# Coin Piles: Necessary Conditions for Emptying Both Piles ## First Necessary Condition: \( 3 \mid (a + b) \) Let \( a \) and \( b \) be the number of coins in the two piles. Without loss of generality, assume \( a \le b \). In every move, we remove a total of \( 3 \) coins from both piles. If the pile becomes empty after an integer number of moves, \( a + b \) must be a multiple of 3, i.e., \( 3 \mid (a + b) \). ## Second Necessary Condition: \( a \ge b/2 \) If \( a < b/2 \), both piles cannot become empty. Even if we remove only \( 1 \) coin from the first pile in every move, after \( a \) moves, the first pile will be empty but the second pile will contain \( b - 2a > 0 \) coins left.