.gif)
Mastermind (board game)
Encyclopedia
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 Israel
i postmaster and telecommunications expert, but the game resembles an earlier pencil and paper game called bulls and cows
that may date back a century or more.
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 are allowed, so the player could even choose four code pegs of the same color. 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 twelve (or ten, or eight) 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 peg indicates the existence of a correct color 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 white-white-black-black and the player guesses white-white-white-black, the codemaker will award two colored pegs for the two correct whites, nothing for the third white as there is not a third white in the code, and a colored peg for the black. No indication is given of the fact that the code also includes a second black.
Once feedback is provided, another guess is made; guesses and feedback continue to alternate until either the codebreaker guesses correctly, or twelve (or ten, or eight) incorrect guesses are made.
The codemaker gets one point for each guess a codebreaker makes. An extra point is earned by the codemaker if the codebreaker doesn't guess the pattern exactly in the last guess. (An alternative is to score based on the number of colored key pegs placed.) The winner is the one who has the most points after the agreed-upon number of games are played.
, near Leicester
, UK
. (Invicta always named the game Master Mind.) They originally manufactured it themselves, though they have since licensed its manufacture to Hasbro
in most of the world, and two other manufacturers who have the United States and Israel manufacturing rights.
Starting in 1973, the game box featured a photograph of a well-dressed, distinguished-looking white man seated in the foreground, with an attractive Asian woman standing behind him. The two amateur models (Bill Woodward and Cecilia Fung) reunited in June 2003 to pose for another publicity photo.
Games Pioneer Harrison Heath, introduced a very similar version of Mastermind, in which discs were used instead of pins, similar to his previous creation Connect 4. However the version was much less successful than its predecessor and only took a revenue of $360,000.
Divide the 1296 possible games into four categories:
The general process is to list all the games that could be correct with the data so far. The list should be sorted by ascending number of duplicates and within each duplicate level alphabetically. Before guess 1, the list is all 1296 games; thus guess 1 is always "abcd." If the reply to guess 1 is "0 0," for example, then the list afterwards comprises the 16 games involving only e and f. Each subsequent guess is the first game remaining in the list, with the following exceptions:
demonstrated that the codebreaker can solve the pattern in five moves or fewer, using an algorithm that progressively reduced the number of possible patterns.
The algorithm works as follows:
Subsequent mathematicians have been finding various algorithms that reduce the average number of turns needed to solve the pattern: in 1993, Mami Koyama and Tony W. Lai found a method that required an average of 4.340 turns to solve, with a worst case scenario of six turns.
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
.
, 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 4-digit number is guessed .
Code
A code is a rule for converting a piece of information into another form or representation , not necessarily of the same type....
-breaking game for two players. The modern game with pegs was invented in 1970 by Mordecai Meirowitz
Mordecai Meirowitz
Mordecai Meirowitz was an Israeli postmaster and telecommunications expert who invented the code-breaking board game Master Mind. Having been rejected by the leading games companies, he managed to interest a small Leicester-based educational toy company, Invicta Plastics, which restyled and renamed...
, an Israel
Israel
The State of Israel is a parliamentary republic located in the Middle East, along the eastern shore of the Mediterranean Sea...
i postmaster and telecommunications expert, but the game resembles an earlier pencil and paper game called bulls and cows
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 paper and pencil game for two players, predating the similar commercially-marketed board game Master Mind....
that may date back a century or more.
Gameplay and rules
The game is played using:- a decoding board, with a shield at one end covering a row of four large holes, and twelve (or ten, or eight) additional rows containing four large holes next to a set of four small holes;
- code pegs of six (or more; see Variations below) different colors, with round heads, which will be placed in the large holes on the board; and
- key pegs, some colored or black, some white, which are flat-headed and smaller than the code pegs; they will be placed in the small holes on the board.
The two players decide in advance how many games they will play, which must be an even number
Even and odd numbers
In mathematics, the parity of an object states whether it is even or odd.This concept begins with integers. An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without remainder; an odd number is an integer that is not evenly divisible by 2...
. One player becomes the codemaker, the other the codebreaker. The codemaker chooses a pattern of four code pegs. Duplicates are allowed, so the player could even choose four code pegs of the same color. 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 twelve (or ten, or eight) 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 peg indicates the existence of a correct color 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 white-white-black-black and the player guesses white-white-white-black, the codemaker will award two colored pegs for the two correct whites, nothing for the third white as there is not a third white in the code, and a colored peg for the black. No indication is given of the fact that the code also includes a second black.
Once feedback is provided, another guess is made; guesses and feedback continue to alternate until either the codebreaker guesses correctly, or twelve (or ten, or eight) incorrect guesses are made.
The codemaker gets one point for each guess a codebreaker makes. An extra point is earned by the codemaker if the codebreaker doesn't guess the pattern exactly in the last guess. (An alternative is to score based on the number of colored key pegs placed.) The winner is the one who has the most points after the agreed-upon number of games are played.
