Can anyone define for me the $$\textit{heart}$$ [over field k of a group acting on a thing that's probably regarded as a vector space at the moment]? I'm not familiar with the term and it's a bit tough to search for.

The OEIS tags certain sequences as "nice" (and I generally agree with it) but I think that "dull," "rotten," and "notniceatall" could also be useful.

"Twisty Puzzle" seems to be common parlance for Rubik's cube and its many descendents, but I think we need to start using the term "scrambly puzzle" for stuff like the 15-puzzle, Segerman's 15+4 puzzle, and other stuff where you mix up pieces but the mechanism isn't twisting.

There was a link on MetaFilter to Quazel, a site which promises to let you practice conversation in the language of your choice with an AI bot. Here's as far as I went:

Quazel [speaking aloud]: "Hallo! Mein Name ist Unterstrich Unterstrich Unterstrich Unterstrich Unterstrich Unterstrich Unterstrich. Wie ist dein Name?"
Me [through mic]: "Bitte duzen Sie mich nicht."
Quazel: "Es tut mir leid, ich wollte dich nicht verärgern."

Puzzle/Problem: An $$n\times n$$ square grid is filled with the numbers 1 through $$n^2$$. A cell in the grid is a local maximum if its value is greater than all of its orthogonally-adjacent neighbors. Let $$M(n)$$ denote the expected value for the number of local maxima when the grid is filled randomly.
(a) Compute $$M(4)$$
(b) Evaluate $$\displaystyle\lim_{n\to \infty} M(n)/n^2$$

(I decided this was too easy to send off to journals, but I don't think it's a total no-brainer.)

Working on implementing a tile-sliding game that I researched with students a while back. It's a little twist on the usual 15-puzzle type of thing, which you can play on your favorite 3-regular graph.

Perhaps one should distribute the ordinal over the addition and say "nth plus 1st"

(I have, in practice, settled on writing (n+1)-st because saying "one-th" aloud is just too abrasive for me.)

How do you "ordinalize" the expression (n+1)? Do you say (n+1)st, or (n+1)th, or... what?

It's a heartbreakingly beautiful day outside; once I would have set work aside and jumped on my bike for the afternoon, without a care.

One has more cares these days.

Are... are these images GENUINELY related to a search for ["142/13" random walk] somehow?

Spot the image that was thrown into this (quite interesting) article solely to distract any border collies who might be trying to learn enough mathematics to start publishing.
en.wikipedia.org/wiki/Curve-sh

I'm not sure there's any better feeling than going back to a project you had to shelve for a long time and finding that your past self LEFT THINGS WELL-DOCUMENTED AND ORGANIZED for future self to pick up and resume without friction.

Nice problem from Mathematics Magazine (#2150): Find the maximum area of a triangle whose vertices lie on the cardiod $$r=1+\cos \theta$$

Installed manim today.

"No, we have 3blue1brown at home"

Neat discussion of colors on the Commodore 64 (happy 40th anniversary year) and other 8-bit computers.
aaronbell.com/secret-colours-o

I would assume that's someone who specializes in forging rings of power.

Topologists: Don't bore us, get to the torus

I enjoyed reading this analysis of a dots-and-boxes position.
puzzling.stackexchange.com/que

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