Numberphile

Red & Black Knights (extraordinary result) - Numberphile

The video explores two distinct knight-moving puzzles on an infinite chessboard. The first scenario involves a single knight that always moves to the lowest unvisited square. Starting at zero, the knight follows a path, marking visited squares. Initially, it seems the knight will continue indefinitely, but eventually, it gets trapped, having visited all possible adjacent squares. This results in a finite sequence of squares. The second, more complex scenario introduces multiple "courteous" knights of two colors, red and black. The rule for placing knights is that a new knight is placed on the first unattacked square encountered along a spiral path. In the first variation of this, only one type of knight is placed. A knight is placed at zero, then at one, and so on, skipping squares attacked by already placed knights. This creates interesting, periodic patterns, like clusters of five knights separated by singles, and vertical lines with alternating clusters of two and four. After a thousand steps, the pattern is visually striking and mathematically precise, showing a periodic structure.