## Teaser 2955: Go forth and multiply

**From The Sunday Times, 12th May 2019** [link]

Adam and Eve have convex hexagonal gardens whose twelve sides are all the same whole number length in yards. Both gardens have at least two right-angled corners and the maximum possible area this allows. Each garden has a path from corner to corner down an axis of symmetry. Adam multiplies the sum of the path lengths by the difference of the path lengths (both in yards) and Eve squares Adam’s answer, getting a perfect fifth power with no repeated digit.

What was Eve’s answer?

See also: **Teaser 2946**.

[teaser2955]

## Jim Randell 5:28 pm

on9 May 2019 Permalink |Once you’ve worked out the shapes involved, you find that Eve’s answer is

8x⁴, and a simple program (or a simple guess) lets us find the appropriate answer.Run:[ @repl.it ]Solution:Eve’s answer was 32768.LikeLike

## Jim Randell 9:08 am

on12 May 2019 Permalink |Here’s the analysis to work out the formula for Eve’s answer:

If we consider a hexagon (with sides of unit length) that has a right angle at a given vertex, then we can’t have another right angle at an adjacent vertex, as it becomes impossible to construct a convex hexagon if we do.

So, we need only consider a pair of right angles separated by 1 or 2 intermediate vertices.

Taking the case with 2 intermediate vertices we get a hexagon that looks like this:

The length of the diagonal being

(1 + √2).In the case with only 1 intermediate vertex we get a hexagon that looks like this:

If the angle

XYZis θ, then the area of the triangleXYZis given by:which is at a maximum when

sin(θ) = 1, i.e.θ = 90°.And the length of the diagonal

d = √3.So, for hexagons with side

x, the two diagonals have lengths:Adam’s value (the sum multiplied by the difference) is:

And Eve’s value is the square of this:

And we are told

Eis an exact power of 5.Choosing

x = 8givesE = 8⁵ = 32768, which has five different digits as required.LikeLike

## Jim Randell 10:46 pm

on12 May 2019 Permalink |For completeness:

The general case of the first hexagon (with right angles separated by two vertices) is a shape like this:

The red line bisects the hexagon into two identical shapes (by rotation), but in general is not a line of reflective symmetry.

Again the area of the triangle XYZ is given by the formula:

and so the maximum area is achieved when

sin(θ) = 1, i.e.θ = 90°.Which gives us the diagram originally presented (where the red line

isa line of reflective symmetry).So both gardens have the same area, being composed of two (1, 1, √2) triangles and two (1, √2, √3) triangles. Giving a total area of (1 + √2).

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