Hint: It’s an integer!

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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:


Image by Sophie Janotta from Pixabay

Integral calculus makes it easy

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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.


Image by author

Why is it the same as the curved strip which surrounds it?

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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?


Image by author with Tinkercad

Any two spheres, gutted so they have the same height, have the same volume. Here’s why.

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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…


Try it first

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 3:


Calculus helps us understand important real-life applications, such as the rate of expansion of an exploding watermelon. (Source)

Think outside the box. (Sorry.)

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:


And remember to read the fine print

Images modified by author from source and source.

“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. …


Life, the Universe and Everything is a 7th degree binomial.

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

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.

My Solution

First, factor the polynomial.


What math fact do you see?

Stop trying so hard

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…

Adam Hrankowski, ADHD

Math Dabbler in BC, Canada. https://youtube.com/c/mathadam https://bitclout.com/u/MathAdam My Virtual Tip Jar: https://ko-fi.com/mathadam Thanks!

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