Things June 2022: Lyric repetition, puzzle design, Outer Wilds

What just happened?

You may have noticed that Things posts/emails slowed in frequency over the years (from weekly to monthly to sporadic) and more recently had effectively stopped!

There’s a whole aspect of this that I plan to unpack later involving my levels of personal creativity and motivation. I posted in August 2021 that I would try to work around this by trying to focus on a post about a single Thing each time. Well, this has started to work, and I’m jumping off from that to a more traditional round up of things!

Beginnings and Endings in public performances (link)

In this post I examined the ways that different cultural forms (movies, gigs, puppet shows etc) signal to an audience the start and end of a performance, and why this is important.

While writing it, I realised that online talks/presentations, which have become much more prevalent during the pandemic, had not reached a good consensus on these difficult problems, and I set out my own list of suggestions of how to start and end them. Honestly, I’m not that satisfied with these and if anyone has any better suggestions I’d love to hear them. Read the whole thing here.

Why I love the ‘Up All Night’ music video (link)

Effectively a dramatically expanded paragraph from a normal issue of things (this one), I explained in some detail what I think is so good about this music video:

This fascinating short film seems weirdly underdiscussed on the internet, so again I’d be very happy to hear anyone else’s thoughts on it! Mine are here.

Repetition for emphasis in lyrics

When a particular word or phrase is sung repeatedly in a song, the meaning changes slightly: it starts to feel like something the singer really really desires.

This was my favourite feature of Frozen 2 (2019)‘s song “Into The Unknown”

Elsa hears a siren-like voice, and in the first verse sings about how she plans to ignore it, culminating in this:

I’ve had my adventure, I don’t need something new
I’m afraid of what I’m risking if I follow you…
Into the unknown
Into the unknown
Into the unknown

‘Into the Unknown’, music and lyrics by Kristen Anderson-Lopez and Robert Lopez

You can read it between the lines, but the repetition of ‘Into the unknown’ and the tone in which it is sung tell us that, deep down, Elsa really does want to follow the voice. This is then validated in the second verse which instead concludes

Don’t you know there’s a part of me that longs to go
Into the unknown?
Into the unknown
Into the unknown!

This device is really nicely exploited in The LEGO Movie 2 (2019) (another animated sequel from 2019, but which came out before Frozen 2). The protagonists encounter Queen Watevra Wa’Nabi who then goes into a musical number to explain that she is not evil, which brilliantly plays out exactly as you would hope from that premise:

Here the repetition is partly from the backing singers (denoted in brackets):

And if you make eye contact with me
I totally won’t have you executed immediately
‘Cause that’d be evil (evil)
Evil (evil)
Evil… and that’s so not me.

‘Not Evil’ by Jon Lajoie

The repetition of ‘evil’ reinforces the unconvincing negatives, giving the impression that she is, in fact, actually evil.

So anyway, all of this is an elaborate build-up to explain a problem I have with ‘Roxanne’ by The Police.

Sting sings earnestly about how much he loves Roxanne, a sex worker, and how he wants her to stop doing that and just be with him. It’s a little odd as there’s nothing in the song indicating that he would support her or that she could do really anything other than just belong to him, but perhaps that’s supposed to be implied.

Specifically he asks her to change by saying “You don’t have to put on the red light”, which is fine and a reasonably delicate turn of phrase. Where it gets weird is in the conclusion of the chorus, and especially the outro. Again denoting backing singers in brackets, this reads as follows:

(Roxanne) You don’t have to put on the red light
(Roxanne) Put on the red light
(Roxanne) Put on the red light
(Roxanne) Put on the red light
(Roxanne) Put on the red light
(Roxanne) Put on the red light
(Roxanne) Put on the red light
(Roxanne) Put on the red light
(Roxanne) Put on the red light
(Roxanne) Put on the red light
(Roxanne) You don’t have to put on the red light
(Roxanne) Put on the red light
(Roxanne) Put on the red light
(Roxanne) Put on the red light
(Roxanne) Put on the red light

‘Roxanne’ by Sting

So Sting sings two times that she doesn’t have to put on the red light, but then seemingly that she should put it on thirteen times, which for me always undermined what I assume was the intended sentiment.

