Three congruent regular hexagons can be drawn in such a way that all of them overlap each other and create exactly ten distinct areas or compartments?

Answer 1

First, a hexagon has 6 sides. Second, congruent means the polygons are the same size and shape. Third, regular hexagon means that all of the angles and the same and the lengths of the sides are the same.

For my explanation, let's work with squares.

If you were to overlap two perfect squares, you would get at 1 area.

Rotate one of those squares, and you will get 8 areas, 4 on the inside and 4 on the outside. Since there is also a center area, we have 9 areas.

Working with two hexagons would give you 1 or 13 areas.

Obviously, adding a third square or hexagon will not achieve 10 areas, so you can stop here.

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If you overlap 3 hexagons you get

3 sections that are unique to each hexagon

1 section in the middle that is part of each hexagon

3 sections that are shared between only 2 hexagons

Those 7 are straightforward - I drew 3 hexagons in powerpoint to visualize it

The last 3 are a matter of interpretation, but they are there. it depends on what is meant by "distinct."

There are an additional 3 sections that are made up of the outlines of the 3 sections that shared between only two hexagons plus the section in the middle.

That gets you to 10. My 2 cents is that this is a poorly worded question because the answer could be 7 or 10 depending on the interpretation of distinct.