Research on things from my childhood

Cellular automata with spaceships

Theorem. If a 2-D Cellular Automata has a spaceship, then its topological entropy is infinite.

Article Lakshtanov E.L., Langvagen E.S.”Criterion of infinite topological entropy for multidimensional cellular automata”, Information Transmission Problems, vol.40, (2), pp. 70-72, 2004.

Article . Lakshtanov E.L., Langvagen E.S.,”Entropy of multidimensional cellular automata” Information Transmission Problems, vol.42, (1), 2006.

Minesweeper and spectral analysis

It is shown that certain configurations of open cells guarantee the existence and the uniqueness of solution. Mathematically the problem is reduced to some spectral properties of discrete differential operators. It is shown how the uniqueness can be used to create a new game which preserves the spirit of "Minesweeper" but does not require a computer.
Lakshtanov main page

ArticleO.German, E.Lakshtanov, Application of harmonic analysis for creating of pen-and-pencil games. Math. Notes, vol. 88, 5-6, 2010.

Arxiv O.German, E.Lakshtanov, "Minesweeper" and spectrum of discrete Laplacians, Applicable Analysis, Vol. 89, No. 12, December 2010, 1907–1916.

Finiteness in the card game of war

The game of war is one of the most popular international children's card games. In the beginning of the game, the pack is split into two parts, then on each move the players reveal their top cards. The player having the highest card collects both and returns them to the bottom of his hand. The player left with no cards loses. Those who played this game in their childhood did not always have enough patience to wait until the end of the game. A player who has collected almost all the cards can lose all but a few cards in the next 3 minutes. That way the children essentially conduct mathematical experiments observing chaotic dynamics. However, it is not quite so, as the rules of the game do not prescribe the order in which the winning player will put his take to the bottom of his hand: own card, then rival's or vice versa: rival's card, then own. We provide an example of a cycling game with fixed rules. Assume now that each player can seldom but regularly change the returning order. We have managed to prove that in this case the mathematical expectation of the length of the game is finite. In principle it is equivalent to the graph of the game, which has got edges corresponding to all acceptable transitions, having got the following property: from each initial configuration there is at least one path to the end of the game.
Lakshtanov main page

Arxiv E.Lakshtanov, V.Roshchina, Finiteness in the Card Game of War, American Mathematical Monthly, V.119(4), pp.318-323, 2012

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