7 Sevens

Find the last two digits in this expression.



Cats. They like to explore. So, if your cat ever finds a tunnel through the centre of the earth, you can bet she’ll jump in.

A tunnel is a hole without a bottom. Your cat will fall at increasing speed toward the centre. Then she will continue to fall… upward. Eventually she’ll reach the other side of the planet.

But when?

Next time this happens, be ready with a catnip mouse. As soon as the cat jumps in, fling the toy mouse into a low orbit about the earth. …

Who doesn’t want a sextant for their birthday? Trigonometry and sky stuff together! (Source)

Have you found the fox yet? Puzzle Link is in the newsletter. (Fox source)

This newsletter contains friend links (no paywall), so feel free to share!

A comment poured in…

A viewer on my YouTube channel requested that I cover this problem from the 1997 International Mathematical Olympiad (Problem #5):

Find all pairs (a, b) of integers a, b ≥ 1 that satisfy the equation
aᵇ² = bᵃ

We’ll look at this on a YouTube Livestream Sunday, June 27, 2021 at 9 a.m. Pacific Time. Be there or be a regular right quadrilateral!

(If you miss the livestream, this link will let you watch the video post-broadcast.)

Calculus Made Easy

Sylvanus Thompson published an accessible introduction to Calculus in 1914. [Purists…

What is the shaded area?

I’ll present this diagram in the form of an algorithm.

PRINT [(1/2)^n]sin[(2^n)x]

It’s an infinitude of sine waves. The first function, sin x, runs from 0 to π. We take a bite out of the area above the x-axis. We do this with a sine wave having half the amplitude and twice the frequency.

Create a third sine wave, half the amplitude and twice the frequency of the previous. Use it to take a bite out of the remaining area.



The remaining shaded area looks like a batwing. What is the size of the batwing?

Have fun with this. We’ll have a look during the MathAdam YouTube Livestream: Sunday, July 25, 9 A.M. Pacific Time. If you miss the livestream, the link will take you to the replay.



What is the angle between these two tangent lines?

This comes from Sylvanus Thompson’s Calculus Made Easy. As presented in the text (page 86):

Selfie of the author with a $2400 CAD paperweight.

Have you seen The Good Place? Hypatia of Alexandria points to the 5 on her shirt. “Is this an S? Or a math?” Centuries of unencumbered bliss in Heaven have rendered her and her companions intellectual zombies.

I was resting in peace myself until a couple of weeks ago. This email fluttered into my inbox.

Fox source. Animation by author.

You have 5 holes and 1 fox. The holes are in a line, left to right, A, B, C, D and E. The fox is in one of these holes.

Mathematicians have been slicing open cones since the 4th Century BC. Here, we study one of these famous cross sections — the ellipse. Along the way, we’ll discover a beautiful proof from Germinal Dandelin of the 19th Century.

The Ellipse — Two Seemingly Unrelated Descriptions

Slice a right circular cone at an angle to the horizontal. The cross-section forms an ellipse.

A fingernail moon: inspiration for poets and mathematicians alike (Source)

This one comes from Michael Penn, who in turn got it from a dead mathematician with a beard. You can check out Penn’s solution on his YouTube channel. The solution below is my own.

Figure 1 shows a right triangle parachuting to the ground with a bedsheet.

Adam Hrankowski, ADHD

I am a maths/physics tutor in BC. You can find my stuff on YouTube: youtube.com/c/mathadam My Virtual Tip Jar: https://ko-fi.com/mathadam Thanks!

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