Cricket Pitch · Rulesets

Rules

Board. A (finite) ordered list consisting of non-negative integers and a single abstract symbol $\circledcirc$ called the roller.

Left moves. Move the roller left over a positive number of positive integers, and reduce each integer rolled over by 1.

Right moves. Move the roller right over a positive number of positive integers, and reduce each integer rolled over by 1.

Convention. Classical (normal or misère play).

Examples

From the board $[1,0,2,1,\circledcirc,2,3]$ , the players have the following options:

Left: $[1,0,2,\circledcirc,0,2,3]$ , $[1,0,\circledcirc,1,0,2,3]$ .
Right: $[1,0,2,1,1,\circledcirc,3]$ , $[1,0,2,1,1,2,\circledcirc]$ .

Notes

This is a short partizan ruleset. It is partizan because only Left can move the roller to the left (and only Right can move it to the right). To see why it is short, consider the following:

finite: the length of the list is finite, and each element (except for the roller) is bounded by the maximum of the list, so the number of subpositions is always finite.
loop-free: at least one bump gets rolled over per move, so the sum of the sizes of the bumps decreases by at least 1 each turn. (So, in fact, the birthday of a position is bounded above by the sum of the sizes of the bumps.)

For every board position, there exists an isomorphic board position that contains no $0$ value. This is because a $0$ can never be rolled over, and so one needs only to retain the bumps reachable by the roller. For example, $[0,2,3,0,4,\circledcirc,1,0,1]$ is isomorphic to $[4,\circledcirc,1]$ .

Transfinite Cricket Pitch

One could imagine generalising the game such that instead of the board position being a finite ordered list, we instead have an infinite sequence. More generally, a board position could be considered as a non-empty totalled ordered index set $I$ and a mapping $f\colon I\to\mathbb{N}\cup\{\circledcirc\}$ such that there exists a unique $i\in I$ with $f(i)=\circledcirc$ .

Left can move from a board position $f$ to a board position $g$ (where $g$ uses the same index set as $f$ ) when the following properties are satisfied:

$g^{-1}(\circledcirc)<f^{-1}(\circledcirc)$ ;
if $i<g^{-1}(\circledcirc)$ , then $f(i)=g(i)$ ;
if $g^{-1}(\circledcirc)\le i<f^{-1}(\circledcirc)$ , then $f(i)=g(i+1)+1$ ;
if $i>f^{-1}(\circledcirc)$ , then $f(i)=g(i)$ .

Right’s moves can be described symmetrically.

The finite Cricket Pitch positions described earlier could be described within this transfinite setting as all board positions $f$ such that there exist only finitely many $i\in I$ with $f(i)\neq0$ .

Note that Transfinite Cricket Pitch is neither finite (surprise, surprise!) nor loop-free. A simple example of why it is not loop-free can be seen with the position $[\dots,1,1,1,\circledcirc]$ , since Left can move the roller one place to the left, resulting in an isomorphic position to the start (i.e. a loop).

In general, Transfinite Cricket Pitch is not a stopper: consider the position $[\dots,3,3,3,\circledcirc]$ , from which there clearly exists an infinite alternating run.