124 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 ...
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  • 68.9k
104 votes
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Can you fold a square into a square of one-fifth the area?

The way to do this is:
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  • 137k
60 votes

Prove that π > 3

How about this? Why does it work? Alternative cut:
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50 votes
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Cut the disk with a hole in four equal pieces

Here's one solution: I assumed the radius of the hole's curvature matches the curvature radius of the circle, the hole's straight side is equal to the circle's radius, and its curved edges meet the ...
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  • 515
48 votes
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Drink a Little Wine, Cut a Little Rug

Quite an interesting cut I had to make to get it to fit. Shift the bottom piece 2 tiles up and one tile right to produce a 10 x 10 square. You can save all 100 square feet. The end result should ...
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  • 969
44 votes

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

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43 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 ...
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38 votes
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Fairly Sharing a Frosted Cake

We will make all of our cuts vertical, so we can treat this as a square which we need to divide into $10$ pieces with equal slices of the area and the perimeter. This is reasonably easy: Choose $10$ ...
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  • 7,711
36 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'...
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33 votes
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Careless smokers

Yes, it's possible, because one can fit 5 disjoint 1x1 squares in a 2.75x2.75 square: four in the corners, and one in the center rotated 45 degrees. The four cigaret holes can't eliminate all 5 ...
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31 votes
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Cutting a 10-by-2 rectangle

Index your rectangle from (0,0) to (10,2). Then cut from These four pieces can be used to make the square. Note that this dissection works without any flipping or even any rotation of the pieces! ...
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30 votes
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Mutilated chessboard

I believe this works as a short proof.
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29 votes
Accepted

Dissection Puzzle - The Umbrella Stand

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29 votes

Dissection Puzzle - The Umbrella Stand

JonTheMon and xnor's solutions assume we have superior equipment and skill, but the question states that we "have a hacksaw". Well with a hacksaw, we must start from an exposed side; we can't start a ...
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  • 391
28 votes

Dissection Puzzle - The Umbrella Stand

As pointed out in a previous answer, cutting a hole in the middle of the table may be unfeasible if everything you have is a hacksaw. Using the existing hole as a starting point, the cut can be ...
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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^...
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  • 9,471
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 ...
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26 votes
Accepted

You find a piece of paper in your bag

I believe this cut should work:
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25 votes
Accepted

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 ...
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  • 12.7k
24 votes

Fairly Sharing a Frosted Cake

One of the possible solutions is: Calculations Lets say square length is L and Height H Frosting on top = 0.5 * 0.4L * 0.5L Frosting on bottom = same as top Frosting on side = 0.4L * ...
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23 votes

Dissection Puzzle - The Umbrella Stand

You cut a square like this: And rotate it 180 degrees. The cut square (or rectangle) simply needs to have its centerpoint be halfway between the hole and the center of the square, and to be large ...
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22 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 ...
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22 votes

Four fanatics and one checkerboard

I'm not sure why you'd need ANY sort of dissection for this.
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  • 2,592
21 votes

Fair share of a square watermelon?

Look at the cube down a space diagonal. It should appear as a hexagon, which can be divided into three rhombuses by line segments from the center to every other corner. If these three cuts are made, ...
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  • 211
21 votes

Holes in the Table

One cut solution for #3.
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  • 311
21 votes
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Can I Haz My Eye Center'd?

One (non-straight) cut: (Or a different image by Carl Löndahl: https://nup.pw/YK9cjY.png)
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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.
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20 votes

Drink a Little Wine, Cut a Little Rug

I think for the sake of simplicity this should be part of the answer
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20 votes

Universal dissection

I think it is possible to do with Which look like
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  • 9,542
20 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.)
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