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 deftly carve these items such that each resulting ring has the same height (*Figure 1*).

I claim that the apple ring and the grapefruit ring have the same volume.

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. “A credit card imprint will be fine.”

“Oh. Yeah. Can do.”

I don’t know what I had thought he meant by “deposit.” My left pinky? Something less mundane.

“Have you operated one of these before?” he asked, writing down my MasterCard number next to my signature.

“Umm… No.” It hadn’t occurred to me…

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 headed, “One angry jolt.” Surely my clever friend would read that as “a cross tic” and discover the code in the initial letters of the documentation. He would then read the instructions to hold down the <Ctrl> key while he keyed in the appropriate combination of letters and numbers.

Once…

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