21-Coin Subtraction Game with Modulo Invariant

Scenario

There are 21 coins on a table. Two players alternate turns. On each turn, a player must remove either 1, 2, or 3 coins. The player who takes the last coin wins the entire game.

Your Task

1. Model the game using combinatorial game theory ($P$-positions and $N$-positions).
2. Identify the modular invariant $n \pmod 4$ that characterizes all losing ($P$) positions.
3. Prove whether Player 1 (first mover) or Player 2 has a guaranteed winning strategy from initial position $N = 21$.
4. Specify the exact opening move and the reactive invariant that guarantees victory.

Constraints

• Finite, impartial game of normal play convention.
• Moves restricted to $k \in \{1, 2, 3\}$.

Examples From $N = 4$, whoever moves first must leave 1, 2, or 3 coins, allowing the second player to take the rest and win.