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Does 2+2=4?

While scrolling through my twitter feed I came across a series of threads all essentially asking the same thing: what is the meaning of the phrase 2+2=5?

If you’ve read Orwell’s 1984, you’ll know this as the dogmatic statement used by the Party to brainwash Smith into mental submission via doublethink. If you haven’t read 1984… I don’t know what you’re doing with your life go read it.
The main take-away, though, is that this phrase has very strong political undertones of anti-intellectualism and governments trying to politicize truth and deny reality to “control the masses”.

Nazi theory, indeed, specifically denies that such a thing as “the truth” exists. There is, for instance, no such thing as “Science”. There is only “German Science”, “Jewish Science”, etc. The implied objective of this line of thought is a nightmare world in which the Leader, or some ruling clique, controls not only the future, but the past. If the Leader says of such and such an event, “It never happened” — well, it never happened. If he says that “two and two are five” — well, two and two are five.

George Orwell, “Looking Back on the Spanish War”

So this is going to be a mess to tackle, as you may already be able to tell. Let’s dive into it.

These are the types of tweets and posts that had people concerned. It started around summer but even now there are conversations and threads continuing.

The main premise of these kinds of posts is that “2+2=4” is a truism only in “white” culture, since those symbols and mathematical concepts were standardized by white societies. For example, we use base 10 but Babylonian societies used base 60 for their number system.

I think there’s a miscommunication taking place here on both sides. On the one hand, the guy who said “math is not universal” probably didn’t intend to say that the way he did. What I THINK he meant was “the mathematical symbols and constructs we use aren’t necessary universal”. And to that, I agree. Of course for someone working modulo 3, the expression “2+2 = 1” might as well be true.

The key distinction here is that there are different ways to represent the same abstract concept. Whether you represent 4 as “10” in base 3 or as “-1” in modulo 5, we are all talking about the same thing here: the 4th smallest positive integer.

But what of the argument that “2+2 = 5”?

I think the argument here enters a realm of philosophy more than pure mathematics at this point, since we have to consider how we construct mathematical knowledge in the first place. There are essentially two schools of thought: mathematical platonism and mathematical formalism.

Mathematical Platonism is the idea that the concepts of “2” and “4” and “addition” exist beyond the realm of human though. That is, if there were no humans or intelligent life there to write it down or imagine it, the concept of “2” and “4” would still exist. This goes not just for basic numbers and operations but even for the more advanced things, like perfect circles, straight lines, a mathematical limit and so on. This is probably the view that most people hold about what mathematics is.

In this scenario, we can say without a shadow of doubt that “2+2=4” and it always will be. Because the integers 2 and 4 are said to exist in a place outside of our brains.

Mathematical formalism, on the other hand, is essentially the idea that math exists only insofar as we dream it up. Whereas mathematical Platonism says that the concept of the derivative always existed in some abstract space and humans merely ‘discovered’ it, formalism tells us that the differentiation did not exist in our universe before Newton and Leibniz thought it up.

One of the interesting ideas brought up by the conflict between these two schools of thought is whether unsolved problems are “True” in some sense. Most of modern math is built from 9 axioms known as the ZFC. Every theorem is practically built FROM these base axioms using pure logic. However, we know some problems, such as the Continuum Hypothesis, are undecidable from these axioms, meaning it cannot be proved or disproved from those 9 statements alone. Now, some Platonists might argue that there exists an abstract universe of sets where the hypothesis is determinately true or false, and our axioms simply don’t reflect that well enough. A formalist would say that the hypothesis is simply undecidable and remains that way, and we can either add an axiom to make it true or false, but we hold that power.

The question of whether “2+2 = 4”, then, is really up to us to decide. If you are a mathematical Platonist, the statement is simply true. Given the existence of “2”, “4” and the addition operation in some higher space, obviously 2+2 is equal to 4. However, if you are more of a formalist, you might be of the opinion that “it depends”. Depends on the axioms we use and the system of arithmetic we are working in. Sometimes it may equal 4, sometimes it may equal 5, other times it may equal 0.


Practically everyone who’s aspired to become a scientist or has delved into scientific research of any kind is aware of the Nobel Prize. It is widely deemed the cynosure of accomplishment in a field, and is awarded to those who make massive breakthroughs in cutting-edge science.

The Nobel Prize

Of greater interest to me personally, however, is the Ig Nobel Prize.

A pun based on the word “ignoble” this prize recognizes and honors the unusual, the niche, the surprising and quite often the outlandish achievements in science. The following is the brief list of my favorite winners, and why I love them.

