## Cut this in half.

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Moonbeam
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### Cut this in half.

Okay - you have a rectangle measuring 12 x 9.

From the top left and bottom right corners, 2 triangles are removed, both measuring 2 x 4, with the 2 on the 12 side and 4 on the 9 side, so you're left with an oddly shaped hexagon of area 100.

The puzzle is to cut this figure in half and re-fit the 2 pieces back together to make a square measuring 10 x 10.
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Yakk
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### Re: Cut this in half.

Spoiler:
10 by 10.PNG (10.07 KiB) Viewed 2805 times

And with 5 more cuts, I could make 2 10x10 squares.
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Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

Hix
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### Re: Cut this in half.

Yakk wrote:And with 5 more cuts, I could make 2 10x10 squares.

I think you need at least 3 dimensions to do that.

mike-l
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### Re: Cut this in half.

Hix wrote:
Yakk wrote:And with 5 more cuts, I could make 2 10x10 squares.

I think you need at least 3 dimensions to do that.

Not sure on the number of cuts, but it does work in 2 dimensions as well.
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Token
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### Re: Cut this in half.

I recall from somewhere that two dimensions requires (in general) countably infinitely many cuts...
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mike-l
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### Re: Cut this in half.

Token wrote:I recall from somewhere that two dimensions requires (in general) countably infinitely many cuts...

You're right, for the closed circle at least. I believe finitely many suffice for the interior of a square, and 4 suffice for the whole plane.
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Qaanol
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### Re: Cut this in half.

How are you defining a “cut”? To my intuition it should be an oriented, simply-connected region of one fewer dimension than the object being cut, which partitions the object into those components above and those below-or-equal-to the cut. I’m not convinced you can get the five pieces needed for BTP using finitely-many cuts of that sort.
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mike-l
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### Re: Cut this in half.

Qaanol wrote:How are you defining a “cut”? To my intuition it should be an oriented, simply-connected region of one fewer dimension than the object being cut, which partitions the object into those components above and those below-or-equal-to the cut. I’m not convinced you can get the five pieces needed for BTP using finitely-many cuts of that sort.

Certainly not. But that's a pretty restrictive definition of cut, it pretty much forces measurability, which of course is going to disallow any kind of 'size' paradox.
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Qaanol
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### Re: Cut this in half.

What definition do you propose?
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mike-l
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### Re: Cut this in half.

Qaanol wrote:What definition do you propose?

A cut is a partition.
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Qaanol
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### Re: Cut this in half.

Any partition, with no restrictions?
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