The DP reveals a pattern: you win if n≡0(mod4). Can we prove this directly? From n≡0(mod4): any move (1, 2, or 3) leaves n′≡0(mod4). Opponent wins. From n≡0(mod4): take nmod4 stones. Leave n′≡0(mod4). Opponent loses.
This is the power of game theory DP: compute the pattern with DP, then prove and simplify to O(1). This pattern appears in many game theory problems. Look for cyclic structures in the win/lose sequence.