Puzzle: Rice Cookers

Returning to the old tradition of puzzles in things: how do rice cookers work?

Now you can ask the internet for the answer to this, but I suggest this is worth figuring out on your own! If you’re not familiar with this excellent device, the key mystery is that you can add any amount of rice (up to some limit), then add 1.5x as much water, and switch the rice cooker on. You don’t have to tell it how much rice you are cooking, but it will cook it perfectly and then let you know when it’s ready. So how exactly does it know when the rice is done?

Puzzle Design

I know many things readers are not only interested in solving puzzles, but also setting them. I found Elyot Grant’s series of videos on the subject pretty fascinating, albeit a bit longer than they could have been (although this is the ‘extended’ version of his GDC talk).

I particularly appreciated some useful terminology he introduced me to for speaking about puzzles:

Fiero vs Eureka

Elyot likes the term ‘Eureka’ for the moment a core understanding of a puzzle kicks in, arguing this does better justice to it than the more prosaic term “aha moment”. In particular he calls it out as distinct from ‘Fiero’, which describes the warm feeling of accomplishment after you have achieved something difficult. Video games often end up falling back on creating Fiero; creating Eureka moments is harder to do but often more rewarding to experience in the end.


This refers to anything incorporated into a puzzle that isn’t essential to the design, but somehow makes it more attractive or pleasing than if it was the purest distillation of what is needed to provoke a Eureka moment. For example, a sliding block puzzle could be shaped like an animal that it already nearly resembles; or the words in a word puzzle could be thematically linked somehow. This all adds to the pleasing sense in which engaging with and solving a puzzle can feel like understanding a message from its creator.


Finally, ‘aporia’ is the term for when a puzzle seems to be impossible. Ideally, the setting and trust in the puzzle’s creator should be sufficient to convince you that there really is a solution, that this isn’t a mistake or a trick. This can make the sensation particularly fascinating: you know a solution exists, you’ve perhaps even proved it doesn’t, so you know there must be some gap in your logic – you just don’t know what it is. For me this happened repeatedly as I played Snakebird (Steam/iOS/Android; referenced in Things April 2017) and is one of the reasons I enjoyed it so much.

Part 1 of the video series is here, and the YouTube description links you to parts 2 and 3:

Media I Recommend

A long time has passed since the last general Things round-up, which means there have been more chances for me to encounter some really excellent things that I recommend to Things readers.

Video game: The Outer Wilds

Available on PC (Steam), Playstation, Xbox

More than most other media, video games often have the problem that they are not at all accessible or engaging for someone not familiar with the form. So even though this is the strongest I’ve wanted to recommend something for a very long time, you do need to be comfortable navigating 3D environments to enjoy this game.

The Outer Wilds is a sci-fi time-loop mystery puzzle-game set in a kind of toy solar-system. That sounds cute, but I need to expand on that: it’s a really solid sci-fi, with the best-realised time-loop I’ve ever seen, a fantastically crafted mystery with brilliant diegetic puzzles set in an excellently designed toy solar-system that is obsessed with piquing and rewarding your curiosity and may make you think differently about death.

Referencing the puzzle terminology above, while it has its moments of Fiero, The Outer Wilds is particularly notable for being built around Eureka moments, with pleasingly diegetic hints to help you figure them out.

This may provide further context:
– Best Game of 2000-2009 according to me: Portal
– Best Game of 2010-2019 according to me: The Outer Wilds

So to be very clear, I recommend playing this game in the strongest possible terms if any of that sounds even remotely appealing to you.

