30 Jul

Problematic Postcards

If you’ve come here as a result of a puzzling postcard you may have come across, welcome to Studio Infinity! We hope you’ll enjoy looking at some of the other content below as well, but here are the three posts corresponding to the problems you can find on those postcards, each of which links to a solution.

Insubordinate Integral Smallish Sequence Troubling Triangle

And, as I’ve mentioned here before, I invite you all to submit a problem or solution to Math Horizons Playground.

30 Jul

Insubordinate Integral

How does the value of the following improper integral compare to 1? I.e., is it smaller, larger, or exactly equal to 1?

(This problem was proposed to Math Horizons Playground by Mehtaab Sawney of Commack High School. And for all of you $\pi$-ists out there, $\tau$ is of course just the radian measure of a full circle, i.e., $\tau=2\pi$.)

Stumped? You can peek at the solution using the password “strapyb”.

30 Jul

Troubling Triangle

What is the area of the pink shaded triangle (as a fraction of ABC)?

For definiteness, the unlabeled points do in fact trisect each of the sides of the triangle.

Stumped? You can peek at the answer using the password “threedian”.

30 Jul

Smallish Sequence

What are the next few terms in this sequence of smallish numbers?

1 1 1 3 1 3 1 1 3 1 1 3 1 1 3 1 1 1 3 2 1 1 3 2 1 3 1 1 3 1 1 1 3 2 1 1 3 2 1 1 3 ? ? ?

(And, of course, what’s the rule generating the sequence?)

Stumped? You can peek at the solution using the password “1p1a2s”.

29 Jul

More Life at a glance

A few days after the event at TCNJ, students at the PROMYS program at Boston University built another “Life sculpture” in which each layer is a generation and time proceeds downwards. Here, we explored questions of how you might know things like whether the resulting “sculpture” would be connected, or whether it would be self-supporting. For these types of questions, what one really needs is to solve the (more computationally thorny) “inverse Life” question: what colonies of cells can give rise to a given configuration in the next generation?

This sculpture begins with a pattern in its top layer that will eventually result in four Life “gliders” proceeding in different directions, which would then serve as four “legs” for the sculpture to stand on. Unfortunately, we didn’t quite have time to build enough generations to see the four gliders diverge.

27 Jul

x-y-t Life

Here is a photo of the first 13 generations of the evolution of the “R” pentomino pattern in John Conway’s Game of Life. Each layer represents one generation, and time proceeds downwards. In each layer, live cells are represented by boxes. The color of the box indicates how many generations that cell has been alive: yellow for one, orange for two, and blue for three or more. These “Life sculptures” were built by middle-school students at The College of New Jersey in 2019 July.

(If these boxes look familiar, you’re right – they are the same ones from the Boxtahedral Tower reused for a completely different construction.)