## Teaser 3001: Tetragonal toy tiles

**From The Sunday Times, 29th March 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?

[teaser3001]

## Jim Randell 6:29 pm

on27 March 2020 Permalink |I came up with a pattern for 13 tiles that has the same symmetries as a square (I’ll make a diagram of it later), and that gave me a way to calculate the sides of the rectangle, in terms of the larger diagonal of the rhombus,

x.Once these are calculated the value of

xcan be determined manually, or here is a quick program to do it:Run:[ @repl.it ]Solution:The length of the square’s diagonal is 3 inches.I arranged 13 shapes (1 square, 4 rectangles, 4 rhombuses, 4 kites) into the following pattern:

All the diagonals of all the shapes are equal (=

x), except for the shorter diagonal of the rhombus (=x/2).In this arrangement the short side of the rectangle is the hypotenuse of a right-angled triangle where the other two sides have length

x/4, so it has a length ofx√(1/8), and so the longer side has a length ofx√(7/8).Setting

x = 3gives dimensions of 1.061 × 2.806 inches. The smaller side being close to 1 + 1/16 inches. Which is “slightly over 1 inch” as required.The exact shape of the kite doesn’t matter (as long as both diagonals are

x), it doesn’t affect the calculation for the rectangle. (In particular the kites could all be rotated through 180° to give a second arrangement).Placing the rhombuses the other way leaves a gap that cannot be filled by the required rectangle, and we don’t have enough shapes to fill the gap with multiple shapes.

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## Robert Brown 8:06 am

on28 March 2020 Permalink |I did a scale drawing. My kite has the same aspect ratio as the one in the original text, which makes the large angle equal to that of the rhombus. I don’t think your program would have found my answer, which has the rectangle 1.03 inches high.

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## Jim Randell 8:13 am

on28 March 2020 Permalink |@Robert: My arrangement looks like this [ link ], so the exact shape of the kite doesn’t affect the calculations. But I didn’t look too hard for alternative patterns (although you would hope the setter would have made sure there was a unique answer to the puzzle).

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## Robert Brown 11:54 am

on28 March 2020 Permalink |Interesting. Each rhombus has a thin & thick ‘corner’. My layout has the thin corners connected to the square, then the kites & rhombuses are all angled to fit round the square. I tried (but failed) to get the rectangles in the corner, to give it ‘all the symmetries of a square’ ! My rectangle is long & thin, with diagonal =9 inches. I see Brian has 3 inches, I wonder what his layout looks like . . .

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## Jim Randell 12:36 pm

on28 March 2020 Permalink |I tried my pattern with the rhombuses turned through 90°, but I found the gap between the remaining vertices was too large to fit any of the other shapes into.

Looking at Brian’s attachment to the Google Sites page it looks like he has found the same layout I did (although I don’t think his kites are the right shape, but that doesn’t change the answer).

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## Brian Gladman 1:15 pm

on28 March 2020 Permalink |That is because the teaser doesn’t explicitly constrain the non-rhombus shapes to have equal diagonals (of course, all except the kite do). The longest rhombus diagonal can be any diagonal of any other shape and I chose to match the longest diagonals of the rhombus and the kite.

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## Jim Randell 1:20 pm

on28 March 2020 Permalink |@Brian: It does say that the longer diagonal of the rhombus should “equal

anydiagonal ofanyother type”, so I think the vertices of the kite must touch the sides of anxbyxbounding box.LikeLike

## Brian Gladman 2:36 pm

on28 March 2020 Permalink |Yes, you interpret it to mean “equal to the diagonals of the other types” (surely a simpler expression of this interpretation), whereas I interpret it to mean a choice between “any of the diagonals of any other type”. Fortunately this clear ambiguity (!) doesn’t have an impact on the answer.

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## Robert Brown 12:33 pm

on28 March 2020 Permalink |So my alternative pattern lacks one of the required symmetries (it has clockwise & counter clockwise versions). We’ve had problems with words before . . . I guess Brian’s layout is similar to yours.

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