• One Mile
• One Pile Dual Switches Control Two Lights

Abstract

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The purpose of this paper is to solve a class of combinatorial games consisting of one-pile counter pickup games for which the number of counters that can be removed on each successive turn changes during the play of the game. Both the minimum and the maximum number of counters that can be removed is dependent upon the move number. Also, on each move, the opposing player can block some of the moving player’s options. This number of blocks also depends upon the move number. There is great interest in generalizations and modifications of simple, deterministic two-player “take-away-games ” — for a nice survey, see chapter 4 of [1]. We discuss here a modification where the player-not-to-move may effect the options of the other player. Modifications of this type have been called Muller twists in the literature. See [4]. In [3], we discuss games in which the number of counters that can be removed depends on the number removed in the previous move. the electronic journal of combinatorics 10 (2003), #N00 1We begin with some notation. The set of integers is denoted by Z, the positive

One Pile is the most direct generalization of Scoring and the simplest of the Subtraction games. On each move a player is permitted to remove any number of objects bounded both from above and below. (In the applet, a move is performed by pressing one of the buttons located on the perimeter of the drawing area. The Min and Max attributes can be modified by clicking a little off their central line.)

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One Mile

Copyright © 1996-2018 Alexander Bogomolny

Theory of One Pile

The Grundy numbers for the various sizes of the pile are easily found with the Mex rule. For example, for Min = 3 and Max = 5, we get

One Pile Dual Switches Control Two Lights

Ambicom driver download for windows 10. The P-positions correspond to the piles whose size S falls into the range 0 ≤ S mod (Min + Max) < Min.

Copyright © 1996-2018 Alexander Bogomolny

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