You have an \(8\times 8\) chessboard coloured in the usual way. You can pick any two adjacent squares (i.e: any \(2\times 1\) or \(1\times 2\) section of the board) and flip the white tiles to black tiles and vice-versa. Is it possible to finish with \(63\) white pieces and \(1\) black piece?
We start with the point \((1,3)\) of the plane. We generate a sequence of points with the following rule: the \(x\)-coordinate of the new point is the arithmetic mean of the \(x\) and \(y\) coordinates of the previous point, and the \(y\)-coordinate of the new point is the harmonic mean of the \(x\) and \(y\) coordinates of the previous points. The harmonic mean of two numbers \(x\) and \(y\) is \(\frac{2}{\frac{1}{x} + \frac{1}{y}}\). Is the point \((3,2)\) in the sequence?
Four black dots are drawn on a whiteboard. On the dots we write the numbers \(10\), \(20\), \(30\), and \(40\) (one number on each dot). We then repeat the following move any number of times: choose one dot, decrease its number by \(3\), and increase the number on each of the other three dots by \(1\). After some number of moves, is it possible for all four dots to show the number \(25\) simultaneously?