Here’s a few notes that may help with your decision to play/finish it:

  • It takes ~15-25 hours to play
  • Note this wild game is called ‘The Outer Wilds’, and should not be confused with ‘The Outer Worlds’, a game that unfortunately came out around the same time
  • I recommend setting aside an hour for your first session
  • To get a bit cryptic, there are a few things that you may find annoying about it, but almost all of those things have ways to make them less annoying!
  • I personally recommend buying the base edition and then buying the DLC if you want more, rather than diving straight in to the complete ‘archaeologist edition’
  • There are moments late on that may test your patience, especially if you don’t work out some of the ways to make things less annoying – I was personally so invested I didn’t mind these at all, but I can appreciate that your mileage may vary. Still, if you enjoy it half as much as I did it will be well worth your time.

TV Series: Russian Doll

Natasha Lyonne in Russian Doll

As it happens, Russian Doll also involves a time loop, but much more of a magical-realism one than the sci-fi of The Outer Wilds. Its most notable feature is Natasha Lyonne as the protagonist Nadia, who has an approach to life not often seen on screen: a woman who says ‘yes’ to most decisions, especially the inadvisable ones, and is remarkably driven and selfish – but still humane. This makes her a particularly excellent protagonist for the time loop situation she finds herself in and I was gripped by this series all the way to the end.

If this sounds appealing I recommend diving straight into it (it’s on Netflix), but if you need more convincing at the expense of slight spoiling, the trailer is here.

(There is now a second series in which she encounters a different magical-realist sci-fi situation, but I found her character a worse fit for it and I was not surprised and delighted in the same way. The first series can certainly stand alone.)

Film: Everything Everywhere All At Once

IMDb: 8.5/10. Rotten Tomatoes: 95%.

The above trailer looked very promising to me, and I sought out the very first screening I could; the amazing part is that I found the film entirely lives up to the trailer, even to the extent that each minute is almost exactly as intense. A mind-boggling experience that truly delivers on the idea of a multiverse (unlike other films I could mention), I found it so fascinating I saw it a second time at the cinema; I enjoyed it even more, and it has joined the ranks of my all-time favourite films.

At the time of writing you may even still be able to catch it in the cinema, which I strongly encourage you to do!

(If you’re interested, others on my ‘all-time favourites’ list include Speed Racer, Scott Pilgrim vs. the World, Inception, Eternal Sunshine of the Spotless Mind, Synecdoche New York, and Lilo and Stitch)

Song: Dan Deacon – Change Your Life

It seems music is more personal than other art forms, and I feel as if the more a particular piece speaks to someone, the less likely it is to work for most other people. With that in mind, I don’t expect many to find Dan Deacon (referenced a few times in past Things) particularly appealing, but if you only try one of his songs, I recommend ‘Change Your Life’ which really captures the frenetic optimism he achieves, and which is what I find most appealing:

Enormous ever-evolving IP: Star Wars

Since I last wrote about it (in September 2018!), a lot has happened in Star Wars, and as you might expect I have a lot of opinions about it. But that will have to be an entire Things in its own right. So, you can look forward to that. Or not.

Transmission ends


Things February 2018: YouTube series, Poor estimation, The Patriarchy

The patriarchal explanation

The Octonauts is a charming kids TV series, based on the picture books of the same name (referenced in the Things Kids Special in 2013). It features eight anthropomorphised animals on a Star-Trek-style mission to explore the ocean and encounter the Interesting Sea Creature of the Week.

I noticed that the same three (male) characters go on exploratory missions almost every episode, even when other (female) characters would be better qualified for the task at hand, which seemed odd for an otherwise progressive kids TV series. Looking for people who agreed with me on the internet, as you do, I stumbled upon this excellent mumsnet exchange:

Bumperlicious:  Why are the girls relegated to mere sidekicks & not even mentioned in the opening titles? Is because they’re girls or because they’re both [foreign]?

TunipTheHollowVegemalLantern:  It is because of the patriarchy.

This turns out to be a highly versatile response which you can use to answer many questions about modern culture! It also has the benefit of sounding like a joke, while also frequently being accurate.

