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.)

Some theory

Copyright © 1996-2009 Alexander Bogomolny

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

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

Copyright © 1996-2009 Alexander Bogomolny