For each arrangement of matches into piles, write the number of matches in each pile in bi...

For each arrangement of matches into piles, write the number of matches in each pile in binary notation, and then line up the digits of these numbers into columns (adding initial zeros where necessary). Show that a position is a winning one if and only if the number of 1s in each column is even. (For example: Three piles of 3, 4, and 7 give

where each column has exactly two 1s.) (Hint: Show that any move from a winning position produces a nonwinning one. Show that there is a move from any nonwinning position to a winning one )