1. Pizza Protection

The 2019 Medicine Prize winner was Silvano Gallus, for his numerous papers about the effect of pizza consumption on the risks of cancer and death from disease. In my book, anyone who gives a solid scientific reason for why we should eat pizza is automatically one of the greatest human beings ever.

Silvano Gallus himself, smiling gleefully in front of what I am sure is a delicious authentic Italian pizza.

2. Miscalculations

The 2011 Mathematics Prize was awarded to a number of different people from different time periods. A couple of example include: Dorothy Martin, who predicted the world would end in 1954; Lee Rang Jim, who predicted the world would end in 1992, and Harold Camping, who predicted the world would end in 1964 and then later he changed his prediction to 2011. These aren’t all of the winners, but I’m sure you can see what the others’ achievements looked like.

I thought this image seeemed fitting

To quote the givers of the prize, they were recognized for “teaching the world to be careful when making mathematical assumptions and calculations.”

3. Pitch Drop

This award isn’t just to an interesting result, it is one of my personal all-time favorite experiments. The 2005 Physics Award went to a pair of researchers in Queensland University for the Pitch Drop Experiment, which was started in 1927. Essentially, it’s a glob of black tar that is being dropped through a funnel. And it’s still going

Nobody has ever been in the room to observe a drop fall.

The experiment averages one drop every 9~10 years. It fascinatingly blurs the line between solid and liquid, having a viscosity 100 billion times that of water. You could observe this for hours, days, weeks, even and see no discernible change in the structure of the pitch. Slowly but surely, however, it flows.

4. Inflated Numbers

So now Zimbabwe is mostly getting by using the US dollar, but remember the time when their currency was so absurdly inflated it jumped up into the trillions? The 2009 Mathematics Prize went to the Governor of Zimbabwe’s Reserve Bank, for “giving people a simple, everyday way to cope with a wide range of numbers.”

Roughly equivalent to 130 USD at July 2008.

Many developing countries target inflation to be around 5-10%, and anything above 15% is serious cause for concern in any economy. In summer of 2008 Zimbabwe’s inflation was an estimated 250 MILLION percent. That’s 250,000,000%. This, kids, is why learning basic economics is important, especially if you’re in charge of an entire country’s money supply.

5. Insensitive Incentive

This is probably the funniest one yet. The 2015 Economics Prize was awarded to the entirety of the Bangkok Metropolitan Police. The reason?

Offering police officers extra cash if they refused to take bribes.

I’ll just let you unpack that statement yourself.

6. How much wood would a woodpecker peck?

The 2006 Prize in Ornithology went to Ivan Schwab from UC Davis. Him and his partners conducted research into the headaches that woodpeckers may or may not receive from constantly, well… pecking wood.

The answer has to do with a small brain and a dense cranium that prevents it from shock and therefore headaches!

Mostly this one appealed to me because I like the idea of a bird-lover caring enough to check if a bird’s head hurt after a long day of knocking on wood. Really shows you the lengths of human compassion.

The Importance of Nonsense

The motto of the Ig Nobel Prize is to honor “achievements that first make people laugh, then make people think. They’re a fun, goofy event that acknowledges fun, goofy projects and occasionally blunders in the name of science. But even though these may seem unimportant, sometimes it is the pursuit of the trivial and nonsensical that lead to the biggest breakthroughs. After all, Fleming discovered Penicillin because he was too lazy to keep a clean lab.

Similarly, 2006 Ig Nobel Prize winner was conducting experiments on mosquitoes to see how they responded to different cheeses. In the pursuit of this seemingly inconsequential endeavor, he found limburger cheese and human feet are practically indistinguishable to malaria mosquitoes.
Now, this result is being used and cheese traps are placed in strategic locations across Africa to combat malaria.

So go out there and experiment with something stupid! You never know where it may lead…

Trolley Problems

1. The Classic Problem

A trolley is heading down a track with 5 people tied down to it. There is an alternate track it can go down with only 1 person tied down to it. You’re standing next to a lever which can redirect the trolley to the track with only one person. Do you pull the lever and make the switch, killing one person to save 5?

My answer: Of course you do. People who say don’t pull the lever are afraid that you pulling the lever will put that person’s blood on your hands, whereas you doing nothing means the death of the 5 people wasn’t necessarily your fault. But for me, you’re still making the choice to do nothing when you have the ability to pull the lever, so it’s just as bad as pulling the lever to kill 5 people instead of 1. So the obvious answer is to pull the lever.

