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Envy-Free Cake Cutting Protocol
Set Equivalence Theory in sudoku
With the New Year amongst us, the dreaded perception of the passage of time* shoots to its peak strength, ushering in a new four digit number which generates some resolutions to follow through with before the big rock makes another full trip around the nuclear mystery ball.
I actually like this particular tradition because it chucks in some promises of ambition++, albeit a few weeks before half the capitalism tokens are hijacked by the government through the process of taxation.
In 2021 I didn't write a great deal, so I've chosen the world's easiest topic to blog about more regularly: stuff in the last while that might be at least mildly interesting to hear about. There's no agreed upon frequency for these posts, so this may be the only entry, or a few more might threaten to materialise as the year progresses.
'A Discrete and Bounded Envy-Free Cake Cutting Protocol for Any Number of Agents' + refreshing the knowledge of various other mathematical, economical, and/or behavioural principles
This one's not exactly new studying but it did lead me down a rabbit hole of bits that I'd learned about in the past but vaguely needed to reinforce, with some satisfying new discoveries sprouting up along the way.
I like to rewatch Lord of the Rings once every couple of years, and also reread the basics of various theories and laws/principles, especially those that apply to our everyday lives in ways we might not expect. There's always something new that jumps out and clicks more pieces together.
In the case of Fellowship of the Ring (extended), this year it was a further appreciation of Merry & Pippin's narrative throughline involving food. Carrots, mushrooms, stolen crops, nice crispy bacon, second breakfast, and Lembas bread.
Let's begin this section with a quote from some lad called Carl:
"The simplest thought like the concept of the number one has an elaborate logical underpinning. The brain has its own language for testing the structure and consistency of the world."
— Clever red turtleneck space person whose personality was stolen by Jeff Goldblum
Fair portion sizes for all
On New Year's Eve I spoke with someone about cutting a cake fairly into three equal portions, or at least in such a way that each eater of the cake would be satisfied with their slice. This reminded me of an excellent paper that detailed how, while this problem was adequately solved in 1960, it wasn't until around 2016 that a solution was found for computing envy-free cake allocation, including fair scheduling, resource allocation, and conflict resolution for any number of people.
If two people want to share any foodstuff that involves dividing/cutting, I find the neatest solution is simply that the cutter is not the chooser. You're going to be incentivised to divvy up something fairly if the other person gets first dibs, because they're obviously going to pick what they believe to be the bigger portion, which means you're happy either way because you believed your initial cut was equal.
When a few people are wanting a fair portion, it's slightly more complicated to execute but a great deal of fun, as demonstrated by mathematician Hannah Fry in this video. Fry is an amazing communicator of ideas and her lectures on the Numberphiles channel are perfect even if you're not equipped with any maths chops.
I wasn't too familiar with many real world applications of Weber-Fechner until this past week, after hopping from cakes to another Hannah Fry video. These laws relate to human perception and how it can change with given stimuli.
For example, as Fry explains, the reason we seem to experience time faster as we get older - and the years seem to fly by - is because we're unable to perceive the difference between small changes in stimulus if we're dealing with progressively larger units.
There's some good classic scenarios here, such as being able to tell the difference between 100g and 120g of weight, but struggling to tell the difference between 200g and 220g.
Similarly, the more years we've been alive, the more we're used to being alive, so a year becomes a smaller and smaller ratio of our total experience of existing, therefore our perception of them as units of time begins to warp.
It's also worth noting separately that comfort and familiarity results in the brain firing off a bit less in general. A day or two spent in our own home can feel like it zooms by because we're not really required to work as hard to focus or be fully alert - the setting is known to us entirely.
I suppose then that the same applies to general life, where over time we're more used to how occurrences tend to occur, so without a lack of new ideas and experiences, a feeling of sleepwalking and time 'speeding up' sets in.
Pink Floyd summed this up pretty well then:
Tired of lying in the sunshine, staying home to watch the rain
You are young and life is long, and there is time to kill today
And then one day you find ten years have got behind you
No one told you when to run, you missed the starting gun
I guess the lesson here is "be present". And also to distrust chocolate companies who abuse this law to reduce the size of their bars without anyone noticing. This whole thing did bring Lord of the Rings back to mind, and why those Elves are so morbidly aloof. To be fair, living for thousands of years does sound like a chore.
I performed a cheeky refresh of dynamic time inconsistency. It's about how our preferences and decisions can change when presented with certain timescales. We often shoot ourselves in the foot because of this inconsistency. This one's definitely abused on the regular by people wanting to extract capital from others.
A quick example of this from that Wikipedia page: students at the start of a term are offered an extra day to study for their exams, at the cost of $50. Usually that offer would be rejected. After all, exams are months away, what's one more day? However, if the same offer were given the night before the exams, that's now a very tempting offer indeed, and one that could surely be peddled for more than $50.
The extra day never changes, and the choice is exactly the same, but it can become difficult to choose something that our present selves don't see the value in, but our future selves would definitely desire.
