A graph of the histogram for frequencies of water sort puzzles with specific numbers of optimal moves to solve. The puzzle setup in this case is containers that are 5 segments high (N=5), with 5 distinct colours (C=5) and 2 empty containers (K=2) at the start/end. The most common number of moves is 19 and the bulk of the distribution is in the range [18, 20]. The distribution is bell-like, but skewed towards smaller values. It is fitted with a skew normal distribution curve that has parameters ξ=20.4, ω=2.38, and α=-1.76.

Finally, adding one extra colour (C=5) is the hardest of all with an average of nearly 19 moves to solve. This was the least surprising to me. Is it a coincidence that this average is 5 moves higher than the C=4 case, or is it a pattern? Is the average 25 if C=6? A question for future research...

A graph of the histogram for frequencies of water sort puzzles with specific numbers of optimal moves to solve. The puzzle setup in this case is containers that are 6 segments high (N=6), with 4 distinct colours (C=4) and 2 empty containers (K=2) at the start/end. The most common number of moves is 18 and the bulk of the distribution is in the range [17, 19]. The distribution is bell-like, but skewed towards smaller values. It is fitted with a skew normal distribution curve that has parameters ξ=19.3, ω=2.56, and α=-1.57.

With one extra segment on each container (N=6) the average increases to more than 17.5! This means adding 6 extra segments (only 4 of which are coloured) is significantly harder than taking 5 empty segments away (as in the previous case). This was also not what I expected.

A graph of the histogram for frequencies of water sort puzzles with specific numbers of optimal moves to solve. The puzzle setup in this case is containers that are 5 segments high (N=5), with 4 distinct colours (C=4) and 1 empty container (K=1) at the start/end. The most common number of moves is 15 and the bulk of the distribution is in the range [14, 16]. The distribution is bell-like, but skewed towards smaller values. It is fitted with a skew normal distribution curve that has parameters ξ=16.6, ω=2.50, and α=-1.49.

With one less container (K=1) the average only goes up by about half a move. I was surprised by this result! I would think reducing how much space you have to put colours would make the puzzle a lot harder, it only makes it a little harder. The distribution for this simulation is shown below.

The simulation is generic in the height of the containers (N), the number of colours (C) and the number of empty containers (K). My online version of the puzzle uses N=5, C=4 and K=2; but I also ran simulations for close variations on these values.

A graph of the histogram for frequencies of water sort puzzles with specific numbers of optimal moves to solve. The puzzle setup in this case is containers that are 5 segments high (N=5), with 4 distinct colours (C=4) and 2 empty containers (K=2) at the start/end. The most common number of moves is 15 and the bulk of the distribution is in the range [14, 16]. The distribution is bell-like, but skewed towards smaller values. It is fitted with a skew normal distribution curve that has parameters ξ=16.1, ω=2.38, and α=-1.77.

I also looked at the distribution of optimal moves by generating and solving many (200k) random puzzles. For the kind of puzzle used in the game (5 segment tall containers; 4 colours; 2 empty containers) the average is around 14.5 and the distribution is approximated by a skew normal distribution.

The algorithm for the optimal number of moves comes from here: kociemba.org/themen/water...

I have referenced this site before because I have talked about this kind of puzzle in the past. Then it was a ball sort problem instead of a water sort (the difference being how many segments move at once)

Screenshot from my water sort puzzle game. There are 6 containers, each of which is 5 segments high. The 4 containers on the left have their segments filled with colours. There are 4 distinct colours and they are all shuffled up in the segments. The goal of the game is to use the empty containers to sort the colours back into individual containers. The solved state of the puzzle will still have 2 empty containers and 4 full ones, but each of the full containers will only have a single colour.

I built another small game! This is a remake of a puzzle game I have enjoyed playing in the past. I also included an algorithm for finding the optimal number of moves. Trying to find the optimal sequence as a human is an additional challenge to the game.

birchmd.github.io/water-sort/

Cover of Once Upon a Prime by Sarah Hart. Three planes intersect in the middle of the cover: one shows text (presumably from a famous work of literature), another shows mathematical imagery such as geometric diagrams and graphs of functions, and the last shows the tail of a whale. The text "The Wondrous Connections Between Mathematics and Literature" is also included.

