Kolmogorov complexity is Turing equivalent to the halting problem

This post describes several Turing equivalent problems to the halting problem: the busy beaver problem, the busy beaver runtime problem, the minimal program problem, and the Kolmogorov complexity problem.

But really, only one part of the proof is hard, and that is proving that the halting problem can be Turing-reduced to the problem of calculating Kolmogorov complexity.

The core of the argument is this: there exists short strings with short minimal programs, whose runtime is insanely long. That's really it.

Aitken's trick for calculating decimal expansions of rationals

Aitken's Method

Professor Aitken was both a mental calculator and a master mathematician, and a detailed account of his mental math abilities is found in An exceptional talent for calculative thinking, (IML Hunter, 1962)

I'd just like to sketch out one amazing trick, which is illustrated thus:


The proof is  $x = 5/23 = 15/69$, then, $x = (15 + x) /70$, so the trick is to calculate $15/70$ by short division, to generate the digits of $x$ one at a time, then immediately feed to the top to continue. A kind of "just-in-time" algorithm!

Short online games recommendation

Each game should be finishable within an hour.

The Company of Myself
https://www.kongregate.com/games/2DArray/the-company-of-myself
Sad philosophical game about a loner.

Missed Message
https://zephyo.itch.io/missed-message
Interactive fiction. Depression, cyclic day, suicide, lesbianism... also 2018 memes.

Fixation
https://www.kongregate.com/games/2DArray/fixation
Prequel to The Company of Myself. Has comics as transitions. Even sadder.

Viricide
https://www.kongregate.com/games/2DArray/viricide
An AI talks about someone of their lifestory. The AI might be depressed just like their programmer.

Geometry quickie: volume of a tetrahedron

I saw this formula on Wikipedia for the volume of tetrahedron and was intrigued.

Any two opposite edges of a tetrahedron lie on two skew lines, and the distance between the edges is defined as the distance between the two skew lines. Let $d$ be the distance between the skew lines formed by opposite edges... 

then it gave a volume formula. I want to derive it. This post gives my thought process, where I keep using the trick of reducing a general problem to a good example. This is a general strategy for solving problems, often written as "wlog".
(I write it as "wolog" and pronounce as "volog", as if it's a German adverb. So that I can write "We assume wolog that...", "... and thus wolog we have...")

How would a true geometer approach this problem? She would first do a dimension analysis. The volume has dimension $meter^3$, and the lengths have dimension $meter$, so it ought to look like
$$Volume = k \cdot AB \cdot CD \cdot h$$
where $h$ is the distance between the two skew lines of $AB$ and $CD$, and $k$ is some dimensionless number yet to be determined.

Let's Read: Jorge Luis Borges's Dreamtigers

In this post, we summarize every single entry in  Dreamtigers (Borges, 1964). We also add commentaries on particularly obscure references.

Dreamtigers, first published in 1960 as El Hacedor ("The Maker"), is a collection of poems, short essays, and literary sketches by the Argentine author Jorge Luis Borges. Divided fairly evenly between prose and verse, the collection examines the limitations of creativity. 

Let's Read: A Problem

In this post, we read through Borges's story A Problem, which is about Don Quixote, a story written by Cervantes:

The story follows the adventures of a nobleman named Alonso Quixano who reads so many knightly stories that he goes insane and decides to become a knight-errant, reviving chivalry and serving his country, under the name Don Quixote de la Mancha. He recruits a simple farmer, Sancho Panza, as his squire. 

His name means "Don Quixote, of la Mancha". La Mancha is a region in central Spain, where Don Quixote lived in.

A Problem

Jorge Luis Borges

Let's read: Dead Men's Dialogue

In this post we read through Borges's story Dead Men's Dialogue, which, other than the usual kind of maddening riddles that writers play, where they only say half of what is said, and I have to fill in the other half, is extra hard to read for me because it is full of references to South American history that I don't know.

So I'll fill in all the blanks here.

Dead Men’s Dialogue

Jorge Luis Borges