Now let’s get onto some interesting ones. I promise they won’t be the classic alternate versions you’re used to.

2. Theseus’ Trolley

Same setup as before, except now there’s about 10,000km between where the track splits and where the second man is tied down. If, during the long journey, a team of engineers on the trolley replace each part bit by bit – replace a plank of wood here, a nail there – until every single original piece has been replaced with a new one, is the trolley that kills the man the same one that you diverted? 

The key here is that the trolley is moving as a single entity the entire time and any given replacement at one time is only a small part. If you replace a car’s tire, for example, you can’t say it’s an entirely new car. But what if you replace the windshield wipers next month? Is it a new car now? What about if you replace the seats inside with leather? Taking one replacement at a time, there seems to be no change in the entity called ‘the trolley’ at all. However, over time, once every single original piece has been replaced, is it still the original

3. Driverless Trolley

The trolley system has been privatized and monopolized by the Edison corporation and all but automated away using AI. You drive an automated trolley down a track and see a track straight ahead that has a person on it.

Very energy-efficient

The trolley is programmed to brake when it detects an obstacle and so you sit back and decide to let the trolley stop itself. Too late, you realise the detection system hasn’t worked, and you go to slam the brakes, but the damage is done. The trolley runs over and kills the person. Who is responsible for his death?

Should the company be held liable for a detection software that didn’t work in this one circumstance? Or are you to blame for not stopping the trolley when you had the chance to? Often these trolley problems are criticized for not actually being applicable or relevant in the real world, and thus the conclusions are not based on anything tangible. This one, however, will only become increasingly pertinent with the rise of automated vehicles, and how we as a society choose to answer this question will have very real legal implications.

4. The Riemann Trolley

A trolley is heading down a track with a countably infinite number of people on it. If it continues down that path, it would literally kill infinitely many people. 

There is another track, however. For every nth individual on the first track, there are n individuals tied together on the second track. In other words, you have 1 + 2 + 3 + 4+… people on the second track.

According to the analytical continuation of the Riemann Zeta Function, this expression is equivalent to -(1/12). Do you trust this track to continue down the path of infinitely many people or do you pull the lever to revive one twelfth of a person? 

If you don’t know about the this expression or the Riemann Hypothesis at all, 3blue1brown has an AMAZING video on this topic which, if you take your time to watch through and understand, is immensely rewarding even for mathematical laymen.

5. The Trolleyologist

A trolley is headed down an empty track. However, you have the option to divert the trolley to another track where a philosopher who constantly asks you trolley questions is tied down. How fast do you pull the lever?

Prometheus and Divine Fire

And in a single word to sum the whole-
All manner of arts men from Prometheus learned

– Aeschylus, Prometheus Bound

This is the first comic I ever made, in it’s terribly pixelated glory:

Clearly I made good use of all the available space

The Greek myths and stories have always held a special place in my heart, ever since I was but a young 9-year-old reading about the adventures of the demigod Perseus Jackson. 5 years after finishing the much-treasured series, I retain that fascination with Greek mythology to this today, and one of my favorite stories is that of Prometheus. 

In case you’ve been chained under a rock and have never heard of this story, this video provides a quick animation of the tale. Short version is, Prometheus steals fire from the gods and gives it to the humans. Zeus was very much unhappy with this, and so he sentenced Prometheus to an eternity of torment chained to a rock. It’s a fun (the Greeks had a different definition of fun) story about the origin of fire that you tell your child before bedtime, and all’s well.

Except I don’t think that’s all there is to it. 

The Ancient Greeks were a society of poets and philosophers. In fact, the words themselves derive from the Greek language (“poet” comes from “poiētḗs” – inventor; “philosophy” comes from “phil” – love and “sophos” – wisdom). Their mythology wasn’t simply an assortment of bedtime stories, nor was it a set of doctrines to live by. Instead, many of their myths were attempts to personify and characterize the forces that governed the world around them.

It is not so surprising, then, that the Greek Gods and Titans were so… human. They weren’t perfect, omnipotent beings like Yahweh of the Abrahamic religions, nor were they abstract spirits or ideas like those present in Aboriginal myths*. Instead, they were embodiments the best of us and the worst of us. They were powerful and intelligent and loving and caring but they could also be zealous and cold and ruthless and tempestuous and prideful and petty. 

*this isn’t to say the Aboriginal and Abrahamic myths aren’t interesting in their own ways, of course.

