Silver Rectangles and the Ways of Kings

Golden rectangles The defining property of golden rectangle is that if you stick a square on its longer side, you get another golden rectangle. The smaller vertical rectangle is similar to the larger horizontal rectangle. This means φ / 1 = (1 + φ) / φ which tells us φ² = 1 + φ and so the golden ratio φ equals (1 + √5)/2. Silver rectangles A silver rectangle is one that if you stick two squares on its longer side you get another rectangle with the same aspect ratio. This tells us σ / 1 = (1 + 2σ) / σ and so σ² = 1 + 2σ and the silver ratio is σ = 1 + √2. Just as you can define a golden ratio and a silver ratio, there’s an analogous way to define a sequence of metallic ratios. Kings and Dellanoy numbers The silver ratio has several connections to the ways of ways kings. By that I mean the number of ways a king can go from one corner of a chessboard to the diagonally opposite corner without backtracking. A king can move one space in any direction. If we start with a king in the bottom…