# 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?

1.  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.
2. 