Marienbad (game)

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The Marienbad game is a variant of the Nim or Misère game made famous by Alain Resnais' 1961 film Last Year in Marienbad .

The rules

Nimsy.jpg

A player places sixteen matches in four rows according to the diagram opposite:

The two players take turns taking matches from one of the rows. When making a move, only matches may be removed from a single row; however, it is up to the player to decide how many: at least one, at most all.

The player who has to take the last match away loses.

Winning strategy

In this game there is a winning strategy for those who follow suit (see Nim game ), for this one writes the number of matches in the individual rows in the dual system : 

1 = 0 0 1
3 = 0 1 1
5 = 1 0 1
7 = 1 1 1

and forms the corresponding column sums, i.e. H. 

S = 2 2 4

In the starting position these column sums are all even .

In order to be able to win the game with certainty, one must observe the following two strategies in sequence:

Initial strategy

You let the opponent begin: Depending on how the opponent moves, you have to remove as many matches on the next move that all column sums are even again afterwards.

If you start yourself, you cannot force victory; one must hope that the opponent does not know the strategy and that, through a mistake, it is possible to produce straight column totals again.

example 1

Assuming the first player takes all seven matches from the fourth row on the first move, then the following applies 

1 = 0 0 1
3 = 0 1 1
5 = 1 0 1
0 = 0 0 0

and give the corresponding column totals 

S = 1 1 3.

Now the second player takes three matches from the third row: 

1 = 0 0 1
3 = 0 1 1
2 = 0 1 0
0 = 0 0 0

and the column totals 

S = 0 2 2

are all straight again.

In this way you continue the game until it comes to a position in which you can only get rows with one match each by one move. - Now you only pay attention to the number of matches and the number of rows and no longer to the dual sums.

End strategy

You move in such a way that after the move there is an odd number of rows of ones. This way of playing forces the opponent to take the last match.

Example 2

One finds the following position: 

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You take four matches from the last row according to the end strategy so that one remains. - The other is forced to take the last one.

Example 1 (continued)

Suppose the first player takes a stick from the second row, so that the following position arises: 

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so the second player takes the match from the first row according to the starting strategy. 

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If the first player takes a match now, the second player takes away the other row entirely according to the end strategy. If, on the other hand, the first takes one row completely, the second takes one off the other. In both cases, the first player has to take the last stick.

literature

Jörg Bewersdorff : Luck, Logic and Bluff: Mathematics in Play - Methods, Results and Limits , Vieweg + Teubner Verlag, 5th edition 2010, ISBN 3834807753 , doi: 10.1007 / 978-3-8348-9696-4 .

Web links

Individual evidence

  1. ↑ Bewersdorff: Glück, Logic 2010, p. 168