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A floor is covered by a tessellation of equilateral triangles, each having three equal arcs inside it. What proportion of the area of the tessellation is shaded?

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Heptagon Has

Age 11 to 14 Short Challenge Level:

It can possess all five properties.
The description does not say that heptagon has to be convex, i.e. all of its interior angles need not be less than $180^{\circ}$. Since $(2 \timesĀ 7 - 4) \times 90 = 900$, the interior angles of all heptagons total $900^{\circ}$. The creation of a heptagon with all the given conditions is possible as the diagram shows. Notice that four of the interior angles are acute and the other three are reflex angles.

This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.