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Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number. Cross out the numbers on the same row and column. Repeat this process. Add up you four numbers. Why do they always add up to 34?

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!