# Sunday Times Teaser 2946 – Pentagonal Gardens

### by Nick MacKinnon

#### Published March 10 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?

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The layouts of the two pentagonal gardens are shown above with the areas of their delineated elements.  The example of a possible square lawn in the right hand garden is shown in outline to avoid obscuring the calculation of the overall garden area (all garden/lawn side lengths are $$a$$ metres). The total garden areas are hence: $(4+\sqrt{3})a^2/4$ $(4+\sqrt{7})a^2/4$ This gives the difference between the areas of the two gardens as: $(\sqrt{7}-\sqrt{3})a^2/4$ and the sum of the two paved areas (the garden areas less a square lawn of area $$a^2$$) as: $(\sqrt{7}+\sqrt{3})a^2/4$ The product of these two areas is hence $a^4/4$ which we are told is a five digit integer with 5 different digits.  Given the presence of the fourth power, we only need to consider even integers between 16 and 24, which is easily done manually.  But here is a Python program to do the job: