An $8\times8$ checkerboard is filled with two-sided coins (that are blue on one side and red on the other side).
The following picture shows three examples of a cross (multiplication sign): the five red coins at the right margin form a standard cross centered at their middle coin; the two red coins in the upper left corner form a cross that is centered at the coin in the corner (the three remaining squares of the cross are outside the board and irrelevant); the three red coins in the middle of the upper half form another cross (with two squares outside the board):
Marco starts with the situation where all coins are blue, and by flipping some coins he wants to reach the situation where all coins are red. Whenever Marco flips one coin $x$, the other coins from the cross centered at $x$ are flipped simultaneously.
- Can Marco reach the situation where all coins are red?
this question is worth 100 points
- What is the number of moves to reach the goal according of $k$ where $k$ red coins are stochatically distributed on the board ?
Hint: Linear algerba is required