## Teaser 2946: Pentagonal gardens

**From The Sunday Times, 10th March 2019** [link]

Adam and Eve have convex pentagonal gardens consisting of a square lawn and paving. Both gardens have more than one right-angled corner. All the sides of the gardens and lawns are the same whole number length in metres, but Adam’s garden has a larger total area. Eve has worked out that the difference in the areas of the gardens multiplied by the sum of the paved areas (both in square metres) is a five-digit number with five different digits.

What is that number?

[teaser2946]

## Jim Randell 1:16 am

on10 March 2019 Permalink |If I’ve understood this correctly there are only two ways to construct the gardens, and this gives a fairly limited set of integer sides. And only one of those gives a viable 5-digit number.

I worked out 2 possible shapes for the gardens, where the lawn forms most of the perimeter:

To make calculating the area easier I have split the lawn of A’s garden in two and inserted the paved area between. The areas are the same as if the lawn was a contiguous square.

If we suppose the line segments making up the perimeter of the gardens are units, then both squares are unit squares and both triangles have a base of length 1.

A’s triangle has a height of

√(7)/2and an area of√(7)/4. E’s triangle has a height of√(3)/2and an area of√(3)/4.The difference in the total areas of the gardens is:

The sum of the paved areas is:

Multiplying these together we get:

So, if the gardens were all of side

x, the numberNwould be:For a 5-digit value for

Nthere are only a few values ofxto try, and this can be completed manually or programatically.This Python program runs in 77ms.

Run:[ @repl.it ]Solution:The number is 16384.LikeLike

## Jim Randell 8:55 am

on10 March 2019 Permalink |I’m pretty sure these are the only two possible shapes.

If you imagine trying to form a set of 5 equal length linked rods into a

convexpentagon with more than one right angle, then it is not possible with 3 right angles, so there must be exactly 2 right angles. And these are either on adjacent vertices or have one vertex between them.Either way, the remaining rods are forced into unique positions, giving the shapes A and E.

Although note that with A there are many ways that a unit square can fit inside the pentagon (not necessarily touching the perimeter).

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