177 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 ...
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  • 136k
173 votes
Accepted

How can 64 = 65?

This is a famous physical puzzle that can be tied to the fibonacci series. To answer the question as posed, the issue is that the two slopes are different ($\frac25$ vs $\frac38$). Note that all ...
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  • 22.3k
153 votes

How can 64 = 65?

The diagram is misleading, as it hides a gap in the middle of the second configuration. This is what we actually get if we rearrange the shapes in question. Notice that the diagonal “bows” slightly, ...
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  • 2,487
123 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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  • 68.2k
116 votes
Accepted

Chaos and Order: a visual puzzle in stained glass

Solved it! Spoilers ahead: P.S. I think this window is brilliant :-)
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  • 9,461
104 votes
Accepted

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

The way to do this is:
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  • 136k
95 votes

Join all circles together only with 6 lines

If you use just one massive line, you can make it pass through the center of all of the circles!
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  • 1,220
86 votes
Accepted

Turning a goat?

Here's the image equivalent: Apologies for the poor quality image. My office laptop has very limited capabilities in image editing.
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  • 11.6k
84 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 ...
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  • 68.2k
78 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 ...
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  • 10.9k
76 votes
Accepted

The Jeweller's Dilemma

It is Proof/construction: Even more strongly,
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  • 23.6k
75 votes

Join all circles together only with 6 lines

Here's an option that uses only 4 lines. You can extend the concept to place another 2 lines if you really want 6 ...
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  • 1,085
73 votes

Join all circles together only with 6 lines

By mapping the puzzle onto a cylindrical topography, I've solved the puzzle using only a single straight line.
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  • 855
69 votes
Accepted

Is it always possible to balance a 4-legged table?

The answer is Here's why: More detailed proof:
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  • 136k
65 votes
Accepted

Cover 63 squares of a chess board

These should do it: Just to show another example:
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  • 5,130
64 votes
Accepted

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 ...
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63 votes

Can you perfectly wrap a cube with this blocky shape?

The shape can be folded like this
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  • 10.7k
59 votes

Prove that π > 3

How about this? Why does it work? Alternative cut:
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  • 11.6k
58 votes
Accepted

The Non-Pythagorean Theorem

Explanation:
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  • 536
57 votes
Accepted

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 ...
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  • 136k
54 votes

Turning a goat?

I've found a simpler solution that doesn't rotate the entire goat! See the image below for a visual explanation.
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  • 2,834
54 votes
Accepted

Create a 3 inch measurement

I can do it in folds, by
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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.
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52 votes
Accepted

Join all circles together only with 6 lines

I think this is one possible solution:
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  • 751
52 votes

The Jeweller's Dilemma

It is possible! How is that polyhedron? WTF!? How!? It is hard to understand? So let's visualize that with $k = 90^\circ$: Let's see some properties of this particular polyhedron: Ok, but how ...
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52 votes

Folding paper into corners

I managed to make Like so:
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51 votes
Accepted

Coffee-break Puzzle: Where does the Driver Sit?

Added an image for clarification:
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51 votes

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 ...
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50 votes
Accepted

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
49 votes
Accepted

Five Angles in a Star

$\hskip 1.5in$ This is an image of an arrow sweeping each of the successive angles in the star. Notice that, after it traces all $5$ angles, its orientation is reversed - meaning it has rotated $180^{...
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  • 7,661

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