You may also like

problem icon


Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

problem icon

Great Squares

Investigate how this pattern of squares continues. You could measure lengths, areas and angles.

problem icon

Square Areas

Can you work out the area of the inner square and give an explanation of how you did it?

Hallway Borders

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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!