Take turns moving a piece orthogonally into the empty square. Win by stalemating the opponent.
 
Lewthwaite's Game is more of a puzzle than a game. It has been proven a win for the second player. In Winning Ways For Your Mathematical Plays, the authors suggest adding a rule: A player may slide a connected row or column of stones (up to four) provided both ends of the moving group belong to the player. Even with this rule change, however, the game remains a simple win for the second player, as the computer will quickly demonstrate.
 
At the World of Abstract Games web site, where the game is profiled, Joao Pedro Neto suggests a "switch rule" whereby a stone adjacent to the empty square can swap with an adjacent enemy stone. This rule addition, coupled with the requirement that the opponent cannot swap the same two stones on the next turn, appears to make the game more competitive.
 
This file includes the original game, a "push" variant incorporating the rule suggested in Winning Ways, a "swap" variant following Neto's suggestion, and a combination push and swap variant.