Nathaniel Johnston
@njohnston.ca
Associate Professor of Mathematics at Mount Allison University
Interested in quantum information theory, Conway's Game of Life, recreational mathematics, and mathematics pedagogy.
🔗 https://njohnston.ca/
▶️ https://www.youtube.com/@NathanielMath
Interested in quantum information theory, Conway's Game of Life, recreational mathematics, and mathematics pedagogy.
🔗 https://njohnston.ca/
▶️ https://www.youtube.com/@NathanielMath
Huge shout-out to authors who put humour, even very mild humour, in their papers. You keep me awake.
August 25, 2025 at 8:57 PM
Huge shout-out to authors who put humour, even very mild humour, in their papers. You keep me awake.
Now, two months later, Musk says that "Grok 3 is becoming superhuman" because Grok 3 obtained just as good as solution (i.e., an absolutely terrible non-solution) to this Putnam problem. Unreal.
February 28, 2025 at 8:24 PM
Now, two months later, Musk says that "Grok 3 is becoming superhuman" because Grok 3 obtained just as good as solution (i.e., an absolutely terrible non-solution) to this Putnam problem. Unreal.
Shameless self-promotion time: if you enjoyed this thread and/or are interested in these sorts of aspects of Conway's Game of Life, have a look at my (free) book "Conway's Game of Life: Mathematics and Construction", co-authored with Dave Greene: www.conwaylife.com/book/
January 1, 2025 at 2:25 AM
Shameless self-promotion time: if you enjoyed this thread and/or are interested in these sorts of aspects of Conway's Game of Life, have a look at my (free) book "Conway's Game of Life: Mathematics and Construction", co-authored with Dave Greene: www.conwaylife.com/book/
#3 (1/4): Oscillator density. In the early days of Life, it was conjectured that the maximum density (i.e., maximum ratio of alive cells to dead cells) of an infinitely large still life is 0.5. This density is easily attained by alternating rows of dead and alive cells.
January 1, 2025 at 2:22 AM
#3 (1/4): Oscillator density. In the early days of Life, it was conjectured that the maximum density (i.e., maximum ratio of alive cells to dead cells) of an infinitely large still life is 0.5. This density is easily attained by alternating rows of dead and alive cells.
#2 (1/4): Still life glider synthesis. A glider synthesis is a way of crashing together 2 or more gliders so as to create another object. For example, in the image below three gliders collide so as to create a lightweight spaceship.
January 1, 2025 at 2:21 AM
#2 (1/4): Still life glider synthesis. A glider synthesis is a way of crashing together 2 or more gliders so as to create another object. For example, in the image below three gliders collide so as to create a lightweight spaceship.
#1 (1/4): Phoenices. A phoenix is an oscillator in which every cell dies in every generation (and thus every alive cell was dead in the previous generation, hence the name). The first known phoenix has period 2: conwaylife.com/wiki/Phoenix_1
January 1, 2025 at 2:20 AM
#1 (1/4): Phoenices. A phoenix is an oscillator in which every cell dies in every generation (and thus every alive cell was dead in the previous generation, hence the name). The first known phoenix has period 2: conwaylife.com/wiki/Phoenix_1
But there's a problem: the answer was found by computing the thing for n = 1, n = 2, and n = 3, spotting a pattern, and hoping that the pattern continues. Seem familiar?
December 20, 2024 at 3:00 AM
But there's a problem: the answer was found by computing the thing for n = 1, n = 2, and n = 3, spotting a pattern, and hoping that the pattern continues. Seem familiar?
Wow, ChatGPT got the correct answer of 10^(n(n-1)/2)! People proclaim this would be scored 10/10 and that the AI apocalypse is here.
December 20, 2024 at 3:00 AM
Wow, ChatGPT got the correct answer of 10^(n(n-1)/2)! People proclaim this would be scored 10/10 and that the AI apocalypse is here.
To see why this line of reasoning falls flat to me, let's look at its answer to this year's question A6. If you don't want to read the technical details of the question, all that really matters here is that the question asks you to compute a formula for something.
December 20, 2024 at 2:59 AM
To see why this line of reasoning falls flat to me, let's look at its answer to this year's question A6. If you don't want to read the technical details of the question, all that really matters here is that the question asks you to compute a formula for something.
And the people reading the "proof" don't notice that it's garbage since they're exhausted from the previous five paragraphs which were almost entirely content-free and irrelevant (but technical and impressive-sounding!).
December 20, 2024 at 2:59 AM
And the people reading the "proof" don't notice that it's garbage since they're exhausted from the previous five paragraphs which were almost entirely content-free and irrelevant (but technical and impressive-sounding!).
This wouldn't be so annoying if ChatGPT were honest about how it came up with and verified (or didn't verify) the solution. But instead, it acts like a grade-grubbing student and throws some crap against the wall, hoping that something will stick.
December 20, 2024 at 2:58 AM
This wouldn't be so annoying if ChatGPT were honest about how it came up with and verified (or didn't verify) the solution. But instead, it acts like a grade-grubbing student and throws some crap against the wall, hoping that something will stick.
ChatGPT gets the right answer, which on the surface seems super impressive. But then you read its "proof" and see that it didn't actually prove anything: it just did a ton of computations, noticed a pattern, and hoped against hope that the pattern continues to hold forever.
