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Hallway Borders

Age 11 to 14 Challenge Level:

Carla, Michael and Andrew from Smithdon HS, Hunstanton sent in a correct solution.

It is important to realise that if the hallway is $x$ by $y$ feet, then the perimeter involves $2x +2y - 4$ tiles, rather than $2x + 2y$. This is because the tiles in the four corners go along two edges each, so are counted twice when we calculate that the perimeter is $2x + 2y$ feet.

Then the solution is based on solving:

$$2x +2y - 4 = \frac{1}{2} xy$$
which after a clever rearrangement looks like

$$ y = 4 + \frac{8}{(x - 4)} $$
so we know $x - 4$ is a factor of $8$.

Ignoring negative solutions leaves the positive solutions of $x=5,\,6,\,8,\,12$ and therefore $y=12,\,8,\,6,\,5$ correspondingly.

Hence the hallway is either $5$ by $12$ feet or $6$ by $8$ feet.

Did you expect to find two possible answers? Well done if you got both!