Mastermind (game)

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Mastermind
Mastermind.jpg
Game data
author Mordechai Meirovitz
publishing company Invicta (1971/72),
Parker Brothers (1973),
Pressman Toy (1981),
Hasbro ,
et al. a.
Publishing year 1971/72
Art Deduction game
Teammates 2
Duration 20 minutes
Age from 8 years

Awards

Game of the Year 1973

Mastermind , also known as SuperHirn , also known in the GDR as Super Code, Variablo and LogicTrainer , is a logic game for two people in which a sequence of colors is to be determined through successive guesses. The game was invented by Mordechai Meirovitz in 1970 and became one of the most successful games of the 1970s. By 2000, over 55 million games had been sold in 80 countries.

history

Mastermind was invented in 1970 by Mordechai (Marco) Meirovitz, an Israeli telecommunications expert living in Paris . After several game companies rejected the game, he presented the game in 1971 at the Nuremberg Toy Fair . The small English company Invicta Plastics from Leicester , founded by Edward Jones-Fenleigh in 1946, bought all rights. Jones-Fenleigh refined the game and then published it as a mastermind in 1971/72 . It was an instant hit and was named First Game of the Year in 1973 by the British Association of Toy Retailers in the UK ; Until then, the British Association of Toy Retailers had awarded a Toy of the Year every year since 1965 .

The game was published in 1973 by Parker Brothers , 1981 by Pressman Toy and many other publishers. Several variants were also published by various publishers.

Mastermind for four players was patented in 1980.

In German-speaking countries it was initially sold under the name SuperHirn and later also as a mastermind . In the GDR it was sold under the logic trainer or super code ( VEB Plasticart ) and in the English-speaking countries it was mainly distributed as a mastermind . A variant with two by three pens, in which two players guess each other, was manufactured in the GDR under the name Variablo ( VEB Plastspielwaren Berlin ).

As of 2000, Invicta licensed the game to Hasbro , Pressman Toy and Orda Industries .

Game board and game principle

The game board is a perforated plate with four rows of round holes arranged one behind the other to accommodate mushroom-shaped colored stones and parallel to it four smaller holes in a square arrangement to accommodate nail-shaped pens.

At the beginning a player (the coder) secretly determines a four-digit ordered color code , which is selected from six colors; each color can also be used several times. The other player (the guessing) tries to figure out the code. To do this, he uses a similar color code as a question; Guess blind on the first move, on the following moves with the help of the answers to the previous moves.

With every move the guesser gets the information how many pens he has placed correctly in color and position and how many pens are in the correct color but are in the wrong position. A hit in color and position is indicated by a black pen, a pen with the correct color in the wrong position is indicated by a white pen. There are also versions with red instead of black pins to indicate hits. All questions and answers remain pegged until the end of the game.

The guesser's goal is to guess the color code with as few questions as possible. The coder may not change his color row and must put the correct number of black and white pens for each question.

There are also variants with two boards , in which each board is a guess and a coder on the other board and both ask alternately.

example

Example position after the 2nd attempt

The color code is sought:

green - red - blue - green

1st guess : red - yellow - red - green

Answer : Once black (because green is correct in the fourth position), once white (because red appears once in the code you are looking for, but not in the first or third position).

2nd guessing attempt : green - green - orange - red

Answer : Once black (because green is correct this time in the first position), twice white (because 1. green appears a second time in the code searched for, but not in the second position and 2. red occurs in the code searched for, but not in the fourth position).

If the correct color code is found, the answer is four blacks. The guessing player usually has twelve attempts to find the correct solution. This number can vary depending on how the game is played; the smaller travel version only offers space for six attempts.

Result space

Mastermind

The number of possible codes is . With 6 color codes, all 4 pens have the same color, with 210 there are exactly 2 colors (of which 90 times 2 + 2 pens and 120 times 3 + 1 pens), with 720 there are exactly 3 colors (2 + 1 + 1 pens) , and with 360 color codes, all 4 pens are different colors. In a mastermind variant with 8 color codes for the four slots, the number of possible codes is 4096.

Super mastermind

The Super Mastermind version, released in 1975, has 5 holes and 8 colors. The number of possible codes is there: .

As with the original mastermind, the player has 12 attempts to solve the task.

