Turns out magic squares of squares are actually *easy* to find in most finite rings (called non-Parker). And the Parker rings (which do *not* work) are the harder ones to find!
Anyway, fun nice weekend project! Here is my code in case anyone cares: colab.research.google.com/drive/1luEjB...
It was _only_ after writing my code that I bothered to look up "magic squares of squares in finite fields" and found a paper from 2019 (arxiv.org/abs/1908.03236 - with a much better algorithm!) and Matt Parker's video from 2021 (youtu.be/FCczHiXPVcA) discussing the idea of finite fields and rings.
![A magic square of squares with the numbers [[27^2, 20^2, 3^2], [57^2, 58^2, 60^2], [8^2, 6^2, 42^2]] where (modulo 11^2) all the rows, columns, and diagonals sum each to 7^2.](https://cdn.bsky.app/img/feed_thumbnail/plain/did:plc:kpgymgvzedujnjklipzlhvnw/bafkreia4wamhon2b6vf3e6av7upiqfrvbqzems3323g6ffqb35sxe7la4i)
A magic square of squares with the numbers [[27^2, 20^2, 3^2], [57^2, 58^2, 60^2], [8^2, 6^2, 42^2]] where (modulo 11^2) all the rows, columns, and diagonals sum each to 7^2.
I was thinking about magic squares of squares (infamously hard to find) and wondered whether, as a stepping stone, modular arithmetic could help. I threw in some "terrible python code" and soon I was finding plenty of them! Here is a fun one where the modulo and the total sum are also squares!
A crescent moon shining above Edinburgh castle, that is lit up with floodlights. In the foreground the dark outline of Arthurs Seat, the mountain in the middle of Edinburgh city. In the background, whispy clouds and the distant mountains of Fife. Credit: https://www.tomduffin.com
"It is the moon, I ken her horn,
That's blinkin' in the lift sae hie;
She shines sae bright to wyle us hame,
But, by my sooth, she'll wait a wee!"
🏴Happy Burns Night🏴
📷: @tomduffinphotos.bsky.social
#RobertBurns #BurnsNight #astrophotography 🔭
Lately I just scroll past posts on /the state of things/ ... it just keeps feeding the outrage and nothing comes out of it.
More cats and fun things is definitely why I'm here on Bluesky!
Indeed! Found about it from your interview on @andrewcopson.bsky.social 's "What I Believe". After reading the story about the Forelius pusillus ants, I just knew I had to know more!
By working together, the earliest humans were increasingly able to overcome the challenges that nature threw at them: food scarcity, water shortages, dangerous predators. But as a consequence, other humans became our primary threat. We were no longer battling against nature, but against each other.
Cooperation is, at heart, a means by which entities improve their own position in the world. In other words, cooperation is favoured if and when it offers a better way to compete. A corollary of this is that cooperation frequently has victims.
Sometimes a male orb-weaving spider manages to force a copulation with a juvenile female, inserting both of his palps into the female's separate sperm- storage organs. If the male succeeds, something strange happens to him: his heart spontaneously stops beating and he dies in flagrante.
Ancestral Homo may have looked much like you or I, but to the sabre-toothed cats and gigantic lions that prowled the savannah, I suspect that our human relatives probably looked a lot like lunch.
A dominant male chimp rules by force. Dominance is taken and imposed upon others, against their will. But ancestral humans did things differently. In a world where anyone could hurl a stone or strike a spear from afar, brawn was no longer a viable means to wield authority. The mutiny had begun.
Book cover of The Social Instinct
The largest collective on Earth is a single international super-colony, spanning countries and continents, of a species called the Argentine ant. Although relations between members of the same super-colony are friendly, competition between ants from different families is ferocious.
When the spiderlings of a black-lace spider mother hatch, she provides a meal of unfertilised eggs for them to eat and, a few days later, actively encourages them to devour her alive.