December 20, 2024 at 2:58 AM
ChatGPT gets the right answer, which on the surface seems super impressive. But then you read its "proof" and see that it didn't actually prove anything: it just did a ton of computations, noticed a pattern, and hoped against hope that the pattern continues to hold forever.
To illustrate what I mean by this, let's look at its answer to question B1, which asks for which sizes of grids it's possible to pick squares in a certain way.
December 20, 2024 at 2:58 AM
To illustrate what I mean by this, let's look at its answer to question B1, which asks for which sizes of grids it's possible to pick squares in a certain way.
ChatGPT, most of the time, simply does not prove its answer is correct. This isn't a matter of "lack of rigor" or however else people try to deflect this point. It's that ChatGPT quite simply ignores the entire crux of the given problem.
December 20, 2024 at 2:57 AM
ChatGPT, most of the time, simply does not prove its answer is correct. This isn't a matter of "lack of rigor" or however else people try to deflect this point. It's that ChatGPT quite simply ignores the entire crux of the given problem.
The problem is that Putnam questions all require you to prove something. Not just compute it or get the right answer, but prove that it's right. Heck, lots of Putnam questions *tell* you the answer, and just want you to prove it correct.
December 20, 2024 at 2:57 AM
The problem is that Putnam questions all require you to prove something. Not just compute it or get the right answer, but prove that it's right. Heck, lots of Putnam questions *tell* you the answer, and just want you to prove it correct.
First, the claims are that ChatGPT o1 scored 80+ points on the Putnam, or 60+ points, or high enough to be in the top 2% of participants, or whatever other impressive-sounding claim someone can make up.
December 20, 2024 at 2:56 AM
First, the claims are that ChatGPT o1 scored 80+ points on the Putnam, or 60+ points, or high enough to be in the top 2% of participants, or whatever other impressive-sounding claim someone can make up.
There's been a bunch of claims (mostly on X) that ChatGPT did great on this year's Putnam math competition. Let's do a thread to talk about it! 🧵
#MathSky
#MathSky
December 20, 2024 at 2:55 AM
There's been a bunch of claims (mostly on X) that ChatGPT did great on this year's Putnam math competition. Let's do a thread to talk about it! 🧵
#MathSky
#MathSky
It looks like Overleaf's new AI writing tools don't like the notes that I make to myself while writing papers.
December 5, 2024 at 4:21 PM
It looks like Overleaf's new AI writing tools don't like the notes that I make to myself while writing papers.
Spoiler: that sum equals...
December 5, 2024 at 11:44 AM
Spoiler: that sum equals...
Had a recent research project where we had to evaluate this hideous sum. Convolution to the rescue! Made a video to talk about how it works (or at least how some similar but simpler sums work):
www.youtube.com/watch?v=aIj6...
www.youtube.com/watch?v=aIj6...
December 5, 2024 at 11:40 AM
Had a recent research project where we had to evaluate this hideous sum. Convolution to the rescue! Made a video to talk about how it works (or at least how some similar but simpler sums work):
www.youtube.com/watch?v=aIj6...
www.youtube.com/watch?v=aIj6...
FoxTrot pioneering this meme:
December 5, 2024 at 2:44 AM
FoxTrot pioneering this meme:
Me, a moron: I don't need a local TeX installation, Overleaf works so well and is so easy!
Overleaf: like every online service that exists, goes down from time to time.
Me: shocked Pikachu.
Overleaf: like every online service that exists, goes down from time to time.
Me: shocked Pikachu.
December 3, 2024 at 1:05 PM
Me, a moron: I don't need a local TeX installation, Overleaf works so well and is so easy!
Overleaf: like every online service that exists, goes down from time to time.
Me: shocked Pikachu.
Overleaf: like every online service that exists, goes down from time to time.
Me: shocked Pikachu.
The On-Line Encyclopedia of Integer Sequences (OEIS) is hiring someone to manage the crazy number of sequences that it has to review. If you're a US resident with a math PhD, give it a look! Applications due Jan. 25, 2025.
neilsloane.com/doc/OEIS.ME....
neilsloane.com/doc/OEIS.ME....
November 25, 2024 at 6:33 PM
The On-Line Encyclopedia of Integer Sequences (OEIS) is hiring someone to manage the crazy number of sequences that it has to review. If you're a US resident with a math PhD, give it a look! Applications due Jan. 25, 2025.
neilsloane.com/doc/OEIS.ME....
neilsloane.com/doc/OEIS.ME....
I wrote a review of Jane Hawkins' new book "The Mathematics of Cellular Automata", which is now online.
tl;dr: It would be fun to teach a CA course from this book to students who have already taken Real Analysis.
www.tandfonline.com/doi/full/10....
tl;dr: It would be fun to teach a CA course from this book to students who have already taken Real Analysis.
www.tandfonline.com/doi/full/10....
November 25, 2024 at 12:18 PM
I wrote a review of Jane Hawkins' new book "The Mathematics of Cellular Automata", which is now online.
tl;dr: It would be fun to teach a CA course from this book to students who have already taken Real Analysis.
www.tandfonline.com/doi/full/10....
tl;dr: It would be fun to teach a CA course from this book to students who have already taken Real Analysis.
www.tandfonline.com/doi/full/10....
Fun fact: my advanced linear algebra textbook has a page about exactly this! Screenshot attached.
November 22, 2024 at 10:49 AM
Fun fact: my advanced linear algebra textbook has a page about exactly this! Screenshot attached.