Mastermind (board game)

Game of the Year 1973

Mastermind, also mastermind in the GDR as super code or Variablo known, is a logic game for two players that is played with colored pins that are inserted into a special frame. The game was invented in 1970 by Mordecai Meirovitz and became one of the most successful games of 1970. By 2000, more than 55 million games sold in 80 countries.

• 5.1 worst-case optimization
• 5.2 Average case optimization
• 5.3 minimax strategy

History

Mastermind was invented in 1970 by Mordechai (Marco ) Meirovitz, a Paris-based Israeli telecommunications expert. After several game companies rejected the game, he put the game before 1971 at the Nuremberg International Toy Fair. Founded by Edward Jones Fenleigh 1946 small English company Invicta Plastics of Leicester bought all the rights; Edward Jones Fenleigh refined the game and it was then 1971/72 published as the mastermind. It was immediately very successful and was the first Game of the Year 1973 Award in the UK by the British Association of Toy Retailers; until then the British Association of Toy Retailers award every year since 1965, had a toy (Toy of the Year ).

The game was released in 1973, Parker Brothers, Pressman Toy 1981 and many other publishers. It also appeared several variants for various publishers.

Mastermind for four players was patented in 1980.

In the German-speaking world it was initially and later marketed under the name Super brain as a mastermind. In the GDR, it was code (VEB Plasticart ) sold and distributed primarily in the English-speaking world as the mastermind under Variablo (VEB Berlin plastic toys ) or Super.

Today, Invicta has licensed the game to Hasbro, Pressman Toy and Orda Industries.

Gameplay

One player ( the encoder ) determines the beginning set a four-digit -level color code, which is selected from six colors; A color can also be used several times. The other player ( the rater ) trying to figure out the code. This purpose it uses a similar color code as a question; the first train blind advised at the other trains using the answers to the previous trains.

In any train towards the rater gets the information, how many pins it has set properly in color and position and indeed how many pins the right color, but are in the wrong position. A hit in both color and position is indicated by a black pen, a color- correct pin in the wrong place by a white pen. There are also versions with red instead of black pins to display hits in color and position. All questions and answers will remain visible until the end of the game.

The aim of the raters is to guess the color code with as few questions. There are also variants with two game boards where everyone is at the same time on a board rater and on the other board encoder and ask both alternately. The task of the encoder after setting the color code is totally predestined; he has no choice, can of course still be wrong, which is why the part of the coder can be entrusted to a computer.

Example

Wanted is the color code:

Green - red - blue - green

First rate trial: red - yellow - red - green

Answer: Once black ( because green is right at the fourth position ), one white ( because red in this code occurs once, but not on the first or third place).

Second rate attempt: green - green - orange - red

Answer: Once black (because green right this time at the first position ), white twice ( because 1 green occurs in this code a second time, but does not occur at the second position and 2 red in this code, but not at the fourth position).

If the correct color code is found, the answer is four times black. The ratende player usually has twelve attempts to find the right solution. Depending on the version of the game can vary this number; the smaller travel version only has room for six attempts.

Outcome space

Mastermind

The number of possible codes. At 6 color codes all 4 pins have the same color, at 210, there are exactly two colors (90 x 2 2 pins and 120 times 3 1 pins ), at 720, there are exactly 3 colors ( 2 1 1 pins ), and at 360 color codes all 4 pins have different colors.

Super Mastermind

When published in 1975 Super Mastermind Version 5 holes and 8 colors are available. The number of possible code is there: Like with the original mastermind the players stand to 12 attempts to solve the task.

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

A good Mastermind player requires powers of deduction and logic skills.

In the context of mathematical investigations, the search strategy in the Mastermind game can be optimized under various aspects. As a criterion either the worst possible case (worst case), the probabilistic average ( average case ) or the game-theoretic minimax value is defined as:

Worst- case optimization

Donald E. Knuth has shown in 1976 that it is possible, any color code of the game (4 points, 6 colors) to determine in a maximum of five trains.

Average case optimization

Kenji Koyama and Tony W. Lai found 1994 Mastermind strategy, which requires an average of 5625/1296 = 4,340 tests, provided one assumes that to be searched combination is equally likely to be drawn at random from the 1296 possibilities. This strategy - and each other, that is, on average, just as fast - requires six tests in the worst case. They proved also 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. By a trick, they have succeeded, to reduce the search space so that a full brute force search with the 1994 hardware available was possible.

Minimax strategy

If one also considers the strategic influence of the encoder, the result for the search length of a game theoretic minimax value of 5600/1290 = 4.341. In this case, the encoder selects its code among all 1290 codes with at least two colors equally probable from. The calculations to have been independently for the first time by Tom Nestor (1985 ) and some years later, Wiener Mike performed.

Variants

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

In addition, there is also a " Grand Master Mind". In this variant, the player must also crack a code made ​​up of four elements, these elements are, however, formed from a combination of a color of a shape. There are five different colors and five different shapes. In addition, there are three " response pins ." Black is ( correct combination and location ) for a completely correct answer, blue for a partially correct answer, and white for a proper color / shape pair in the wrong place.

Word Mastermind calls itself a variant in which the encoder instead of a color code specifies a vierbuchstabiges important word of. The Ratende must try to guess the word specified by also supporting meaningful words.