The other half is attractive young men

And so with this in mind, Prometheus’ myth takes on a new meaning. The story tells us that humanity began to prosper and grow bountifully after receiving fire stolen from the Gods. It allowed us to transform from docile beings of dirt and clay to people of passion, ambition and aspirations. The “fire” then can now be seen as so much more than a flame; it’s a representation for our innate drives that compel us to think and feel and love and hate. 

Here the myth becomes less a tale about the origin of fire and more of a declaration by the Greeks that we hold this Godly flame. In other words, if there is anything that can be called divine in our universe, it is us. If there are beings that can be called Gods out there, then we have the same drives and capacities and dreams as them. Prometheus is said to have fashioned humans in the image of the Gods, but the Greeks fashioned their Gods in the image of humans.

Creation of Man by Prometheus
Prometheus making man out of mud.

There’s something about this idea that is fascinating to me. The Ancient Greeks seemed to understand and capture what I feel many other societies did not: the search for the divine is misguided, for it is inside us. Not in some temple or altar or in some mountain in the sky. We are free to be masters of our own lives and captains of our own ships. 

The very same fire that fuels the Gods fuels us. That’s why we can create, craft, inspire and imagine and it’s also why we can be mean and cruel and rapacious and irascible. We take pride in humanity’s accomplishments, and we take fault for its atrocities. 

And there’s my take on one of my favorite myths! If anything I hope I’ve given you a motive to look more into the fascinating world that is Greek mythology, and feel a little better about yourself, knowing you have divine fire in you. To top it off, here’s a redraw of that very first comic:

Now that looks so much more presentable…

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Newcomb’s Paradox

I feel like math and english are often seen by people on two opposite ends on a spectrum of subjects. On one side you have the “strict” and “rigorous” math, where there is only “one right answer.” On the other hand, you have “free” English Literature, where any answer can be correct “as long as you can justify it.”

But this had always seemed just plain incorrect to me. Literature is NOT a wishy-washy dream subject where every answer can be right. If you think J.B Priestley’s An Inspector Calls is a pro-capitalism play, you’re wrong. If you think Steinbeck’s Of Mice and Men is a celebration of opportunity and freedom in America, you’re just plain wrong. There’s no way about it, it’s just wholly incorrect.

And math questions don’t always have only one right answer. Quadratics have 2 solutions, and trigonometric identities can have infinite solutions within an unbounded domain. And in some, strange, strange cases, we can have 2 solutions that are contradictory. The existence of either one should disprove the other, but both seem to be true. This is the crux at the heart of Newcomb’s Paradox.

Now, we consider a scenario. Two closed boxes lay in front of you, labelled A and B. Now, you can either take only Box B or both boxes A and B. You can’t choose only Box A because of… reasons. Anyways, overseeing your choice is a master psychologist and human reader. Let’s call this guy Kevin because… you know… reasons. He tries to predict your decision, and allocates money into the boxes accordingly before you actually make the choice.  

Now, before anything begins Box A contains a $1,000 cheque. Sweet, a brand new PC, I hear you think to yourself. But here’s the thing. If Kevin predicts you’ll take both Box A and B and guarantee yourself that sweet grand, he’ll leave Box B empty. However, if he predicts you’ll take only Box B, he’ll slip in a cool $1,000,000 cheque. Yeah, forget the brand new PC, you can go buy a brand new MANSION with that kind of money. And Kevin is pretty smart, so his predictions are right 90% of the time.

To recap: if you choose both boxes, you’re guaranteed $1,000 but only have a 10% chance of getting that extra $1,000,000 in Box B. If you choose only Box B you’ll have a 90% chance of getting a quick one million, but you’re missing out on that $1,000 in Box A. The table below nicely summarises the possibilities:

What do you do to maximise your winnings?

The correct answer is…non existent.

Ok, well looking at the table the answer may seem obvious. No matter what the Kevin predicts, your winnings are ALWAYS higher if you choose both boxes than if you don’t. After all, he makes his prediction and puts the money in the boxes before your choice. So whatever is in Box B, your earnings are greater if you choose Box A as well. No-brainer. Both boxes is CLEARLY the right choice. And you’re right!

Except… the thing is Kevin is 90% accurate. You taking both boxes means there’s only a 10% chance of Box B having a million bucks. But Box B would have a 90% chance of being filled with money if you don’t take Box A. Let’s use Expected Value to calculate our expected earnings. Now EV = k=0nakP(ak)

What that mess of symbols means is that you multiply the value of each outcome by the probability of that outcome occurring, and then add all of them up.

