In how many different ways can you place \(12\) chips in the squares of a \(4\times 4\) chessboard so that
there is at most one chip in each square, and
every row and every column contains exactly three chips?
On a TV screen the number \(1\) appears. Every minute that passes by, the number that is currently on the screen increases by the sum of its digits. For example: if at some point the number \(12\) appears on the screen, the next number will be \(12+(1+2)=15.\) Will the number \(123456\) ever appear on the screen?
Seven vertices of a cube are labelled with the number \(0\), and the remaining vertex is labelled with \(1\). You are allowed to repeat the following move: choose an edge of the cube and increase by \(1\) the numbers at both ends of that edge.
Is it possible to reach eight numbers that are all divisible by three?
Each number on the number line (not only whole numbers!) is painted black \((B)\) or white \((W)\). Is it always possible, regardless of how the number line is painted, to find a line segment such that both its endpoints and its middle point are painted with the same colour?