Problems

In draughts, the king attacks by jumping over another draughts-piece. What is the maximum number of draughts kings we can place on the black squares of a standard $8\times 8$ draughts board, so that each king is attacking at least one other?

In order to encrypt telegraph signals it is necessary to divide every possible 10 character ‘word’ – an arrangement of 10 dots and dashes – into two groups, so that any two words in the same group differed by no fewer than three characters. Find a method of doing this or prove that no such method exists.

What is the maximum difference between neighbouring numbers, whose sum of digits is divisible by 7?

Four lamps need to be hung over a square ice-rink so that they fully illuminate it. What is the minimum height needed at which to hang the lamps if each lamp illuminates a circle of radius equal to the height at which it hangs?

Is it possible to place the numbers $1,2,\dots 12$ around a circle so that the difference between any two adjacent numbers is 3, 4, or 5?

Airlines connect pairs of cities. How can you connect 50 cities with the fewest number of airlines so that from every city you can get to any other city by taking at most two flights?

In a corridor of length 100 m, 20 sections of red carpet are laid out. The combined length of the sections is 1000 m. What is the largest number there can be of distinct stretches of the corridor that are not covered by carpet, given that the sections of carpet are all the same width as the corridor?

Two lines on the plane intersect at an angle $\alpha$. On one of them there is a flea. Every second it jumps from one line to the other (the point of intersection is considered to belong to both straight lines). It is known that the length of each of her jumps is 1 and that she never returns to the place where she was a second ago. After some time, the flea returned to its original point. Prove that for the angle $\alpha$ the value $\alpha /\pi$ is a rational number.

On a circle of radius 1, the point $O$ is marked and from this point, to the right, a notch is marked using a compass of radius $l$. From the obtained notch $O_1$, a new notch is marked, in the same direction with the same radius and this is process is repeated 1968 times. After this, the circle is cut at all 1968 notches, and we get 1968 arcs. How many different lengths of arcs can this result in?

A White Rook pursues a black bishop on a board of $3\times 1969$ cells (they walk in turn according to the usual rules). How should the rook play to take the bishop? White makes the first move.