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Sampling products from finite groups
I was recently reminded of a very simple chestnut about sampling elements from finite groups. This is the kind of problem you give your intro group theory students to test whether or not they are paying attention, or at least to keep them busy; nothing special. But the way it was worded suggested a harder version of the problem that immediately piqued my interest, and though I haven’t fully solved it yet, it ended up tying itself up in a neat little bow by the end of it...
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The forbidden subword method
I haven’t felt like writing anything hard or long, but I have felt like writing, so here is something easy that I have had cause to think about over the past couple days. This is a post about a basic tool in the enumeration toolkit, which I have used countless times since I learned of it, but strangely I have never heard anyone else talk about it. There are less general techniques that are well-known, and there more general techniques that are significantly more complicated to use, but this Goldilocks zone seems woefully underdisturbed...
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Chris Barker's Iota-Jot-Zot family of esolangs
The alternative, more poetic title that I was tempted to give this post was A Most Esoteric Tragedy. I relented for three reasons. Morally: I detest clickbait. Practically: it would unhelpfully obscure the topic on the rare occasion that someone might actually find this ‘blog helpful. (Ha!) Most of all, emotionally: it would be comically hammy for me to entertain the idea that the contrivance contained in this analysis could be called a tragedy. But hopefully, the fact that I spent the entire flavour text being snobby about this alternative title sufficiently conveys the self-indulgence and megalomania I feel for having come up with it...
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pu pi toki pona: A case study in orthodoxy
toki pona is a minimalist constructed language, created by linguist Sonja Lang between 2001 and 2014. It features 9 consonants and 5 vowels in its phonemic inventory (cf. most dialects of English at 24 and 14-25), and those 14 phonemes combine into a 123-word vernacular. The standard reference document on the conlang is currently Lang’s book from 2014. There are many things to find fascinating about toki pona, but the one I’d like to explore in detail right now is this: toki pona dedicates one of its 123 words to that specific book, despite not having a word for “book”...
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Introducing: Comments!
For a long time, I’ve wanted comments for my Jekyll-based ‘blog. However, the existing alternatives, Disqus and Staticman, were not satisfactory, so I decided to roll up my own implementation. It has certain drawbacks that might make it unsuitable for most applications, but I would like to tell its story here anyway, because it would not have been possible without the ‘blog posts of others...
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The cheapest path problem and idempotent semirings
Let me propose a interesting theoretical variant of the shortest path problem. You have a directed graph with source and target vertices, and every edge has a cost to traverse it. However, not every cost is in good old American dollars—some are in alternative currencies, and you don’t know what the exchange rates will be until the day of your trip...
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