[{"id":247278,"date":"2026-06-30T09:36:14","date_gmt":"2026-06-30T14:36:14","guid":{"rendered":"https:\/\/www.johndcook.com\/blog\/?p=247278"},"modified":"2026-06-30T09:36:14","modified_gmt":"2026-06-30T14:36:14","slug":"silver-kings","status":"publish","type":"post","link":"https:\/\/www.johndcook.com\/blog\/2026\/06\/30\/silver-kings\/","title":{"rendered":"Silver Rectangles and the Ways of Kings"},"content":{"rendered":"

Golden rectangles<\/h2>\n

The defining property of golden rectangle is that if you stick a square on its longer side, you get another golden rectangle.<\/p>\n

<\/p>\n

The smaller vertical rectangle is similar to the larger horizontal rectangle. This means<\/p>\n

\u03c6 \/ 1 = (1 + \u03c6) \/ \u03c6<\/p>\n

which tells us \u03c6\u00b2 = 1 + \u03c6 and so the golden ratio \u03c6 equals (1 + \u221a5)\/2.<\/p>\n

Silver rectangles<\/h2>\n

A silver rectangle is one that if you stick two<\/em> squares on its longer side you get another rectangle with the same aspect ratio.<\/p>\n

<\/p>\n

This tells us<\/p>\n

\u03c3 \/ 1 = (1 + 2\u03c3) \/ \u03c3<\/p>\n

and so \u03c3\u00b2 = 1 + 2\u03c3 and the silver ratio is \u03c3 = 1 + \u221a2.<\/p>\n

Just as you can define a golden ratio and a silver ratio, there’s an analogous way to define a sequence of metallic ratios<\/a>.<\/p>\n

Kings and Dellanoy numbers<\/h2>\n

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.<\/p>\n

A king can move one space in any direction. If we start with a king in the bottom left corner of the board, the no-backtracking requirement means the king can move up, right, or up and right.<\/p>\n

The number of paths a king can take from one corner to the opposite corner of an n<\/em> \u00d7 n<\/em> chessboard is the n<\/em>th central Delannoy number D<\/em>n<\/em><\/sub>. more generally Dellanoy numbers are defined for an m<\/em> \u00d7 n<\/em> chessboard, but I’ll stick to the case\u00a0m<\/em> =\u00a0n<\/em> called the\u00a0central<\/em> Dellanoy number, or just Dellanoy numbers for short.<\/p>\n

The first Delannoy number is 1 because there’s only one way for a king to get from one corner to the other: do nothing, because the opposite corner is the same corner. The second Delannoy number is 3 because the king can move up then right, or right then up, or move diagonally up and right.<\/p>\n

For a 3 \u00d7 3 grid things are significantly more complicated, and D<\/em>3<\/sub> = 13. For an 8 \u00d7 8 grid the number of paths is 48,639.<\/p>\n

Generating function<\/h2>\n

How would you estimate the number of paths on an n<\/em> \u00d7 n<\/em> board for large values of n<\/em> without calculating it exactly? You might start by finding a generating function for the Delannoy numbers, which works out to be<\/p>\n

(x<\/em>\u00b2 \u2212 6x<\/em> + 1)\u22121\/2<\/sup><\/p>\n

The radius of convergence r<\/em> for the generating function series is the distance from 0 to the closest singularity of the generating function, which is the smaller root of<\/p>\n

x<\/em>\u00b2 \u2212 6x<\/em> + 1<\/p>\n

which is<\/p>\n

3 \u2212 \u221a8 = (3 + \u221a8)\u22121<\/sup> = (1 + \u221a2)\u22122<\/sup> = 1\/\u03c3\u00b2<\/p>\n

i.e. the radius of convergence is the reciprocal of the silver ratio squared.<\/p>\n

Asymptotic estimate<\/h2>\n

The radius of convergence gives us a first approximation to the asymptotic size of the series coefficients. Since we’re working with the generating function of the Delannoy numbers, these coefficients are the Delannoy numbers. That is,<\/p>\n

D<\/em>n<\/em><\/sub> ~ r<\/em>\u2212n<\/em><\/sup> = (\u03c32<\/sup>)n<\/em><\/sup> = \u03c32n<\/em><\/sup>.<\/p>\n

That’s as good as you can do just knowing the radius of convergence. A more careful analysis would refine this estimate by dividing by a factor proportional to \u221an<\/em>.<\/p>\n

Related posts<\/h2>\n