Red & Black Knights (extraordinary result) - Numberphile

TL;DR

Placing red and black knights alternately on an infinite spiral chessboard produces unexpected, self-organizing color-segregated quadrants rather than random or periodic patterns.

Key Points

• 1.The single-color knight problem produces a perfectly periodic pattern. One knight placed at every unattacked square on an infinite spiral board creates clusters of five separated by singles, with a precise 2-4-2-4 repeating structure along axes — regular and unsurprising.
• 2.Two-color knights follow a turn-based placement rule. Black knights avoid squares attacked by red knights; red knights avoid squares attacked by black knights — each color is indifferent to its own attacks, creating two competing 'armies' claiming territory.
• 3.The idea came from Jonas Carlson in Sweden. He sent Neil a letter describing the problem, calling the result 'totally unbelievable,' and shared images up to 100,000 cells showing unexpected strip-like color separation.
• 4.At 100,000 cells, strips and islands of pure color emerge. One quadrant becomes almost entirely red, another mostly black, with mixed 'undecided' zones — a structure nobody would predict from the simple local placement rule.
• 5.By 64 million cells, black dominates two full quadrants and red dominates two others. Thin indecisive strips separate them; Michael Branicki produced visualizations showing this permanent, self-reinforcing segregation that grows indefinitely.
• 6.Neither color gains a first-mover advantage, and the result has no known explanation. Despite black playing first, territory evens out; the emergence of solid quadrants from a local rule remains mathematically mysterious, and a three-color extension is teased in a bonus video.