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A pentagon puzzle

Puzzling Asked on April 28, 2021

Consider the following pattern made of regular pentagons:

If the pattern continued, will it form a complete loop or will the pentagons overlap?

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5 Answers

The shapes will

Proof

Image

Answered by hexomino on April 28, 2021

It will

This kind of generalises.

Answered by loopy walt on April 28, 2021

I will try to provide as simple a solution as possible. Hopefully no pictures will be necessary.

It is well known that

Therefore,

Answered by Half_Bit on April 28, 2021

There are 180 degrees * (n-sides minus 2) in an n-sided polygon. For a pentagon, each angle is therefore 540 degrees / 5, or 108 degrees

The amount of change in angle for a pair of adjacent pentagons will be (108 + 108) - 180, or 36 degrees.

Take 360 degrees in a circle and it is evenly divisible by 36 degrees, so the completed series of pentagons will consist of 10 pentagons and they will perfectly join up.

Answered by kpkpkp on April 28, 2021

The answer is

And I'm going to (at least try to) proof that this generalizes to all regular polygons:

Examples

Square Heptagon Octagon

Answered by danzel on April 28, 2021

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