YouTube series of note

The Vox ‘Earworm‘ series by Estelle Caswell digs into various aspects of music, including a few really interesting pieces on long-term musical trends in the US pop charts. I particularly enjoyed the episodes on the triplet flow in rap, the fade-out, and especially repetition:

The ‘Fictional Fight Commentary‘ series does exactly what you hope: pitch-perfect delivery from Auralnauts creators Craven and Zak commenting on various famous fight scenes from movies. My personal favourite is the Revenge of the Sith Anakin/Obi-wan fight, but the Batman vs. Superman one is also great.

(Incidentally, Auralnauts had the dubious honour of having a video in which they removed the original music track get flagged as copyright infringing – for using the music track they removed, by the copyright holders of the song that was no longer present. Also incidentally, they made the generic film trailer I put in the last Things).

Every Frame A Painting was an incredible series of videos about movies and I’m amazed I haven’t featured it in Things before. It’s narrated by Tony Zhou, jointly written and edited with Taylor Ramos – although this latter part was only revealed in the post-mortem, the reasons explained in more detail by Taylor herself here, but more tersely one can say It Is Because Of The Patriarchy.

Aiming for quality over quantity, all 28 EFAP videos are brilliant, but as you have to start somewhere, I particularly recommend Jackie Chan – How to Do Action Comedy, The Spielberg Oner, and The Marvel Symphonic Universe.

Terrorism using the media as a megaphone

As I understand it, the aim of a regular terrorist is to create terror disproportionate to the amount of power they actually have. So perhaps the media response to these incidents should be a little more tempered, since that attention is exactly how terrorists gain a disproportionate response. I’m not sure there’s any good way to balance that though, especially with social media doing much of the amplification.

Still, I found it interesting that this quite wide-ranging study found a correlation between media coverage and subsequent violent incidents.

 “one additional New York Times article about an attack in a particular country increased the number of ensuing attacks in the same country by between 11% and 15%”

Poor estimation as a feature, not a bug

When doing indoor rock-climbing, or during my brief dalliance with extremely amateur parkour, I noticed that I and others tend to underestimate what we can physically achieve. We could jump further than we thought; we could stretch to reach a hand-hold that seemed too far; we could gain greater lift by running and kicking off a wall than we expected, and so on.

Now, I know that “evolutionary psychology” theories are usually untestable and often useless, simply servicing to reinforce one’s existing prejudices. Still, it’s easy to imagine that this physical capability bias (which I haven’t found named, but presumably must be) would work as a survival trait: creatures that overestimated their ability to achieve physical feats would presumably be at a mortal disadvantage in the long run.

This led me to wonder about another tremendously strong bias we have that runs the other way: the Planning Fallacy, best summed up by Hofstadter’s Law:

It always takes longer than you expect, even when you take into account Hofstadter’s Law. — Douglas Hofstadter

Superficially this bias seems like it should be a disadvantage. However, since many humans can benefit from one human’s work, perhaps we collectively benefit, as people work on things (and eventually, sometimes, complete them) that they wouldn’t have tried if they had correctly anticipated how long it would take, to society’s benefit.

It’s a trivial example, but I naively thought I could tell a particular story in a comic at one page a week in two years; it ended up taking more like seven. I probably wouldn’t have started it if I had realised that, but I’m glad that I did it, and now people – or at least, fans of two particular video games from 1999 – benefit from it.


When I first read about Bitcoin, I figured it was a cute idea but it didn’t scale, so couldn’t really work as an alternative global currency. Some of the scaling issues are now becoming apparent. As is so often the case, Charlie Stross has an insightful take on the matter.