Set Equivalence Theory in sudoku
These past few months have been properly sudoku-centric for me, and I'm not ashamed at how nerdy that sounds. Classic sudoku (where the only rules are digits 1-9 must appear in every box, row, and column) is just the tip of the iceberg. But this exhilarating topic will be explored fully in another, dedicated blog post, because I've found some truly insane puzzles and rulesets, plus amazing niche communities of puzzle setters, and it deserves its own lengthy waffle.
Just to tease some initial sudoku logic to the folks at home - and by folks at home I mean the four people who read my blog - here's a little gem that applies to 100% of all sudokus in existence. It deals with identical sets.
The following seems pretty counterintuitive at first and may be too specific for efficient use against newsagent sudoku books, but sometimes the milk can successfully be poured before the cereal with far more advanced puzzling, especially when the breakfast bowl is invisible and keeps teleporting around the table.
Here we have a blank grid with two sets, orange and blue. As we know, the digits 1-9 must appear once in each row and column, therefore the digits 1-9 appear once each in the orange set, and once each in the blue set, because they both cover exactly one column and one row.
In short, orange currently contains one set of all the possible digits, and blue contains one set of all the possible digits. They are identical.
Now we remove the overlapping cell that's in both the blue and orange set. Whatever it was (1, 2, 3, 4, 5, 6, 7, 8, or 9), it's been removed from both sets. So orange and blue are still identical.
Or another way to think of it, orange and blue "weigh" the same. They still contain the same digits, because whatever digit has been removed from the overlapping cell has been deleted from both sets simultaneously.
Expanding this logic is very simple. We just add more chunks of 1-9 to the sets. Now orange has four sequences of 1-9, and so does blue.
Symmetry doesn't really matter beyond just balancing the weight. The blue set being off-centre and the orange set being split into two islands has no impact on what's to come.
Let's remove the overlapping cells.
This next image now looks less inviting than the first example, but the system works. Blue and orange are completely identical. If you find any completed sudoku puzzle, it is 100% guaranteed that the cells I've highlighted in blue will match the orange cells.
This can be useful when narrowing down potential candidates. If you know orange already contains two 8s, but blue currently has zero 8s, you know that at least two more 8s need to find their way into blue at some point.
Of all the set combinations, this one is definitely my favourite. It's a little more complex to set up because it involves a "double" orange bag with some orthogonal overlapping, but let's not worry about that. The finished product boasts the same basic logic.
Who is Phistomefel? They're a puzzle setter with 80 published sudokus on Logic Masters.
What is a ring? It's like a circle made of whatever.
Feast your peepers on that absolute heisted duffel of loot. Orange is always the same as blue, in all sudokus. What sets the ring apart is that it can genuinely hit the nuclear button on pesky ultra hard sudokus, at times bypassing intended logic chains that human setters have crafted. Just remember that it probably won't help you in any sudoku magazine you'll find knocking about in airport book stores, because minimal advanced tactics are required to churn through those.
Those tactics usually consist of Snyder notation, empty singles/doubles, X-wings, and Y-wings. Chuck those into a search engine if you care to speedrun your puzzling.
Everyday sudoku tactics aside, I find the SET logic to be a very satisfying thing indeed. If the entire universe were a giant sudoku grid and we all lived as numbers, it would be a proper Sixth Sense moment if someone told us about Phistomefel's Ring.
Preview of more to come
Stewart Lee's pedal bin list
I used to think keeping a detailed list of every movie I'd seen was a little obsessive, until I discovered the impressive and mildly disturbing mammoth-chungus of a thing that is Stewart Lee's yearly roundup. The guy records movies, albums, books, comics, TV, theatre, comedy, radio, historic sites, food, and my favourite: lists of people or things he likes and doesn't like.
"On the pedestal" and "in the pedal bin".
If the perfect joke has an identical setup and punchline, then these lists are the perfect troll. They're pure bait, luring not-very-noteworthy people into writing cringey response pieces about being placed next to Donald Trump and The Taliban in the "dislike" section.
But the amusing truth is that no retort can land a critical hit on such a troll, because it's so simplistic and childish. It has no attack surface. It's too slippery to be grappled. The more complex your counterargument, the dumber you look.
In fact if someone makes a straightforward, context-free list of people they don't like, and you respond trying to act smart or righteous in any way, you lose. It's a solved game, an X in the middle of noughts and crosses, a disc right in the centre of a Connect 4 board.
On top of that, Stewart is a professional comedian. You're not going to win - as Simon Amstell pointed out on Nevermind The Buzzcocks when Donny Tourette tried to lay some verbal shitposting on Bill Bailey.
A bunch of the pedal bin crew apparently spaffed out some tweets, but the following people lost the game in style by choosing to write entire articles about it: Simon Evans, Martha Gill, Julie Burchill.
I particularly like the repeated argument that putting Donald Trump in the pedal bin is "too obvious" and "audience-pleasing". It's like saying Avengers: Endgame is too obvious to put on a list of watched movies because everybody else has also seen it.
That's it for today's "things I found interesting". There may or may not be another one in the near or distant future.