An army of checkers pieces tries to jump five rows up the board. Can it ever make it?
On an infinite checkerboard, fill every square below a horizontal line with a checkers piece, as many as you like. A piece can jump over an adjacent piece into an empty square beyond, removing the piece it jumped, exactly as in checkers, but moving in any of the four grid directions. Using only these jumps, can you ever get a single piece to land five rows above the starting line?
Reveal the answer
No, it's mathematically impossible, no matter how many pieces you start with or how cleverly you sequence the jumps. John Horton Conway proved this in 1961 (reaching four rows up is achievable) by assigning each square a value based on powers of the golden ratio and showing the total value of all pieces can never increase, while reaching the fifth row would require exceeding the total value available even in an infinite starting army. The proof is a classic early example of a 'potential function' argument, now a standard technique across combinatorics and computer science.