Hang a cube from one of its vertices. Now, if you slice it horizontally through its center, you get a...

Continue reading...# hard

## Continuum Hypothesis

We have seen in the Fun Fact Cantor Diagonalization that the real numbers (the “continuum”) cannot be placed in 1-1 correspondence with...

Continue reading...## Banach-Tarski Paradox

Did you know that it is possible to cut a solid ball into 5 pieces, and by re-assembling them, using...

Continue reading...## Face Derivatives and Computer Vision

One challenge in robotics is the problem of computer vision: how do you program a computer to interpret and “understand”...

Continue reading...## Pretzel Unlinking

Imagine that each of the ropes in the two sets of links in Figure 1 are solid (with thickness) and...

Continue reading...## Unbelievable Unlinking

Imagine that the two objects in Figure 1 are solid (with thickness) and made of very flexible and stretchy rubber....

Continue reading...## Red-Black Pairs Card Trick

Here’s a terrific mathematical card trick that will impress your friends. When you do this trick, the effect of the...

Continue reading...## Prime Number Theorem

Fix some number N. What fraction of the integers less than or equal to N are prime? Thinking about it,...

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