Dividing into congruent polygons

Geometry Level 2

The picture shown demonstrates that you can divide a square into four congruent polygons.

Is it possible to divide a square into five congruent polygons?

No Not enough information to determine Yes

Denton Young by Denton Young , 47, USA

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2 solutions

It may be surprising, but watch this

An image is worth a thousand words An image is worth a thousand words

21 Helpful 7 Interesting 16 Brilliant 2 Confused

Very good. Splitting it horizontally into five equal pieces would also work.

Denton Young - 2 years, 4 months ago

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I wonder if there is a less trivial solution

Juan Eduardo Ynsil Alfaro - 2 years, 4 months ago

I read pentagons for polygons. Ed Gray

Edwin Gray - 2 years, 4 months ago

I love how the solution to this problem is so simple, yet it is so easy to miss it if you start to overthink it.

Jeremy Galvagni - 2 years, 4 months ago

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That is very true!

Tirth Patel - 2 years, 4 months ago
Denton Young
Jan 14, 2019

Yes. Use four parallel lines and split it into five rectangles,each with the same height as the square and 1/5 of the base.

7 Helpful 3 Interesting 4 Brilliant 0 Confused

Curiously, (not to mention somewhat tangentially), as a corollary to Monsky's Theorem , a square cannot be dissected into an odd number of congruent (or even equal area) triangles. I would not have come across this fact if it were not for my initial overthinking of your question, so thanks for posting it. :)

Brian Charlesworth - 2 years, 4 months ago

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You're welcome. That was an interesting read: I had never heard of Monsky's Theorem until your comment.

Denton Young - 2 years, 4 months ago

Incidentally, I got the idea for this problem from a similar problem published a few decades ago by L. Vosburgh Lyons, who did not use a square but used a polygon for which you can use a similar trick with the dissection into 4 parts to cause people to overthink things, when the dissection into 5 (or any other number) of parts is as easy is the solution I posted. The picture is designed to make you overthink it. :)

Denton Young - 2 years, 4 months ago

Brilliant observation Brian!

Vinav Mehta - 2 years, 4 months ago

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