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 ...
- 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 ...
- 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, ...
- 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 ...
- 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 :-)
- 9,461
104
votes
Accepted
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!
- 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.
- 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 ...
- 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 ...
- 10.9k
76
votes
Accepted
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 ...
- 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.
- 855
69
votes
Accepted
Is it always possible to balance a 4-legged table?
The answer is
Here's why:
More detailed proof:
- 136k
65
votes
Accepted
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 ...
- 1,981
63
votes
59
votes
58
votes
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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 ...
- 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.
- 2,834
54
votes
Accepted
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.
- 591
52
votes
Accepted
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 ...
- 8,828
52
votes
51
votes
Accepted
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 ...
- 43k
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 ...
- 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^{...
- 7,661
Only top scored, non community-wiki answers of a minimum length are eligible