Reaching row 5 in Solitaire Army

Introduction

Solitaire Army, also known as Conway's Soldiers, is a mathematical game similar to peg solitaire. The setup is: you have an infinite peg solitaire board extending in all directions, and the starting position has a peg in every single position on the x-axis or south of it. In other words, every position (x,y) with y ≤ 0 contains a peg.

Legal moves are precisely those of peg solitaire: you may move a single peg exactly two spaces in any of the four orthogonal directions, provided the space it jumps over contains a peg and the space it ends up in is empty. The peg jumped over is removed.

The aim is to advance one peg as far north as possible, using up as many other pegs as you need to help get it there. Reaching y=1, for example, can trivially be done in one move; reaching y=3 is reasonably easy; reaching y=4 takes nearly twenty moves even with optimal play.