184 votes
Accepted

How can this shape perfectly cover a cube?

The shape Using this, we can make a guess for how the cube might be folded: Once that fold is done, the shape looks more like this: A drawing of the finished product: And an animation of the whole ...
Deusovi's user avatar
  • 146k
177 votes
Accepted

Is this duplo train track under too much tension?

First, we can check that there is no angular misalignment. Since 12 curved pieces are needed to make a full circle, the number of left pieces minus the number of right pieces must be a multiple of 12. ...
2012rcampion's user avatar
  • 18.9k
134 votes
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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
  • 77.4k
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
  • 146k
87 votes
Accepted

Can you perfectly wrap a cube with this blocky shape?

This seems to work: Below, I printed out the shape, and cut off the excess. The white parts are for glueing; if everything works out as planned, all of them will be covered by the coloured bits ...
Bass's user avatar
  • 77.4k
83 votes
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What percentage is grey?

A visual approach...
caPNCApn's user avatar
  • 19.2k
80 votes

Simple geometry. Or is it?

I know the answer is already given but I'd like to show an easy explanation of why the 2 planes are coplanar. Take this image: Consider two pyramids sitting side by side, and draw a line between ...
Ivo's user avatar
  • 11.2k
67 votes
Accepted

Cover 63 squares of a chess board

These should do it: Just to show another example:
hdsdv's user avatar
  • 5,190
67 votes

Prove that π > 3

How about this? Why does it work? Alternative cut:
loopy walt's user avatar
  • 21.4k
66 votes

Can you perfectly wrap a cube with this blocky shape?

The shape can be folded like this
Weather Vane's user avatar
  • 14.5k
65 votes
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Six pyramids in a cube

The answer is because the volume of a pyramid is proportional to its height, and we know that each pair of opposite pyramids together has the same total height. Therefore, all three pairs of pyramids ...
Eric Tressler's user avatar
58 votes
Accepted

The Non-Pythagorean Theorem

Explanation:
T. Verron's user avatar
  • 536
58 votes
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Help the prisoners

You can't. Color them like a checkerboard - the top-left-front cell is black, and the ones adjacent to it are white, and the ones adjacent to those are black... Each prisoner in a white cell must ...
Deusovi's user avatar
  • 146k
56 votes
Accepted

A new way to cut a pizza

This paper by Joel Haddley and Stephen Worsley answers a slightly different question - finding monohedral disc dissections where not all pieces touch the centre - but the results generally apply to ...
Jaap Scherphuis's user avatar
54 votes
Accepted

Create a 3 inch measurement

I can do it in folds, by
DooplissForce's user avatar
54 votes
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Release the "Q" ball

Here is one possible solution to this puzzle
caPNCApn's user avatar
  • 19.2k
53 votes

Folding paper into corners

I managed to make Like so:
Jonathan Allan's user avatar
53 votes

Odd-looking circle

He has made a rapid escape from the scene because he actually didn't know what a "circle" was. No wait.
Faruk D.'s user avatar
  • 591
51 votes
Accepted

Coffee-break Puzzle: Where does the Driver Sit?

Added an image for clarification:
Carl Löndahl's user avatar
50 votes
Accepted

How many matchsticks need to be removed so there are no equilateral triangles?

@hexomino's answer is correct and well-reasoned, as always. Here's another approach, which to me feels much more.. "axe-to-the-head" is what I'd call it in my native language, so I thought ...
Bass's user avatar
  • 77.4k
48 votes
Accepted

Odd-looking circle

phenomist's user avatar
  • 13.6k
47 votes
Accepted

Find the perimeter (seemingly unsolvable problem)

Here's a non-visual solution which some may find more easy to understand than a visual solution:
Tanner Swett's user avatar
  • 2,319
46 votes
Accepted

A colorful dodecahedron

Partial Answer: Solution: Other Solutions: Fun Stuff:
DqwertyC's user avatar
  • 8,206
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
  • 10.3k
43 votes
Accepted

Magnets on a whiteboard

This is a solution
Xoff's user avatar
  • 814
40 votes

Two chunky pixelated X's locked in mortal combat!

The combined area of the X's is The solution:
Daniel Mathias's user avatar
38 votes
Accepted

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
  • 77.4k
38 votes

Find the perimeter (seemingly unsolvable problem)

To me the most visually intuitive solution is as follows: First of all, Then take
caPNCApn's user avatar
  • 19.2k
38 votes
Accepted

Can you tile a 25 x 25 square with a mixture of 2 x 2 squares and 3 x 3 squares?

I think the answer is Consider the following image: Generalizing this result, the question "For which $n$ can an $n \times n$ square be tiled with $2 \times 2$ and $3 \times 3$ squares?" ...
Bubbler's user avatar
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