Review: Once Upon a Prime by Sarah Hart

The new-to-me content was sparse; I was already familiar with all the mathematical ideas and many of the historical anecdotes. The literary examples where these things appear are interesting though. It's not the book's fault that I love math so much!

⭐⭐⭐⭐

Cover of Someone You Can Build a Nest In by John Wiswell. A shadowy figure with a scary smile, wild hair, tentacles and glowing eyes looms over a woman. Chains, books and spiders adorn the background.

Review: Someone You Can Build a Nest In by John Wiswell

I didn't enjoy the first 2/3s because it looked like a typical liar revealed trope. I never like that one and I was dreading how it would resolve. But, without giving anything away, the last third of this book included my favourite parts.

⭐⭐⭐

Thanks to @lynnesargent.bsky.social for the beta testing and encouraging me to pursue this project ❤️

The goal of the game is to use your understanding of rates of change to reproduce various curves. I think it is a fun, interactive way to engage with concepts from calculus and differential equations. What do you think? Feedback is welcome!

I made an incremental/idle game! It's in the style of classics like Kittens Game and Universal Paperclips. The central element is a plot that visualizes how quantities in the game are changing (with respect to time and each other). You can check it out here:

birchmd.github.io/de-idle/

Cover of The Atlas Six by Olivie Blake. It shows image reminiscent of a diagram of the solar system with the sun at the center being the iris of an eye and pierced by a sword. The phrase "Knowledge is Carnage" are written at the top.

Review: The Atlas Six by Olivie Blake

I liked the opening of this book; recruiting powerful magical individuals to compete for spots in a secret society is very fun. I feel like the book drags in the middle though relative to this opening. It is more character-focused than I would prefer.

⭐⭐⭐

Cover of The Alloy of Law by Brandon Sanderson. The image shows two men: one with a bowler hat and shotgun slung over his shoulder, the other in a suit with a long overcoat and goggles around his neck.

Review: The Alloy of Law by Brandon Sanderson

It's been a long time since I read the original Mistborn trilogy and I remember quite enjoying those books. I think I liked this even more though - in a guilty pleasure sort of way. I'm a sucker for high action, witty remarks and powerful magic.

⭐⭐⭐⭐

Cover for A Grim Reaper's Guide to Cheating Death by Maxie Dara. A grey parrot is perched on a grim reaper's scythe and someone in a hooded black cloak holds a magnifying glass which contains the text "A S.C.Y.T.H.E Mystery".

Review: A Grim Reaper's Guide to Cheating Death by Maxie Dara

I didn't like this one as much as the first book. The parrot was funny and the additional world-building was interesting. But overall I didn't find the mystery as engaging and it didn't have as many touching emotional scenes.

⭐⭐⭐

@3blue1brown.com always produces incredible math explainers, but this one is particularly excellent. I love the overlap between math and art in it.

Cover of Play Nice by Jason Schreier. The title is shown in a blue font in the same style as the Blizzard Entertainment logo. There is an axe stuck in the 'E' as if it had been thrown there; perhaps by a fantasy video game character.

Review: Play Nice by Jason Schreier

I'm not much of a gamer (I haven't played any of Blizzard's games), but I am a programmer that has worked at startups so I could sympathize with some of the grind culture. This is an engaging non-fiction book that I quite enjoyed!

⭐⭐⭐⭐

In set-theoretic foundations, the statement “every fully faithful and essentially surjective functor is an equivalence of categories” is equivalent to the axiom of choice. But with the univalence axiom, it is just _true_

-The HoTT Book

Cover of Braiding Sweetgrass by Robin Wall Kimmerer. The subtitle text reads "Indigenous wisdom, scientific knowledge and the teaching of plants." Braids of sweetgrass appear above and below the title.

Review: Braiding Sweetgrass by Robin Wall Kimmerer

The book as a whole is a little repetitive, but in fairness the message of gratitude towards the life-sustaining earth and humanity being a productive part of ecosystems rather than apart from (or worse destroyers of) them deserves repeating.

⭐⭐⭐⭐

Cover of Saint Death's Daughter by C. S. E. Cooney. The title is woven into a woman's hair along with several items, including flowers, bones and small animals.

Review: Saint Death's Daughter by C. S. E. Cooney

I enjoyed the world building in this most of all. I love stories that include footnotes! Some of my favourite aspects include necromancers being allergic to violence, the pantheon of gods and how they manifest in different kinds of magic.

