134 votes
Accepted

Prove that π > 3

I didn't have a knife with me, so I only used my unit circle cookie cutter to split each square like this: I then rearranged the parts into this shape: Since the angle covered by this shape is ...
Bass's user avatar
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107 votes
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Can you fold a square into a square of one-fifth the area?

The way to do this is:
Deusovi's user avatar
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67 votes

Prove that π > 3

How about this? Why does it work? Alternative cut:
loopy walt's user avatar
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44 votes
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Dissecting the exotic bulbfish

Cut along the red lines and move the pieces as indicated by the yellow arrows. As is usual with this kind of dissection, it helps if you look at the area to work out the length of the side of the ...
Jaap Scherphuis's user avatar
44 votes

Can you fold a square into a square of one-fifth the area?

Paul Panzer's user avatar
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38 votes
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Pythagorean quilts

The optimal solution is which is achievable (for example) like this: For another (or perhaps, the other) way to achieve the minimal number of pieces, you can check out OP's self-answer below. Here'...
Bass's user avatar
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34 votes

Pythagorean pentagons

6 pieces, no flipping required: History: 7 pieces, no flipping required: 8 pieces, requires two flips 9 pieces, requires one flip very similar: 10 pieces, no flipping required not very elegant, ...
loopy walt's user avatar
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33 votes
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Mutilated chessboard

I believe this works as a short proof.
Tyler Seacrest's user avatar
32 votes

Pythagorean pentagons

My efforts to minimize the number of pieces did not improve upon loopy walt's best, but here is another 7-piece solution without flipping: Sixteen pieces, with some flipping required.
Daniel Mathias's user avatar
28 votes
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Tiling a rectangle with nine squares

The dimensions are and the tiling looks like this: Working out the dimensions of the rectangle is quite easy. We know its total area is $4209$ (i.e., $2^2 + 5^2 + 7^2 + 9^2 + 16^2 + 25^2 + 28^2 + 33^...
r3mainer's user avatar
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28 votes
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Maximum Pieces of Cake in Four Cuts

If all cuts are straight cuts and the cake is a rectangular prism or cylinder, it's not possible. From Wikipedia's page on the Cake Number: In mathematics, the cake number, denoted by Cn, is the ...
Engineer Toast's user avatar
27 votes

Dissecting the holey octomino into a square

Well, from my viewpoint this is a four-piece dissection, since parts of each piece don't move relatively to each other. They are even connected, to some extent. However, I would completely agree that ...
Thomas Blue's user avatar
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26 votes
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Squaring a cross

Solutions are as in the following diagram: Addition: My original solution for E has 5 cuts and 9 separate pieces - a simpler solution is shown below, with only four cuts (to the cross) and five ...
Penguino's user avatar
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26 votes
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You find a piece of paper in your bag

I believe this cut should work:
Deusovi's user avatar
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24 votes

Pythagorean pentagons

OK, I'll give you my solution for reference. And just for fun, here is another seven piece solution. And finally my own 6-piece solution
Florian F's user avatar
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24 votes
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A pizza dilemma

I found a solution where every slice is strictly larger than 8 square inches: I found a solution, and after comparing it to @spherical-wug-in-a-vacuum's solution, I found a solution which uses fewer, ...
isaacg's user avatar
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23 votes

Four fanatics and one checkerboard

I'm not sure why you'd need ANY sort of dissection for this.
Braegh's user avatar
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22 votes
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Cutting a cross made of 5 equal squares by 2 straight cut into 4 figure to together form a square

Here is a good way of seeing how this dissection comes about.
Jaap Scherphuis's user avatar
21 votes
Accepted

Can I Haz My Eye Center'd?

One (non-straight) cut: (Or a different image by Carl Löndahl: https://nup.pw/YK9cjY.png)
hmakholm left over Monica's user avatar
21 votes
Accepted

Near-fill with 3x1 long triominos, how to do a different void square than the center square?

The trick to this puzzle is to: (And here are those tilings: the center was already given, and the rest are obtainable from these by rotation.)
Deusovi's user avatar
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21 votes

Cut this shape into 3 pieces and fit them together to form a square

Here is a solution that works in the general case of two squares of any size placed next to each other.
Jaap Scherphuis's user avatar
20 votes

Universal dissection

I think it is possible to do with Which look like
elias's user avatar
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20 votes
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Restore the square

Apologies for the crude drawing but I think you need to do something like this Some deduction that went into this answer
hexomino's user avatar
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20 votes
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Cut this shape into 3 pieces and fit them together to form a square

A slightly different graphic:
Beastly Gerbil's user avatar
19 votes
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Professor Halfbrain and the dissection of a rectangle

hvd's user avatar
  • 949
19 votes
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Obtain four equal parts with a single cut

Okay, I don't know how people here create images that fast, but I get a more sheet-like solution.
William Nathanael's user avatar
19 votes
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5 points on a ball, divide the ball into 2 halves so that one half as exactly 4 points

The claim is
loopy walt's user avatar
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18 votes
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Piece of Cake for King Solomon

To divide the cake into n equal pieces, Proof: Here is a drawing to illustrate how it works:
Jaap Scherphuis's user avatar
17 votes
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Dissecting the holey octomino into a square

I seriously doubt that this can be done in 4 pieces or less. It would be a miracle if it was possible, but it obviously isn't a walk in the park to prove. Regardless, to get people started, I have ...
greenturtle3141's user avatar
17 votes

How can the white cross be cut into 5 smaller pieces that can be resembled into the two smaller red crosses shown?

Here is a visual answer. And the pieces joined together
Paul Panzer's user avatar
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