I've been playing with the new o3-mini-high model which came out this weekend, marketed as "Great at coding and logic". A surprising start to its chain of thought here.

(Some of you may enjoy thinking about this question yourself.)

https://twitter.com/RadishHarmers/status/1884770279341035981

ChatGPT is not happy with me. How it started, how it ended. I unfortunately can no longer share a link to the full conversation but I'll share the salient points in posts below.

o1-mini (the latest OpenAI offering, ChatGPT with advanced reasoning skills) can be quite impressive, when you know the correct answers to accept and incorrect answers to reject. Here's a simple question, which some of you may enjoy figuring out.
Its answers in more detail. Can you figure out which is correct?

Ok. Let's do this. Here's the full solution for the four people who care.

https://twitter.com/littmath/status/1834273354628424080

The previous discussion was here. You may want to read that first before this thread, but you don't need to. This new thread will be self-contained.

https://twitter.com/RadishHarmers/status/1834630204813091265

I find this clear now, like so (argument discovered by others such as @NoahJSnyder and @MtgJulian; I am only reframing or rewording it):

https://twitter.com/littmath/status/1769408478139785497

Consider a random walk in which one takes equally likely steps of one unit up or one unit down, but with different distributions of speeds. (E.g., maybe up steps take one hour, while down steps have probability 1/2 of taking 2 hours, 1/4 of taking 3 hours, 1/8 of 4 hours, etc).

@llllvvuu @littmath @NoahJSnyder @MtgJulian Summarizing the proof you all devised:

A "tied" string has equally many HHs and HTs. A "flippy" string is tied of size > 1 starting and ending with H, with no such proper prefix. Reversal is a length-preserving involution exchanging flippy strings starting with HH vs. with HT. @llllvvuu @littmath @NoahJSnyder @MtgJulian Say S has base P if S is of the form PHQ, PH is tied, and HQ has no flippy prefix. As there are equally many Q of a given length starting with H vs. T, our above involution tells us there are equally many strings of a given length whose base is followed by HH vs. HT.

I condemn Hamas killing 1400 Israelis.

I condemn Israel killing 1400 Gazans.

I condemn Israel killing 1400 Gazans.

I condemn Israel killing 1400 Gazans.

I condemn Israel killing 1400 Gazans.

I condemn Israel killing 1400 Gazans. I condemn Israel killing 1400 Gazans.

One last desperate attempt to illustrate why unique prime factorization isn't just an obvious tautology:

https://x.com/RadishHarmers/status/1282128425499742208

Suppose you worked at a bank that only had quarters and dimes. There's lots of amounts of money you can pay out: You can make payments of a dollar, or $1.50, or $2.60, or 45¢, lots of things.

I gave one explanation but it's not my favorite. It's just the one that tied in with the discussion of fractions. I'll give another explanation I like much better:

https://twitter.com/RadishHarmers/status/1282422228030754824

What are the last digits of the multiples of 4? Well, it goes like this: 0, 4, 8, 2, 6, 0, 4, 8, 2, 6, 0, 4, 8, 2, 6, ….

I often think there are two most basic, most ubiquitously applicable elementary mathematical insights that, for whatever reason, have not been included in standard curricula and which laypeople remain generally unaware of: 1. The greatest common factor of a collection of whole numbers is always obtainable by adding/subtracting multiples of those numbers (Bézout's theorem; Euclid's algorithm).