History
Since 1971, the rights to Mastermind have been held by Invicta Plastics of OadbyOadby
Oadby is a town within the borough of Oadby and Wigston, in Leicestershire, England. It is to the east of Wigston Magna, and to the southeast of Leicester. Oadby forms part of the Leicester Urban Area, and is situated on the A6 road....
, near Leicester
Leicester
Leicester is a city and unitary authority in the East Midlands of England, and the county town of Leicestershire. The city lies on the River Soar and at the edge of the National Forest...
, UK
United Kingdom
The United Kingdom of Great Britain and Northern IrelandIn the United Kingdom and Dependencies, other languages have been officially recognised as legitimate autochthonous languages under the European Charter for Regional or Minority Languages...
. (Invicta always named the game Master Mind.) They originally manufactured it themselves, though they have since licensed its manufacture to Hasbro
Hasbro
Hasbro is a multinational toy and boardgame company from the United States of America. It is one of the largest toy makers in the world. The corporate headquarters is located in Pawtucket, Rhode Island, United States...
in most of the world, and two other manufacturers who have the United States and Israel manufacturing rights.
Starting in 1973, the game box featured a photograph of a well-dressed, distinguished-looking white man seated in the foreground, with an attractive Asian woman standing behind him. The two amateur models (Bill Woodward and Cecilia Fung) reunited in June 2003 to pose for another publicity photo.
Games Pioneer Harrison Heath, introduced a very similar version of Mastermind, in which discs were used instead of pins, similar to his previous creation Connect 4. However the version was much less successful than its predecessor and only took a revenue of $360,000.
Algorithms
With four pegs and six colors, there are 64 = 1296 different patterns (allowing duplicate colors).Six-guess algorithm
The following algorithm solves the (six-color) game in six or fewer guesses. It has a general procedure and a few listed exceptions. In this section the six colours are referred to as letters a through f.Divide the 1296 possible games into four categories:
- 0 duplicates (e.g. abcd)
- 1 duplicate (e.g. aabc)
- 2 duplicates (e.g. aabb or aaab)
- 3 duplicates (e.g. aaaa)
The general process is to list all the games that could be correct with the data so far. The list should be sorted by ascending number of duplicates and within each duplicate level alphabetically. Before guess 1, the list is all 1296 games; thus guess 1 is always "abcd." If the reply to guess 1 is "0 0," for example, then the list afterwards comprises the 16 games involving only e and f. Each subsequent guess is the first game remaining in the list, with the following exceptions:
- Guess 2 is always "bcde"
- Guess 3 is always "cdef"
- If the list for guess 4 starts with a game on the left side of the list below, then use the game to its right instead:
- "acfb" → "dcad"
- "aebf" → "edfd"
- "aefb" → "eacc"
- "afbe" → "bfcd"
- "bafe" → "eadc"
- "beaf" → "edae"
- "befa" → "eeda"
- "eabf" → "fdfb"
- "aadb" → "babd"
- "abae" → "bbcc"
- "aeaf" → "cffd"
- "cafa" → "fdfa"
- "aaee" → "dddf"
Five-guess algorithm
In 1977, Donald KnuthDonald Knuth
Donald Ervin Knuth is a computer scientist and Professor Emeritus at Stanford University.He is the author of the seminal multi-volume work The Art of Computer Programming. Knuth has been called the "father" of the analysis of algorithms...
demonstrated that the codebreaker can solve the pattern in five moves or fewer, using an algorithm that progressively reduced the number of possible patterns.
The algorithm works as follows:
- Create a set S of remaining possibilities (at this point there are 1296). The first guess is aabb.
- Remove all possibilities from S that would not give the same score of colored and white pegs if they were the answer.
- For each possible guess (not necessarily in S) calculate how many possibilities from S would be eliminated for each possible colored/white score. The score of the guess is the least of such values. Play the guess with the highest score (minimaxMinimaxMinimax is a decision rule used in decision theory, game theory, statistics and philosophy for minimizing the possible loss for a worst case scenario. Alternatively, it can be thought of as maximizing the minimum gain...
). - Go back to step 2 until you have got it right.
Subsequent mathematicians have been finding various algorithms that reduce the average number of turns needed to solve the pattern: in 1993, Mami Koyama and Tony W. Lai found a method that required an average of 4.340 turns to solve, with a worst case scenario of six turns.
Mastermind satisfiability problem
The Mastermind satisfiability problem is a decision problemDecision problem
In computability theory and computational complexity theory, a decision problem is a question in some formal system with a yes-or-no answer, depending on the values of some input parameters. For example, the problem "given two numbers x and y, does x evenly divide y?" 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
ArXiv
The arXiv |Chi]], χ) is an archive for electronic preprints of scientific papers in the fields of mathematics, physics, astronomy, computer science, quantitative biology, statistics, and quantitative finance which can be accessed online. In many fields of mathematics and physics, almost all...
article that the Mastermind satisfiability problem is NP-complete
NP-complete
In computational complexity theory, the complexity class NP-complete is a class of decision problems. A decision problem L is NP-complete if it is in the set of NP problems so that any given solution to the decision problem can be verified in polynomial time, and also in the set of NP-hard...
