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# An Interesting KS3/KS4 Problem from @m4thsdotcom

In January I noticed that Steve Blades (@m4thsdotcom) was posting some interesting problems, many designed to stretch and promote thinking in students studying for their GCSEs.

I was particularly drawn to this question in the middle of January

Initially I thought this was harder than it was and was intending to use the information about the interior and exterior angles to form two simultaneous equations in $$x$$ and $$n$$ where $$n$$ is the number of sides of the underlying polygon. However, there is of course a much simpler way if you use the fact that the interior and exterior angles of the polygon must add up to $$180^{\circ}$$.

Using this fact you can obtain $$7x+1+28x+4 = 180$$ and so $$35x = 175$$ giving $$x = 5^{\circ}$$. Hence the exterior angle of the polygon is $$7 \times 5 + 1 = 36^{\circ}$$ meaning that the polygon has $$10$$ sides and is a decagon.

I would be fascinated to see how students go about solving this question.

Steve has also collected a paper worth of challenging problems in a “Grade 9” paper available here. I particularly like this question about functions: