Camel
Let’s describe a new “chess” piece and call it “camel-tone”. The piece moves jumping: horizontally or vertically – over two chessboard squares, or diagonally – over one square. The picture shows a part of the board with a camel-tone, placed in the center and the positions (marked by x), where it can go with one move. Of course, it cannot go outside the playing board, which happens to be a big square, divided into N×NN\times NN×N little squares. In this task NNN is always divisible by 555.
The camel-tone starts at the square in the top-left corner of the board. The game consists of making a sequence of moves on the board, visiting every square exactly once. Moreover, after N2−1N^2-1N2−1 moves the piece should be exactly one move away from its starting position. This is a so-called “camel-tonian cycle”!