How to slice it?

Geometry Level 5

We want to cut a regular hexagon into several regular polygons such that

each of the 6 original vertices lies on just 1 regular polygon;
all pieces are regular polygons (but need not have the same shape or size);
the number of these pieces is at least 2.

What is the minimum possible number of pieces that the original hexagon can be cut into?

The answer is 12.

1 solution

Geoff Pilling
Dec 8, 2016

Here is how to do it with $12$ :

And, without a rigorous proof, I believe $\boxed{12}$ will be the smallest, since you will need to have hexagons in each corner, and at least one triangle on each edge to connect the hexagons.

I also believe that this is the only arrangement (not counting rotations).

Thanks @Paul Hindess for correcting my original solution!

Great. That was my conclusion too. Here is my argument:

At each corner, we need a hexagon. This is unique to the corner.
On each edge, we need a polygon that doesn't touch the corner, since 2 hexagons do not meet nicely. This is unique to the edge.

Hence, there are at least 12.

Calvin Lin Staff - 4 years, 6 months ago

I think I spent more time figuring out exactly what the problem was. But I agree with Calvin's analysis, which is how I came to the solution.

Michael Mendrin - 4 years, 6 months ago

How does the phrasing look now?

Calvin Lin Staff - 4 years, 6 months ago

It looks like you have improved it significantly... Thanks, Calvin!

Geoff Pilling - 4 years, 6 months ago

It'll be interesting to see how many people will be solving this now. It looks awfully clear to me, especially with that helpful revised graphics. Now that I look at it, I say, "oh yeah".

Michael Mendrin - 4 years, 6 months ago

@Michael Mendrin – True... The graphic could likely make the problem a bit easier....

Geoff Pilling - 4 years, 6 months ago

Why the answer is not 6. If we make a regular triangle is each side of the hexagon, then that will fulfill your requirement, I think. N.B. I ain't understanding what did you mean in the first requirement.

Omar Sayeed Saimum - 4 years, 5 months ago

Yeah, the first requirement would be violated, since in your construction one vertex (of the original hexagon) would be shared by two regular triangles.

Geoff Pilling - 4 years, 5 months ago

How to slice it?

The answer is 12.

1 solution

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