg unless they correspond to the same number of duplicate colours in the hidden code. For example, if the hidden code is red-red-blue-blue and the player guesses red-red-red-blue, the codemaker will award two colored key pegs for the two correct reds, nothing for the third red as there is not a third red in the code, and a colored key peg for the blue. No indication is given of the fact that the code also includes a second blue.
Once feedback is provided, another guess is made; guesses and feedback continue to alternate until either the codebreaker guesses correctly, or all rows of the decoding boards are full.
Traditionally, players can only earn points when playing as the codemaker. The codemaker gets one point for each guess the codebreaker makes. An extra point is earned by the codemaker if the codebreaker is unable to guess the exact pattern within the given number of turns. (An alternative is to score based on the number of key pegs placed.) The winner is the one who has the most points after the agreed-upon number of games are played.
Other rules may be specified.
History.
The game is based on an older, paper based game called Bulls and Cows. A computer adaptation of it was run in the 1960s on Cambridge University’s Titan computer system, where it was called 'MOO'. This version was written by Frank King. There was also another version for the TSS/8 time sharing system, written by J.S. Felton and finally a version for the Multics system at MIT by Jerrold Grochow.
The modern game with pegs was invented in 1970 by Mordecai Meirowitz, an Israeli postmaster and telecommunications expert. Meirowitz presented the idea to many major toy companies but, after showing it at the Nuremberg International Toy Fair, it was picked up by a plastics company, Invicta Plastics, based near Leicester, UK. Invicta purchased all the rights to the game and the founder, Edward Jones-Fenleigh, refined the game further. It was released in 1971–2.
Since 1971, the rights to "Mastermind" have been held by Invicta Plastics. (Invicta always named the game "Master Mind".) They originally manufactured it themselves, though they have since licensed its manufacture to Hasbro worldwide, with the exception of Pressman Toys and Orda Industries who have the manufacturing rights to the United States and Israel, respectively.
Starting in 1973, the game box featured a photograph of a white man in a white jacket seated in the foreground, with a young Asian woman of high caste standing behind him with the golden symbols of office visible on her sari, denoting the power and intellect behind the throne. The two amateur models (Bill Woodward and Cecilia Fung) reunited in June 2003 to pose for another publicity photo.
Algorithms and strategies.
Before asking for a best strategy of the codebreaker one has to define what is the meaning of "best": The minimal number of moves can be analyzed under the conditions of worst and average case and in the sense of a minimax value of a zero-sum game in game theory.
Best strategies with four pegs and six colors.
With four pegs and six colours, there are 64 = 1296 different patterns (allowing duplicate colours).
Worst case: Five-guess algorithm.
In 1977, Donald Knuth demonstrated that the codebreaker can solve the pattern in five moves or fewer, using an algorithm that progressively reduces the number of possible patterns.
The algorithm works as follows:
Average case.
Subsequent mathematicians have been finding various algorithms that reduce the average number of turns needed to solve the pattern: in 1993, Kenji Koyama and Tony W. Lai performed an exhaustive depth-first search showing that the optimal method for solving a random code could achieve an average of 5625/1296 = 4.3403 turns to solve, with a worst-case scenario of six turns.
Minimax value of game theory.
The minimax value in the sense of game theory is 5600/1290 = 4.341. The minimax strategy of the codemaker consists in a uniformly distributed selection of one of the 1290 patterns with two or more colors.
Genetic algorithm.
A new algorithm with an embedded genetic algorithm, where a large set of eligible codes is collected throughout the different generations. The quality of each of these codes is determined based on a comparison with a selection of elements of the eligible set. This algorithm is based on a heuristic that assigns a score to each eligible combination based on its probability of actually being the hidden combination. Since this combination is not known, the score is based on characteristics of the set of eligible solutions or the sample of them found by the evolutionary algorithm.
The algorithm works as follows:
Complexity and the satisfiability problem.
In November 2004, Michiel de Bondt proved that solving a "Mastermind" board is an NP-complete problem when played with "n" pegs per row and two colors, by showing how to represent any one-in-three 3SAT problem in it. He also showed the same for "Consistent Mastermind" (playing the game so that every guess is a candidate for the secret code that is consistent with the hints in the previous guesses).
The "Mastermind satisfiability problem" is a decision problem that asks, "Given a set of guesses and the number of colored and white pegs scored for each guess, is there at least one secret pattern that generates those exact scores?" (If not, then the codemaker must have incorrectly scored at least one guess.) In December 2005, Jeff Stuckman and Guo-Qiang Zhang showed in an arXiv article that the "Mastermind satisfiability problem" is NP-complete.
Variations.
Varying the number of colors and the number of holes results in a spectrum of "Mastermind" games of different levels of difficulty. Another common variation is to support different numbers of players taking on the roles of codemaker and codebreaker. The following are some examples of "Mastermind" games produced by Invicta, Parker Brothers, Pressman, Hasbro, and other game manufacturers:
The difficulty level of any of the above can be increased by treating “empty” as an additional color or decreased by requiring only that the code's colors be guessed, independent of position.
Computer and Internet versions of the game have also been made, sometimes with variations in the number and type of pieces involved and often under different names to avoid trademark infringement. "Mastermind" can also be played with paper and pencil.
There is a numeral variety of the Mastermind in which a 4-digit number is guessed.
The game was compiled into for the Switch under the name "Hit & Blow".