So if we choose both boxes: $1000 x 1 + $1,000,000 x 0.1 = $101,000

This is because the probability of getting $1000 in Box A is guaranteed, but we’re relying on the 10% chance that Kevin is wrong and predicted we’d choose only one box.

But if we choose one box: $1,000,000 x 0.9 = $900,000

Because there’s a 90% chance that Kevin predicted we’d choose just one box. And since the expected earnings for choosing just Box B are much higher than choosing both boxes, we choose Box B. No brainer. And you’re right!

Except…the dominant strategy clearly dictates that both boxes have more value than just one. And herein lies the crux of the Newcomb Paradox. Two different, mathematically valid lines of reasoning that result in contradictory conclusions. And down comes crumbling the facade of math’s indestructible logic and rigor. 

While it may seem to nullify the beauty that comes from the universal truths of math, I prefer to think it somewhat adds to it. To know that math isn’t just this black or white, yes or no, right or wrong. It can be mysterious, nuanced, paradoxical even. That, to me, is the true beauty of Newcomb’s Paradox.

Shuffling Cards

So last week I entered this programming competition and one of the questions was about permutations of letters. Essentially, there was some string of letters and we had to count how many permutations of that string were substrings of a larger string we were given.

The details don’t really matter, the main point is that at some point pretty much everyone in the competition thought about using the permutations function to solve it. Basically what it does is it takes some string as an input and outputs all of the permutations of that string. For example if you gave it ‘abc’, it would output: [‘abc’, ‘acb’, ‘bac’, ‘bca’, ‘cab’, ‘cba’].

Now this was all well and good for small 3 letter strings, but what a lot of us didn’t initially realise was the sheer number of permutations that came with even slightly longer strings. Even when you just increase the input to strings with 10 characters, the function outputs over 3 and a half MILLION strings. Somehow none of us realised how impossible using this function would be, and I think it comes down to the fact that humans have a pretty terrible idea of how fast factorials grow. So let’s talk about it using cards.

Alright, let’s suppose you’re shuffling a deck of cards. Every time you shuffle, the 52 cards fall in some arrangement. We can count the total number of arrangements possibly quite easily. For our first card, we have 52 choices. For the second, we have 51 (since we’ve already used one card), for the 3rd, 50, and so on until we reach the final card. The total product is 52 x 51 x 50 x… 3 x 2 x 1 = 52! (That’s 52 factorial, I’m not just really excited about the number 52) This is what we mean by the number permutations of the deck. This is also the number of strings that the ‘permutations’ function would’ve output if we had given it a 52-letter input. 

The number of possible permutations of a deck

Now, we can write 52! in standard form; it’s 8.06 x 10^67. Simple enough. But that doesn’t even come CLOSE to representing how massive that is. So let’s try and visualise it.

We’re gonna make some assumptions. Let’s say around the world, there are 5000 decks of cards being shuffled every second. Doesn’t seem like too unreasonable an estimate. Supposing every new shuffle made is an arrangement that’s never been made before, how long do you reckon it’ll take to get through all of them at that rate? A couple of years? A couple of centuries? Millennia? The lifetime of the UNIVERSE?

Not. Even. Close.

5000 shuffles a second means 1.61 x 10^64 seconds. That is a LONG time. By contrast, the age of the universe is only about 4.3 x 10^17 seconds. 

Let’s do a thought experiment in an attempt to visualise this length of time. Suppose you’re waiting until that momentous day when, at last, every single possible arrangement of 52 cards will have been done. What can you do to pass the time? I mean, apart from reading Astronomical Blunder Comics? 

Well, you could type out every single word in the Oxford English Dictionary, of course! But don’t worry about being a bad or slow typer. For our purposes, you’re just gonna have to type at the gentle pace of one word every billion years.

Yes, you read that correctly. One word. Every billion YEARS. The sun will swallow up the Earth in its entirety and collapse into a white dwarf long before you’ve finished your first page. And remember while all this is going on, you’ve still got 5000 new arrangements a SECOND being made.

Roughly equal to one word every billion years.

So, there’s about 600,000 words in the OED as of now, and that’s the copy we’ll use. After painstakingly typing out all 600,000 words at that snaillike rate, go take a trip to the Sahara Desert and remove a singular grain of sand (Let’s just pretend the Earth still exists after this time. Nothing else about this scenario is normal either). Delete all those words you spent all that time typing up, and restart from the beginning, at the same rate. When you’ve once again compiled the entirety of the OED, take out another grain of sand from the Sahara.  Repeat until the Sahara is no longer a desert, just a landscape of bedrock. That should be about 1.5 septillion grains of sand, which means 1.5 septillion copies of the OED, all typed at 1 word every billion years.