Unendurable Line

I found this video quite satisfying:

Marie Kondo

In an age of abundance, we now routinely run into the problem of having too many things, and things like loss aversion make it difficult for us to deal with the issue. This scene from Labyrinth (1986) makes a lot more sense to me now than it did as a child:

Clare introduced me to a comic that summarised the ‘KonMari’ method (by Marie Kondo) of dealing with clutter. It isn’t really one method so much as a series of ideas and principles to apply to the task of decluttering. I think it’s particularly effective because it intuitively gets at some of the biases that make this a difficult problem. Here’s a few choice principles:

  • Don’t sort things by location, sort by type. Get all your (say) books in one place, then figure out what to do with them.
  • Don’t think about ‘what can I throw out’. Think about ‘what should I keep’. Only keep things that ‘spark joy’.
  • Don’t buy more storage units. This is sort-of tidying up, but isn’t decluttering. It’s just a way of getting the clutter out of your sight.
  • The correct time to read a book is when it’s you just acquired it. If you have many books you have not read for years, let them go. When it’s time to read one, you can easily get it again.

Most deeply, Marie Kondo observes that a lot of useless clutter takes us out of the present moment: we over-attach to the past with belongings that are no longer useful to us, and we over-attach to a theoretical future version of ourselves by keeping things we aspire to use but never will.

This echoes a message from the amazing show The Encounter (recommended to me by Tarim), in which a climactic insight is that “objects hold us stuck in time”.

There’s a KonMari method book or alternatively a comic version.

[This just in, thanks to Anisa – there’s going to be a Marie Kondo Netflix series – T.M. 15/2/18]

Emergent Stop Motion

Like a more accessible version of Thomas Sauvin / Lei Lei’s short film Recycled, Oliver KMIA’s ‘Instravel’ compiles recurring patterns in Instagram holiday photos to create stop-motion-style animation:

– Transmission finally ends


Things 124: Puzzles & Polaroids, Bond is now Bourne, Cooking Tips

OctopusFruitbat Game Write-upPuzzles & Polaroids at the British Museum
Clare and I were asked by Stubble & Glasses (who I also happen to work for) to design and run a company game event in a professional capacity, because some of them had enjoyed some of our earlier game events. So we formed a company called OctopusFruitbat, came up with something that combined puzzle-solving with creative instant-photo-taking, and it went a bit like this. If you’re ever looking for someone to come up with a similar event, of if you want advice on doing something yourself, please do get in touch!

Also don’t forget, this Friday I’ll be running Competitive Sandwich Making (which last year went like this) as part of the amazing all-weekend festival of gaming that is Hide&Seek’s Weekender at the Southbank Centre.

I’ve been nurturing the idea that films that follow the “They Made Him, Then Tried To Kill Him, Now He Must Fight Them” storyline are rising to such prominence that it must be some kind of Hero’s Journey for the modern age (I’m thinking Bourne, Hitman, Grosse Pointe Blank, Kill Bill, Blade, Ultraviolet…). The closest I could find on TV Tropes was Contract on the Hitman, which doesn’t quite nail it.

On metafilter, wuwei draws it out more explicitly by contrasting James Bond with Jason Bourne:

Who is our generation’s James Bond? Jason Bourne. He can’t trust his employer, who demanded ultimate loyalty and gave nothing in return. […] Bourne survives as a result of his high priced, specialized education. He can do things few people can do […] and like the modern, (sub)urban professional, Bourne had to mortgage his entire future to get that education. They took everything he had, and promised that if he gave himself up to the System, in return the System would take care of him. It turned out to be a lie.

(You can read the post in full here).

Is there any evidence that there really has been such a transition, that corporations are now violating the social contract in some way that they weren’t before? The three charts in this article do seem to actually endorse this idea – Corporate Profits Just Hit An All-Time High, Wages Just Hit An All-Time Low.

Here are some pictures with captions that have some amazing food-preparation tips, for example:

Previous PuzzleThe Shrinking Empires
Last time I asked why Empires seemed to be getting geographically smaller. I’ve actually asked this question when interviewing analysts, and get two kinds of answers.

The most common answer is that population density is increasing, and apparently human political power tends to stabilise around the 10m-50m range. For example, the Roman Empire was pretty big, but probably only covered ~60 million or so people, ten times fewer than those living in the same geographical area today (according to Citation Needed, but hey, it sounds about right).

