Find these numbers. Ali-Baba is trying to get into a cave. At the entrance stand

Find these numbers.

Ali-Baba is trying to get into a cave. At the entrance stands a square table with a vessel in each corner. In each vessel there is a herring, which may be placed head-up or tail-up. From the outside, the positions of the herrings are not visible. Ali-Baba may put his hands into any two vessels, feel how the herrings are positioned, and set them however he likes (he may leave them as they were or flip one or both). This operation may be repeated several times. However, after each move the table is spun rapidly, so when it stops, it is impossible to tell which vessels were previously touched. The cave door opens if all herrings are in the same position. Help Ali-Baba find a strategy to enter the cave.

Tower of Hanoi. The “Tower of Hanoi” puzzle consists of three pegs, with seven rings of decreasing size stacked on one of them. It is allowed to remove one ring at a time from any peg and place it on any other peg, but it is forbidden to place a larger ring on top of a smaller one. Is it possible, following these rules, to transfer all rings to another peg?

On an infinite squared sheet of paper, 100 cells are coloured black and all others white. In one move, you may switch the colour of any four cells forming a 2×2 square. Prove that it is possible to make all cells white in several moves if and only if every row and every column contains an even number of black cells.

In each cell of an infinite squared sheet of paper, a natural number is written. It turns out that each number is equal to the arithmetic mean of its four neighbouring numbers. Prove that all the numbers are equal to each other.