jsiehler @jsiehler@mathstodon.xyz

If you haven't stumbled across [Fields medalist] Richard Borcherds' YouTube channel yet, it's a trove of mathematical lectures at various levels.
https://www.youtube.com/channel/UCIyDqfi_cbkp-RU20aBF-MQ/videos

Grid Paint (https://grid-paint.com/) is annoying in requiring a sign-in, but is otherwise a delightful and minimal browser-based tool for playing with polyforms, among other uses.

I often use unsolved/open problems in math as a discussion point with, especially, younger students (late elementary or middle school), but also college-age students with little math exploration outside the calculus-focused school curriculum). Too many of my examples are from number theory. So I'm soliciting problems that are comprehensible to young students, invite play and curiosity, but aren't primarily about integers.

Fibonacci-length-substring-word-chains.
baccalaureate-accurate-acute-act-at
computational-optional-tonal-ton-on
orchestration-restrain-stain-tan-an

Nice little bit of card-shuffling mathematics, but also an excellent presentation that takes advantage of the medium.
https://fredhohman.com/card-shuffling/

(That's little to do with Colin's amusing equation, just a loosely-related observation. With experience we tend to choose prime or 'd' notation with appropriate foresight to express things clearly, but it's not an easy skill and we don't usually talk about the fact that we're doing it.)

Maybe Mandelbrot images are clichéd by now, but these videos from Maths Town are a pleasant way to space out and calm down for a few minutes.

https://www.youtube.com/watch?v=huEvvr8Heao

Nothing deep or special, just a pretty little tiling I found. And I colored it in all by myself!

Equals signs and parentheses look terrible when not sufficiently symmetric, and I still find it very hard to draw them tolerably by hand on my tablet... but can't stand it if they're wonky.

Fortunately equals signs and parentheses don't come up much in math! Hahaha. sob.

Without turning to the computer to brute force it, can you figure out which remainder class (mod 7) occurs most frequently among the numbers formed as permutations of 123456789?

Oh, according to the appendix, that one is actually proven, not just conjectured.

Interesting paper on computer-generated conjectures regarding continued fraction representations of mathematical constants.
https://arxiv.org/pdf/1907.00205.pdf

In particular, Figure 2 shows a really splendid (conjectural) representation of \(e\). I gave up trying to figure out how to type it readably here.

Any nice _browser-based_ tools out there for simply drawing (and maybe dragging around) polyominoes on a grid? (Intended audience would be students trying to investigate polyomino problems such as packing rectangles and so on)

Is there a term for an $n$-regular graph which can have its vertices $n$-colored so that each vertex has all $n$ colors among its neighbors? (For example, a cycle whose length is a multiple of 4 works for n=2, and the triangular prism for n=3).

Note, looking into this is officially preparation for one of my fall classes. Work work work.

Matthew Seymour has done more incredible work in service of the connection game Hex, this time in the form of a beautifully-implemented collection of 500 "white/black to play and win" puzzles of the sort that chess and go enthusiasts take for granted as study tools:

http://www.mseymour.ca/hex_puzzle/hexpuzzle.html

An uncommon sort of sliding block puzzle that I spent some time developing a little while ago. No browser-based version of this yet, just a screenshot from Mathematica.

The most recent Mathematical Gazette has some really lovely articles. If you have access to the Gazette at all, I can recommend these two:
https://www.cambridge.org/core/journals/mathematical-gazette/article/ratios-in-heronian-triangles/09561344BC7C9068244414E2DB38202D

https://www.cambridge.org/core/journals/mathematical-gazette/article/threesquare-theorem-of-gauss-and-legendre/F5E4C8F7CD8BEC11CCB4F89488414648

It seems to me that there's a tendency to want to average by flat, as opposed to averaging by person. In that way, it reminds me of problems like "you cycle out at 10 mi/hr and back the same route at 24 mi/hr, what's the average speed for the round trip?"