I found this challenge on a YouTube thumbnail. I have yet to see the video.

The challenge is to show that the expression below is an integer:

To find the volume of a unit sphere, approximate the sphere with numerous discs, each of thickness dR (*Figure 1*). We’ll do this with the northern hemisphere only. Then we can double our result.

Grab a perfect sphere. Enclose it in a tube of paper (*Figure 1*). I claim that the rectangular sheet required to make the tube has the same area as the surface of the sphere.

How can this be?

You have two spheres of different size — say, an apple and a grapefruit. (A proton and a planet will do.) You are going to core each of these fruits. The shape that remains from each sphere is what we will call the napkin ring.

You…

This delight comes from a Putnam problem practice page posted by U of T.

`Given any prime, p > 3, show that p²≡1(MOD 24)`

Let’s put it this way. For a prime, ** p**, greater than

Suppose we have a product of functions, *uvw*. Each component — *u*, *v* and *w* — changes with some additional variable, time, *t*. How will an incremental increase in time, *dt*, affect our product?

Imagine your original function, *uvw*, as a *u × v × w* box:

“You’ll have to leave a small deposit,” said the shopkeeper. “For security.”

I had just scribbled my mark at the foot of a three-page document, set in 6-point Courier Bold.

I set aside the stylus and met the shopkeeper’s eye. I must have looked worried. He smiled. …

The University of Toronto has posted this page and a half of brain teasers. How many can you solve?

Here’s the first:

`Show that `*n⁷ − n* is divisible by *42* for every positive integer *n*.

First, factor the polynomial.

I once designed a video game for a friend of mine. It featured him running back and forth, Donkey-Kong style, striking objects with the smoke from his pipe. My friend loved the game. But he missed the Easter Egg.

I had included with the game some awkwardly-written documentation. It was…

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