(given to François Caddet and Philippe Cloarec). We play on a chessboard with a side of 100. A move consists of choosing a rectangle formed by squares of the chessboard and inverting their colors. What is the smallest number of moves that allows the entire chessboard to be turned black?
. The answer is 100. We can achieve this by transforming one line out of two and then one column out of two. We cannot do less because inverting the colors of a rectangle can only change the number of separations between white and black squares on the edge of the chessboard by at most 4, and rectangles containing a corner can only change it by at most 2.