One Pile

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.)


This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


What if applet does not run?

Some theory

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Copyright © 1996-2017 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

Pile size 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Grundy number 0 0 0 1 1 1 2 2 0 0 0 1 1 1 2 2 0 0 0 1 1

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

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  • |Contact| |Front page| |Contents| |Algebra|

    Copyright © 1996-2017 Alexander Bogomolny

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