.
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 BrothersParker Brothers
Parker Brothers is a toy and game manufacturer and brand. Since 1883, the company has published more than 1,800 games; among their best known products are Monopoly, Cluedo , Sorry, Risk, Trivial Pursuit, Ouija, Aggravation, and Probe...
, Pressman
Pressman Toy Corporation
Pressman Toy Corporation is a toy manufacturer based in New York City which was founded in 1922 by Jack Pressman. It currently focuses on family games and licensed products. Its slogan is "Games people play...
, Hasbro
Hasbro
Hasbro is a multinational toy and boardgame company from the United States of America. It is one of the largest toy makers in the world. The corporate headquarters is located in Pawtucket, Rhode Island, United States...
, and other game manufacturers:
| Game | Year | Colors | Holes | Comments |
|---|---|---|---|---|
| Mastermind | 1972 | 6 | 4 | Original version |
| Royale Mastermind | 1972 | 5 colors × 5 shapes | 3 | |
| Mastermind44 | 1972 | 6 | 5 | For four players |
| Grand Mastermind | 1974 | 5 colors × 5 shapes | 4 | |
| Super Mastermind (a.k.a. Deluxe Mastermind; a.k.a. Advanced Mastermind) | 1975 | 8 | 5 | |
| Word Mastermind | 1975 | 26 letters | 4 | Only valid words may be used as the pattern and guessed each turn. |
| Number Mastermind | 1976 | 6 digits | 4 | Uses numbers instead of colors. The codemaker may optionally give, as an extra clue, the sum of the digits. |
| Electronic Mastermind (Invicta) | 1977 | 10 digits | 3,4,or5 | Uses numbers instead of colors. Handheld electronic version. Solo or multiple players vs. the computer. Invicta branded. |
| Walt Disney Mastermind | 1978 | 5 | 3 | Uses Disney characters instead of colors |
| Mini Mastermind (a.k.a. Travel Mastermind) | 1988 | 6 | 4 | Travel-sized version; room for only six guesses |
| Mastermind Challenge | 1993 | 8 | 5 | Both players simultaneously play code maker and code breaker. |
| Parker Mastermind | 1993 | 8 | 4 | |
| Mastermind for Kids | 1996 | 6 | 3 | Animal theme |
| Mastermind Secret Search | 1997 | 26 letters | 3-6 | Valid words only; clues are provided letter-by-letter using up/down arrows for earlier/later in the alphabet. |
| Electronic Hand-Held Mastermind (Hasbro) | 1997 | 6 | 4 | Handheld electronic version. Hasbro. |
| New Mastermind | 2004 | 8 | 4 | For up to five players |
| Purble Shop (Advanced Mode) | 2006 | 5 | 5 |
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
Online skill-based game
Online skill-based games are online games in which the outcome of the game is determined by the player's physical skill or mental skill...
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
Paper and pencil game
Paper-and-pencil games are games that can be played solely with paper and pencil .In some board games, including some abstract strategy games like Gomoku, a piece once played will not be moved on the board or removed from the board...
.
There is a numeral variety of the Mastermind in which 4-digit number is guessed .
See also
- Board gameBoard gameA board game is a game which involves counters or pieces being moved on a pre-marked surface or "board", according to a set of rules. Games may be based on pure strategy, chance or a mixture of the two, and usually have a goal which a player aims to achieve...
- Bulls and cowsBulls and cowsBulls and Cows -- also known as Cows and Bulls or Pigs and Bulls or Bulls and Cleots -- is an old code-breaking paper and pencil game for two players, predating the similar commercially-marketed board game Master Mind....
- JottoJottoJotto is a logic-oriented word game played with two players, a writing implement, and a piece of paper. Each player picks a secret word of five letters , and the object of the game is to correctly guess the other player's word first...
- Purble Shop
- Deduction board gameDeduction board gameDeduction board games are a genre of board game in which the players must use deductive reasoning and logic in order to win the game. While many games, such as bridge or poker require the use of deductive reasoning to some degree, deduction board games feature deductive reasoning as their central...
External links
- Mastermind at BoardgamegeekBoardGameGeekBoardGameGeek is a website that was founded in January 2000 by Scott Alden and Derk Solko as a resource for the board gaming hobby. The database holds reviews, articles, and session reports for over 45,000 different games, expansions, and designers. BoardGameGeek includes German-style board games,...
- Mathworld article on Mastermind
- https://sites.google.com/site/goldendrongo/home/games Source code for mastermind game in turbo c 2 and turbo pascal 3 for windows.
- http://qt-apps.org/content/show.php/Mastermind?content=69525 Source code for mastermind game in Qt C++.