1 Farbe  A A A A A        8        8
2 Farben A A A A B      280
         A A A B B      560      840
3 Farben A A A B C     3360
         A A B B C     5040     8400
4 Farben A A B C D    16800    16800
5 Farben A B C D E     6720     6720
                      32768    32768

Variant with gaps as an additional color

It is often agreed that slots may be left free. These gaps apply like an additional (imaginary) color. This increases the number of possible codes for the standard mastermind and for the super mastermind .

Strategies

A good mastermind player needs combination and logical thinking skills.

As part of mathematical studies, the search strategy in the mastermind game can be optimized from different points of view. Either the worst possible case (referred to as criterion worst case ), the probabilistic average ( average case ) or game theory minimax value basis:

Worst-case optimization

Donald E. Knuth showed in 1976 that it is possible to determine every possible color code of the game (4 digits, 6 colors) in a maximum of five moves. To do this, the guessing person begins with a color code that contains two colors twice. This procedure, but also any color code advised below, can be derived according to a uniform procedure with a minimax character : Always guess a color code in which the maximum of the number of color codes that are still conceivable according to the various possible answers to the current guessing attempt, is as small as possible.

Average case optimization

Kenji Koyama and Tony W. Lai found a mastermind strategy in 1994 that requires an average of only 5625/1296 = 4,340 attempts, assuming that the combination to be searched for is likely to be randomly drawn from among the 1296 possibilities. This strategy - and any other that is just as fast on average - takes six tries in the worst case. They also proved that there can be no faster strategy.

Kenji Koyama and Tony W. Lai found their strategy through a brute force calculation of all possible strategies. Using a trick, they succeeded in reducing the search space so that a complete brute force search was possible with the hardware available in 1994.

Minimax strategy

If the strategic influence of the coder is also taken into account, a game-theoretical minimax value of 5600/1290 = 4.341 results for the search length. The coder selects his code from all 1290 codes with at least two colors with the same probability. The calculations for this were carried out independently for the first time by Tom Nestor (1985) and a few years later by Mike Wiener.

variants

In addition to this classic mastermind, there is the extended version Super Mastermind with five slots and eight colors. In German-speaking countries it is called SuperHirn professional ( Parker ).

There is also a “Grand Mastermind”. In this variant, the player also has to crack a code made up of four elements, but these elements are formed from a combination of a color and a shape. There are five different colors and five different shapes. There are also three “answer pens”. Black stands for a completely correct answer (combination and correct position), blue for a partially correct answer, and white for a correct color / shape pair in the wrong position.

Word Mastermind is the name of a variant in which the coder creates a four-letter meaningful word instead of a color code. The person guessing must try to guess the laid word using meaningful words.

literature

  • Leslie H. Ault: The Mastermind Handbook . Ravensburger Buchverlag, Ravensburg 1982, ISBN 3-473-42802-7

Web links

Individual evidence

  1. Toby Nelson: A Brief History of the Master Mind Board Game . (English)
  2. The 70’s ( Memento from May 9, 2008 in the Internet Archive ) at the Toy Retailers Association (English)
  3. Mastermind at abstractstrategy.com (English)
  4. a b Mastermind in the BoardGameGeek game database (English)
  5. Non-Invicta Mastermind at Toby Nelson (English)
  6. Patent US4241923 .
  7. Mastermind history ( Memento from August 12, 2007 in the Internet Archive ) at Invicta (English)
  8. Donald E. Knuth: The Computer as Master Mind . In: Journal of Recreational Mathematics , Vol. 9 (1), 1976–77 ( PDF )
  9. a b Jörg Bewersdorff : Luck, Logic and Bluff: Mathematics in Play - Methods, Results and Limits . 6th edition. Vieweg + Teubner Verlag, 2012, ISBN 978-3-8348-1923-9 , doi: 10.1007 / 978-3-8348-2319-9 , chapter 2.15
  10. Kenji Koyama, Tony W. Lai: An Optimal Mastermind Strategy . In: Journal of Recreational Mathematics , Volume 25, 1993, pp. 251-256
  11. Newsgroup note from Mike Wiener ( Memento from August 5, 2009 in the Internet Archive )
  12. ^ Donald Knuth: Selected papers on fun and games . Center for the Study of Language and Information, Stanford 2011, ISBN 978-1-57586-584-3 , p. 226
  13. Jörg Bewersdorff : Luck, Logic and Bluff: Mathematics in Play - Methods, Results and Limits . 7th edition. Springer-Spektrum, 2018, ISBN 978-3-658-21764-8 , doi: 10.1007 / 978-3-658-21765-5 , chapter 3.13