It has become modish to assert that female-biased care is culturally entrenched, rather than biologically influenced, but a cursory look at our neighbours on the tree of life shows that we cannot lay all the blame for a mother's work never being done at the feet of the patriarchy.
📚 I've recently finished “The Social Instinct” by @nicholaraihani.bsky.social. It was a wonderful and thought provoking book, on how cooperation evolved in nature, and how crucial it is for our survival.
💡 Anyway, here a few of my favourite quotes from the book. What do you think of them?
Interesante!
🤣
A sum of cubes 1^3 + 2^3 + ... + 9^3 = (1 + 2 + ... + 9)^2 = 45^2 = 2025.
... and rearranged as a sum of cubes!
A square where each side is made of (1 + 2 + ... + 9) units. Therefore, the square has 45^2 = 2025 units total.
Most interestingly, the square of a triangle number can always be broken up groups, ...
A (9 + 8 + ... + 1) triangle on top of a (1 + 2 + ... + 9) triangle, forming a 9 x 10 rectangle.
Calculating the sum of numbers from 1 to 9 is easy. If you duplicate the triangle, you obtain a 9 x 10 rectangle. So the sum must be (9 x 10) / 2 = 45.
The sequence 1 + 2 + ... + 9 = 45 which looks like a triangle.
Moreover, 45 is a triangle number. This means it can be arranged as a sum of consecutive numbers, looking like a triangle.
A square of side length 45 and area 2025, with a party face emoji on the side.
This year is a square number, because 2025 = 45².
Happy New Year at the @nhm-london.bsky.social 🎉!
😧 I'm old! (1980)
Rolled the entire sequence of 419 digits! 🤣 Ha, no chance!
Good work by the way!
(And she's deputy editor at @ourworldindata.org, so she knows what she's talking about!)
Anyway, can't recommend enough @hannahritchie.bsky.social ”Not the End of the World”. For all of us dreading what the future may bring due to climate change; she brings some (data based!) rays of hope and focus on actions that can actually have an impact.
My 2024 Books - Not the End of the World, by Hannah Ritchie - Love Triangle, by Matt Parker - Project Hail Mary, by Andy Weir - A theory of Everyone, by Michael Muthukrishna - On the Edge, by Nate Solver - Infectious Generosity, by Chris Anderson - The World Behind the World, by Erik Hoel - What I Believe, by Andrew Copson - Unruly, by David Mitchell - Free Will: the basics, by Meghan Griffith
Not a ”new year's resolution” but, as it turned out, this was probably the year I've read the most books! (“those are rookie numbers”, I know!)
¡Y #TalDiaComoHoy de 1968, Bill Anders toma desde la órbita lunar del Apolo VIII esta maravillosa foto-“Earthrise”- una de las más famosas de todo el siglo XX!
¡¡Feliz Navidad!!🎄🎁 😘
El efecto perspectiva: el astronauta William Anders, autor de esta famosa instantánea, dijo a la vuelta de la misión Apolo 8: «Fuimos hasta la Luna a explorar la Luna, pero lo que descubrimos fue la Tierra». #LaRevuelta
Mi nuevo escrito sobre un estudio que llama a la cautela en el uso de agentes de inteligencia artificial.
#InteligenciaArtificial #Ética #LLMs
A chart in three parts showing data on child mortality to make the points that "The world is awful. The world is much better. The world can be much better. All three statements are true at the same time."
The world is awful. The world is much better. The world can be much better.
All three statements are true at the same time. Understanding this is key to solving big global problems.
We believe data & research can help us understand both the problems we face & the progress that’s possible. 🧵
I think the point is that it remains relatively straightforward to find simple prompts where the model fails. And it will remain so until the more general problem, e.g. compositionality, is actually solved.
Coooooool shot of a hot planet
My top podcasts of the year! lists.pocketcasts.com/b4a6928d-ee1... #pocketcasts #playback2024
Si, estuvo genial la plática! También la disfruté mucho y me animó a comprar tu libro 😉
