Conway's soldiers, also known as the "solitaire army" game (Berlekamp et al. 1982) or "checker-jumping problem" (Honsberger 1976) is a one-player game played on an infinite checkerboard in which every square below a certain horizontal line is occupied by a piece (or "soldier"). Play proceeds by jumping horizontally or vertically over other pieces onto an empty space, where jumped-over pieces are then removed. Surprisingly, no matter how the game is played, it is impossible for any soldier to advance more than four rows beyond the initial line (Honsberger 1976, Berlekamp et al. 1982).
The minimum numbers of pieces needed to reach level 
in Conway's soldiers solitaire game for 
, 1, ..., 4 are given by the finite sequence 1, 2, 4, 8,
20 (OEIS A014225), with configurations attaining
level 
illustrated above.
The protagonist Christopher in the novel The Curious Incident of the Dog in the Night-Time remarks that Conway's soldiers is "a good maths problem to do in your head when you don't want to think about something else because you can make it as complicated as you need to fill your brain by making the board as big as you want and the moves as complicated as you want" (Haddon 2003, pp. 148-149).