⭐⭐⭐⭐

My Lean code just got real horny for some reason

```lean
do
let d ← d
let t ← t
let f ← f
pure ⟨d, t, f⟩
```

I promise this is a real snippet that makes perfect sense in context.

Master of none no longer, I have now completed all the color mastery achievements!

Cover of The Ends of the World by Peter Brannen. The history of life on earth is visually represented from modern times all the way back to the deep past with layers of fossils of different kinds.

Review: The Ends of the World by Peter Brannen
This non-fiction book about the 5 mass extinctions in earth's history is well written with many colourful descriptions of the extreme conditions involved. The parallels to human caused climate change are relevant even 9 years after being published.
⭐⭐⭐⭐

I'm starting to learn Lean because (a) math is fun, (b) programming is fun and (c) knowing your program is correct because math is amazing. I am very impressed with the quality of the Lean docs and tutorials. I have finished the natural numbers game and it was great!
adam.math.hhu.de#/g/leanprove...

Deck 1 Abigale, Eloquent First-Year (ECL) 204 3 Moon-Vigil Adherents (ECL) 184 1 Vinebred Brawler (ECL) 201 1 Overgrown Tomb (ECL) 266 2 Prideful Feastling (ECL) 238 1 Gangly Stompling (ECL) 226 1 Wildvine Pummeler (ECL) 203 1 Champions of the Perfect (ECL) 171 1 Dawnhand Dissident (ECL) 98 1 Blossoming Defense (ECL) 167 1 Evolving Wilds (ECL) 264 1 Requiting Hex (ECL) 116 1 Pummeler for Hire (ECL) 190 1 Selfless Safewright (ECL) 193 1 Lys Alana Dignitary (ECL) 180 1 Morcant's Loyalist (ECL) 236 1 Bogslither's Embrace (ECL) 94 2 Lys Alana Informant (ECL) 181 1 Eclipsed Elf (ECL) 218 1 Iron-Shield Elf (ECL) 108 1 Assert Perfection (ECL) 164 7 Swamp (SLD) 1470 8 Forest (UNF) 244 Sideboard 1 Wildvine Pummeler (ECL) 203 1 Wanderbrine Preacher (ECL) 41 1 Eclipsed Boggart (ECL) 217 1 Bile-Vial Boggart (ECL) 87 2 Moonglove Extractor (ECL) 109 1 Mischievous Sneakling (ECL) 235 1 Safewright Cavalry (ECL) 191 1 Stoic Grove-Guide (ECL) 243 2 Reckless Ransacking (ECL) 152 2 Barbed Bloodletter (ECL) 86 1 Surly Farrier (ECL) 196

7-2 record in Premier Draft Lorwyn Eclipsed at rank Gold 1.

7 wins on #MTGArena in a #MTGLorwyn Premier Draft with a sweet elves deck! Having good rares helped of course, but Moon-Vigil Adherents was still the main win condition.

Mastery Level: 60 Constructed Rank: Platinum 4 Limited Rank: Platinum 4 Games Played: 429 (more than over 80% of players) Quests Gold: 35000 (more than over 90% of players) Daily Wins Gold: 26250 (more than over 90% of players) Boosters opened: 101 Favorite Color: Red Favorite Format: Standard Ranked Favorite Format Wins: 86

#MTGArenaStats during Avatar: The Last Airbender

Cover of Ella Enchanted by Gail Carson Levine. A young girl is pictured with the skyline of two kingdoms in the background.

Review: Ella Enchanted by Gail Carson Levine

I always hate the third act breakup trope, but otherwise this was a nice, whimsical story with a surprising amount of philosophical depth. I also like that the relationship between Ella and the prince is fleshed out before the Cinderella retelling.

⭐⭐⭐

Cover of Race the Sands by Sarah Beth Durst. The image shows a metal lion running across a landscape of sand dunes.

Review: Race the Sands by Sarah Beth Durst

Wow this book was good! The author follows some tropes while subverting others and all the choices aligned perfectly with my preferences. I'll keep this spoiler-free and just say the ending is epic.

⭐⭐⭐⭐⭐

Izzet Elementals in the top 8 at the pro tour with the full play set of Roaming Throne, you love to see it! I'm definitely rooting for Eduardo Sajgalik now. And he's Canadian too which is extra awesome!