A more interesting idea is that it has something to do with technology and inequality. I once heard it said that technology is not politically neutral – for example, Nuclear Power requires greater centralisation of government power than, say Wind Power – and I find this an appealing idea. Perhaps, for example, improved forms of communication give greater power to the people, who are then better able to resist tyrants with aspirations of empire-building through war.

But the more I dig into this, the more it starts to look like post-rationalisation, because I can imagine giving examples to prove the opposite. If everyone can manufacture guns cheaply, is it easier to terrify your populace with asymmetric power you can give your enforcers, or is “a well armed population the best defense against dictatorship”? If you improve transport, is it easier to avoid conscription, or easier to wage war? If you combine Moore’s law with the internet to create continuous public surveillance, do you end up creating a single global culture with no crime, or do you permanently enforce the power structures that exist at the point of implementation? Well, that’s a question for another day.

PuzzleGoogle Correlations
Google Correlate lets you find closely correlating Google search term trends, which sometimes gives silly results by coincidence, and sometimes reveals something very interesting. The question is, how many of these correlations can you explain?


Things 121: Kitten Cam, Chain World, Hand-Waving Explanations

Tim Game Thing
If you ever watched Knightmare and wanted to be the kid in the vision-restricting helmet asking “Where Am I?”, or if you wanted to be the team responding “You’re In A Room” and then frantically shouting “Sidestep left”, or perhaps if you wanted to come up with your own ideas for dungeons for those players to explore, then I recommend you come along to Hide & Seek’s free Sandpit event this Friday 25th May (6.30pm-10pm at the Royal Festival Hall near Waterloo), where I’ll be running a game that gives you an opportunity to do exactly those things.

There will also be a lot of other games going on, which sound pretty amazing, so read all about it, and come along.

VideoCat and Kitten webcams
Which is more compelling: a high resolution webcam of some kittens, or low-resolution webcams you can move around and remotely operate cat toys through (if you’re prepared to use Internet Explorer and Silverlight)?

It turns out that even though the latter sounds superficially cooler and more engaging, it’s nowhere near as good as high-resolution kittens.

LinkJason Rohrer’s Chain World
Jason Rohrer has made some very interesting games (I particularly like Sleep Is Death (or at least the idea of it – I haven’t played it yet), Inside a Star-filled Sky, and Passage is apparently pretty amazing (and very short) if you don’t play it stupidly the first time like I did), so it’s not particularly surprising that he came up with the winning concept in a competition to pitch an idea for a game that in some way represented the abstract idea of religion, and which, when actually released into the world, generated some pretty fascinating results.

Heavily Caveated Film Recommendation
The Green Hornet is a Seth Rogen action/comedy vehicle directed by Michel Gondry. If you analyse that sentence you may realise that this was recipe for disaster. With Seth Rogen starring and also writing (with long-time friend and collaborator Evan Goldberg), there was inevitably going to be some creative conflict with a director as idiosyncratic and driven as Michel Gondry (the man behind some of my favourite music videos).

Trailer: (2’06” is the moment that screams “Gondry!”)

Refreshingly, listening to the commentary makes this very clear, as Gondry, Goldberg and Rogen frequently reference the arguments behind almost every good idea in the movie, and occasionally break into new arguments about them (“We had to convince you that this would work!” “No no no! This was my idea! You guys didn’t want to let me do it!”).

This conflict doesn’t particularly damage the movie, but I ended up wishing Gondry had had more things go his way, because all the really weirdly brilliant parts are unmistakeably his. For example, at one point, the bad guy wants to send a message out through his criminal network, and this is how Gondry represents it:

It’s also great to hear his enthusiasm for his own ideas in the commentary, as when this scene starts: “This is awesome, look! Two cameras from one camera! How… did… the hell it happen?!” (Answer: they got the actors to stay in position while shooting one part, then came back to them and picked up shooting from the same position, then forced the shot to match in digital post-production, which is fine, but just count the splits and think about actually pulling that off).

