Problem
COM-B2-M05-P001 Two Coins per Move
#1
★★☆☆☆ Level 2 of 5
There are \(25\) coins on a table. Initially exactly one coin shows its black side. In one move exactly two coins must be flipped. Prove that it is impossible to reach a position in which all coins show their white side.
Track the parity of the number of black coins.
In one move, the number of black coins either increases by \(2\), decreases by \(2\), or stays the same. Therefore its parity is preserved.
Initially the number of black coins is \(1\), which is odd. In the target position it is \(0\), which is even. The invariant values differ, so the position is unreachable.
A basic entry problem: one invariant, one impossibility.