Now that you’ve reduced the Sahara to just large rocks and gravel, let’s help out the environment a bit. Let’s remove one kilogram of CO2 from the atmosphere (we’re also going to assume no-one is going to be polluting earth anymore because they’ll all be too busy waiting for all the card arrangements to be reached). So, re-fill the Sahara, delete all the words and start typing again. 

It’s a thought experiment, don’t read too deeply into it…

Finish the OED, then remove a grain of sand until you drain the Sahara, then remove a kilogram of CO2. Once ALL of the world’s CO2 has been removed and the atmosphere is completely clear of any carbon…

There will be 1.60 x 10^64 seconds left on the clock. We won’t have even gone through 1% of all the possible combinations. Only once we do that entire process 160 more times, will we have gone through every single permutation. Which means putting back all the carbon in the atmosphere, filling the sahara with sand, starting typing on a blank new document at the rate of 1 word every billion years, and repeat. One hundred and sixty times. 1.61 x 10^64 seconds is a LOOONG time.

This seems almost ridiculously false, because a deck of cards really isn’t all that much; it’s something you can fit in the palm of your hand. Yet the combinations that can be derived from it are barely comprehensible or countable from a human perspective. 

So just know that every time you rearrange your 52 cards, there’s a pretty good chance that it will be the first, and probably only, time that specific arrangement is made for all of human history. If you’re ever feeling unspecial and want to create a historic new moment that will likely never occur again…

Shuffle a deck of cards. 

Sex, Graph Theory, and why you’re taught math

Math, am I right? 

Calculus, trigonometry, geometry, algebra, statistics, proofs. Who needs all of that, and why is everyone forced to learn it at school? Except for mathematicians, of course. And perhaps the odd theoretical physicist or computer scientist. 

Well, ask the sociologists behind this book. More specifically, a claim they made based on surveys saying that men in the U.S tend to have more opposite-sex partners than women. By a LOT. Supposedly the average American male has 74% more female partners than females have male partners. Ugh, men and their promiscuous playboy ways. Typical.

Actually, not quite so. In fact, there’s no way the ratio can be that different, and perhaps if the sociologists knew a bit of math beforehand, they’d be able to dismiss the results of the surveys much more easily. Let’s go through it with a little bit of graph theory.

If you’ve ever done one of those connect-the-dot puzzles, you’ve already indulged in what graph theory is all about. Formally, a graph is a pair G = (V, E) where V is a set of vertices and E ⊆ {(u,v) | u, v ∈ V}. But really, it’s all about dots and lines connecting them. 

ProTip if you want to make your child seem like a prodigy, just say this every time he does a connect-the-dots puzzle.

We call a dot on a graph a vertex and a line from one dot to another (or from one dot to itself!) an edge. So to visualise the population of sexually active men and women, let’s create two columns of vertices as below:

Note that the diagram doesn’t show exactly how many men and women there are.

Now, if a man and a woman are partners, let’s connect them with an edge (for now, we’ll just assume random, arbitrary partnerships and disregard same-sex relationships). Each vertex can have one or more edges connected to it.

Notice that every edge is connected to BOTH the men and women column

OK, so now we’ve visualised the scenario using graphs. Going back to our goal, we’re trying to find the average number of partners of the opposite sex for both men and women. Converting that into our graph theory lingo, that means finding the average number of edges connected to a vertex for each column. Well, to do that you just add up the number of edges connected to all the vertices in one column, and divide by the number of vertices. 

Let’s call the total number of men Vm and the total number of edges connected to the men column Em. Similarly, the total number of women will be Vw and the total number of edges connected to the women column will be Ew. So the average number of opposite partners for men is Em/Vm and for women it’s Ew/Vw.

So the ratio of average partners of men to average partners of women is (Em/Vm)/(Ew/Vw). But Em and Ew are the same. Obviously. A five-year-old looking at the graph could tell you that, since each edge is connected to one vertex from the men column, and one vertex from the women column. In simple terms, it takes two to tango. So the ratio of opposite partner relationships between men and women is simply the ratio of the NUMBER men to women: Vm/Vw

When you look at it this way, it seems ridiculous that men have 74% more opposite sex partners on average. If that were true, it would imply that there are 74% more sexually active men than women in the United States. And even if THAT were true, the number tells us nothing about the promiscuity of men or women at all, it’s just a ratio between the NUMBER of men and women present.

