Place as many soldiers as you like below the line; they advance by peg-solitaire jumps. Reaching row 1 takes 2 soldiers, row 2 takes 4, row 3 takes 8, row 4 takes 20 — and row 5 cannot be reached by any finite army, ever. The proof weighs the entire battlefield with powers of the golden ratio and watches the total refuse to grow.
Every jump is a purchase: you spend the soldier who gets jumped to advance the soldier who jumps. Armies only ever shrink, so the question is what the shrinking buys. Row 1 costs 2 men, row 2 costs 4, row 3 costs 8 — and then the pattern lies to you, because row 4 costs 20, not 16.
Conway's trick is to price the battlefield before anyone moves: each cell costs x^d for distance d from the star, with x chosen so that even a perfect jump only breaks even — you gain the landing cell but pay for both soldiers. Any imperfect jump is a straight loss. The star costs 1.000, and for a row-5 star, everything you could ever muster adds up to just under 1.