Sunday Times Teaser 3001 – Tetragonal Toy Tiles
by Stephen Hogg
Published Sunday March 29 2020 (link)
Thirteen toy tiles comprised a square, rectangles, rhombuses (diamonds on a playing card are rhombuses) and kites (as shown in the diagram). All of each different type were identical. A rhombus’s longer diagonal was a whole number of inches (equalling any diagonal of any other type). Its shorter diagonal was half this. Also, one side of a rectangle was slightly over one inch.
A pattern I made, using every tile laid flat, had all the symmetries of a square. After laying the first tile, each subsequent tile touched at least one other previously placed tile. Ultimately, any contact points were only where a vertex of a tile touched a vertex of just one other tile; only rhombuses touched every other tile type.
What, in inches, was a square’s diagonal?
Here is the pattern:
If the length of the square’s diagonal is \(d\), the rectangles have diagonals of length \(d\) and shorter sides of length \(\sqrt{2}(d/4)\). Hence their longer sides are of length \(\sqrt{14}(d/4)\). So we are looking for an integer length \(d\) such that the length of one of these sides is just larger than an inch.
Applying simple geometry to your picture, if the common diagonal = A, the rectangle width is
0.25 * √2 * A = 0.35355A. So it’s obvious that A = 3, width = 1.06066.