In a bag we have \(99\) red balls and \(99\) blue balls. We take balls from the bag, two balls at a time:
If the two balls are of the same colour, then we put in a red ball to the bag.
If the two balls are of different colour, we return a blue ball to the bag.
Regardless, after each step, one ball is lost from the bag, so eventually there will be only one ball. What is the colour of this last ball?
You have an \(8\times 8\) chessboard coloured in the usual way. You can pick any \(2\times 1\) or \(1\times 2\) piece 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 \(S=(1,3)\) of the plane. We generate a sequence of points with coordinates \((x_n,y_n)\) with the following rule: \[x_0=1,y_0=3\qquad x_{n+1}=\frac{x_n+y_n}{2}\qquad y_{n+1}=\frac{2x_ny_n}{x_n+y_n}\] Is the point \((3,2)\) in the sequence?
Four black dots are drawn on a whiteboard. We write the numbers \(10,20,30,\) and \(40\) on top the dots and then play the following game: choose any of the four dots, subtract three from the number written on it, and then add one to the number written on top of each of the three other dots. After playing this game for some number of steps, can we make all four dots have the number \(25\) on top of them?