The real problems with this film (to save you the trouble, it’s a 6.0 on IMDb and 44% on Rotten Tomatoes) are I think twofold:

1) Goldberg and Rogen consciously chose to reverse many of the staples (some would say clichés) of a superhero movie, which is admirable, but as is so often said, you need to be very familiar with the “rules” of any art form if you want to break them and still have the thing work, so this sentence ends in just the way you would expect.

2) There’s a fine art to crafting an action movie in which the audience can root for the protagonists, even when they’re harming or even killing bad guys. I don’t know how you pull this off, and apparently neither did anyone making The Green Hornet, because these moments frequently feel wrong.

All of which was fascinating, and I don’t regret watching the movie at all.

Answer 2A and 2B – Temperature/Pressure Pairs
Do there exist a pair of points opposite one another on the earth with exactly the same temperature and pressure? I say yes, and here’s my hand-waving explanation for that.

Stage 1
Let T(x) and P(x) denote the temperature and pressure of point x.

Choose two opposite points, A and B (if T(A) = T(B), do this stage with pressure instead of temperature). Consider a path from A to B (blue arrow in Fig 1a) and the corresponding opposite path from B to A (red arrow in Fig 1a).

In Fig 1b, we chart the temperature of the opposite points as we move around these paths. Just as with the Joss Whedon ascent/descent puzzle, since each line of temperature is continuous, they must cross at least once, and this represents a point at which the temperature is equal on opposite sides of the globe (points C and D in Fig 1b).

Stage 2
We now imagine rotating the circlular path we’ve considered about points A and B (Fig 2a). As we do this, the lines of temperature move continuously (Fig 2b), but since they always start and end at points A and B, the end temperatures remain fixed, and so the point at which the lines cross also moves continuously.

This starts to trace out a line of equal-temperature-opposite-points on the globe (turquoise and purple lines in Fig 2a).

Stage 3
If we continue that rotation all the way around, we trace out a continuous wiggly line of equal-temperature-opposite-points around the globe (Fig 3a, purple and turquoise lines).

We now consider two opposite points on this line, C and D (Fig 3a). We consider the pressure as we trace out a path from C to D (purple line) and the corresponding opposite points on the path from D to C (turquoise line). We can now apply exactly the same argument as in Stage 1 – the two lines of pressure (Fig 3b) are continuous and so must cross at some point, X. At that point, both the temperature and pressure are equal on opposite sides of the globe.

While this is not a proof, it’s reassuring that the Borsuk-Ulam theorem does prove that the result is true, although that’s no garantee that this reasoning is completely sound.

Dom Camus supplied a similarly hand-waving explanation but with a much more audacious line of attack, so if you liked the above, you should definitely check it out.

Phil has a pretty elegant approach, although he didn’t sound that convinced by it. He points out that “On any closed curve (not just great circles), the same sort of argument [as in Fig 1 above] holds […] I claim you can always construct a closed curve containing all its own opposite points and along which the temperature is everywhere equal at opposite points [like the turquoise/purple path in Fig 3]. Otherwise there would be some closed curve containing all its own opposite points, along which the temperature was never equal at opposite points, a contradiction.” You can then repeat Step 3, as above, to reach the same conclusion.

Puzzle Level 3The Crumpled Map
The final level in this series (suggested to me by Tarim) is one I don’t have any kind of hand-waving proof of, but I would love to see one.

Imagine you are on a lovely desert island with a selection of your favourite music, a pencil and paper, and an uncannily perfect map of the island. Actually, only the map is important.

If you lay the map on the ground and align it perfectly with the island itself, it’s quite easy to visualise that there must be a single point on that map which corresponds precisely to the bit of the island directly below it.

Now imagine folding and crumpling the map in any kind of way (but not tearing it) and throwing it anywhere on the ground. Wherever it happens to land, will there still be a point on the map that exactly corresponds to the place on the island directly below it?