Now, it seems almost comical that the sociologists didn’t notice this simple fact, but that’s the issue: people largely underestimate the role of mathematics, in everything. Math isn’t just adding and multiplying numbers so you can calculate your bills or fill out your taxes. It’s a way to build rigorous arguments and present and analyse things in different ways. We aren’t taught math because the formulas will come in useful one day; most of them won’t. It’s necessary because it strengthens our logical thinking and reasoning abilities, something you need regardless of your interests or profession. 

So maybe don’t be so hard on math. And this may be a math nerd saying that so you can call me biased, but all you have to do to prove me wrong is build a rigorous argument using logic and base axioms, and we’ll agree to disagree.

‘The Office’ and High-School Stress

So, it’s been about one day since school started and I’ve already experienced the crushing pressure of high-school about 12 times! Because obviously if I don’t get all my work done I won’t get a good grade, which means I won’t get into a good college, which means I won’t get a good job, which means my life will be spent unsatisfied and unhappy. Right? Well, recently I’ve started thinking maybe not. And what prompted this shift in perspective, (asks no-one)? An office. But not just any office. THE Office.

In my humble but always correct opinion, The Office is one of the greatest sitcoms, if not THE greatest, to ever grace our television and computer screens. It’s ridiculously funny but not obnoxious about it with a laugh track(maybe I’ll make a future post about my hatred of laugh tracks), it has well-developed characters that grow and, most importantly, it taught me a lot about the stresses of high school.

One of my favorite things about The Office is that the premise is so seemingly monotone and drab it’s a wonder how anyone thought of making any TV show out of it, much less a timeless and legendary piece of comedy. It’s hard to imagine something more bland than the lives of a few workers in a regional paper company, but the way the show reveals the colorful, vibrant, hilarious lives that these average people lead is truly something to marvel at. 

Jim doesn’t just sell paper, he plays pranks at work and designs games for his coworkers and starts his own business. The humble receptionist Pam goes to art school, invents her own job, and paints murals for the building and the city. Dwight owns a beet farm and comes into work dressed as a Belsnickel. Each and every one of the people who work there is an interesting character that we grow invested in and become intrigued about, despite them being nothing but paper salesmen and accountants in a mid-sized distributor. 

Creator Greg Daniels used a visual metaphor to describe the atmosphere and feel he wanted the show to have. He described the show as a paved over concrete parking lot with one single flower peeking through a crack in the pavement. What I think he meant by that is that the show is a satire on this bleak office environment that the majority of Americans inhabit, but it still has this soul. That one flower coming through the crack symbolises the heart of the show, the potential for growth, the beautiful little things that go unnoticed in our busy lives. 

Fun fact: This shutterstock image is also visual metaphor for my social life.

In high-school it’s easy to feel like if you’re not doing “enough” (whatever that means) to be outstanding and a valedictorian that your life will eventually end up boring and meaningless, nothing but a grey, empty, flat parking lot. I used to think like that, anyways. But I think The Office showed me that wasn’t necessarily true. You could have the funniest, loveliest, most interesting life in the world even as a salesman at a regional paper company. All you have to do is slow down and smell the flowers in the cracks between the pavement.

So enjoy life. Take challenges one at a time and appreciate the little things that make it fascinating. Don’t make it hard for no reason. And as Michael Gary Scott would say, that’s what she said

The Pareto Principle

The world’s top 40 most populous countries (out of nearly 200) make up 80% of the human population. The most frequently used 18% of words account for over 80% of the content of any text. The richest 20% of the global population own about 80% of all the money in the world. What’s going on here?

This, in essence, is the Pareto Principle, often dubbed the 80-20 rule. It’s a theory that for any event, 20% of the causes are responsible for 80% of the effects. Although, the numbers don’t have to be 80-20, and they don’t have to add to 100. They can be 70-25 or 90-15, for example. The main idea is that not all causes are born equal, and that there is an uneven distribution on how much of an effect each cause has. Let’s look at some more ridiculous examples.

The most well known and researched example is Zipf’s Law. Now, specifically Zipf’s law states that in any given text, graphing the frequency each word appears in and the rank of that word in terms of frequency on a log-log scale should produce a straight 45 degree line

Source:, using data of their own website’s word distribution.

Now, this very much should seem surprising to you, unless you’re a robot. We’ve always thought of language and literature as the very opposite of math: it’s subjective, fluid, idiosyncratic. It seems wholly unbelievable that both Shakespeare’s Hamlet and Darwin’s Origin of Species – two works written by different authors at different time periods for entirely different purposes – both follow such a simple mathematical rule.

A log-log graph of word rank vs word frequency in Alice’s Adventures in Wonderland, Hamlet, David Crockett and The Origin of Species, all showing a linear trend for the majority of words.
Source: Ferrer-i-Cancho, Elvevag, 2010

Another big example: Economics. In a lot of cases, and I mean a LOT, Pareto’s principle applies. In the early 20th century Vilfredo Pareto, the Italian economist after which this phenomenon was named, showed that roughly 80% of Italy’s land was owned but just 20% of its people. In 2018, statistics showed that the top 20% of earners in the US payed between 80-90% of the total federal income tax collected. 

Let’s go back again to that statistic about the richest 20% of the world owning 80% of all global assets. Pretty standard example of what I’ve shown you so far. But what if we apply it again? The richest 20% of that 20% should own 80% of that 80%. In other words, the richest 4% should own 64% of the world’s wealth. Apply it again, and we get the richest 0.8% should own just over HALF of the all the money in the world. And Oxfam statistics for 2015 and Credit Suisse’s 2013 report BOTH confirm this, showing the top 1% owning over 50% of global wealth.

Source: Oxfam Statistics

Again, it really should blow your mind that this simple distribution should apply in so many cases, even when iterated. Economics is notorious for its over-generalization and abstraction of the complexity of markets, businesses and entire countries. Money is circulated and flows through a multitude of facets for numerous reasons by billions of people all over the globe simultaneously. Yet we can predict the distribution of the world’s wealth across multiple scales using but a simple power distribution. 

Now, it’s all well and good using the Pareto Principle to notice large trends in the world, but it can help you out a whole lot in your day-to-day life too. Look at it this way: 80% of your problems can be solved by changing 20% of the causes. 

Let’s take a classic example: trying to lose weight. It can be challenging and there’s no shortage of different advice from different people. But just remember that 80% of your calories will likely come from 20% of your food. Thus, just by identifying and cutting a tiny part of your food, you can dramatically reduce your caloric intake. You don’t have to halve your breakfast, lunch and dinner, you can just not eat that 200 calorie chocolate bar.

And businesses LOVE this principle. Several “rules of thumb” in business are based on this simple power law: 80% of your complaints and your income are from 20% of your customers.

And lastly, 20% of what you read and watch ends up taking 80% of your memory. I surely hope this post finds itself into that 20%.

Blunder #1

So, here I am. A (very) lost and confused teenager who has little to no idea what he’s doing, but he knows he likes science, jokes, and science jokes. I like to think of myself as a pretty funny guy, but the empirical evidence for it is dishearteningly low. So I decided to create a new outlet for my insuppressible desire to tell terrible science, math and literature jokes: a comic of stickmen. And now, I am resorting to write blog posts that likely no-one will ever read. 

Wow, I have no life.

Moving on from my depressing lack of social eminence, my comic has got more attention that I had initially anticipated. I mean, I was expecting like 5 people to follow me, and 4 of them would be my friends, and the other would be an instagram bot. But it would seem there are others that actually connect with my sense of humor so much they are willing to expend the Herculean effort of tapping twice and liking my photo, and then tapping once MORE and following me. I didn’t really need validation to continue this comic in the first place, but I can’t deny it feels good having some.

So, about the comic. When it comes to art, my 8th grade art teacher can attest to my startling lack of any artistic proficiency, hence why I use stick figures. If I’m trying to impress someone, I’ll probably make up something about “adopting a minimalist aesthetic to place emphasis on the idea rather than the artwork,” but in reality it’s just because I can’t draw. Then, of course, is the question of the astronauts.

For as long as I can remember, I have been obsessed with space and the cosmos, and since then it’s pretty much become part of my personality and aesthetic. That means listening to space-themed rock bands like Starset, having astronauts as my profile photos on every social media and of course, having astronaut helmets as the main motif of my comic. 

I’ve always held an enormous amount of respect for astronauts. They’re the pioneers of mankind’s exploration, daring to venture into frontiers unknown, unwavering in the face of the cosmos’ enormity and indifference. In a way that’s sort of what I’d like to be. Not on a scale as groundbreaking and trailblazing as them, but perhaps emulating some of those qualities in my own life. I’d like to discover things never discovered before, prove theorems never proved before, make jokes never made before. Of course, the path and journey won’t be easy, and I’m bound to mess up and fall down now and then. But I guess this is where I start. An innocuous, little stick-figure comic. My very first Astronomical Blunder.