Author Archives: georgeaitch

Happy Birthday! Another Maths Paradox To Test Your Mind

Continuing on our theme of head-breaking maths, I’d like to present another example of probability gone wrong.

For this experiment, we’re going to need to gather twenty two of your friends at random. Hold a party in your kitchen (or any way you can snugly fit twenty three people).

Well done Mr Popular, but did you know that there is half a chance that one of you shares a birthday?

This is the Birthday Paradox; that in a set of twenty three people selected randomly, there is a 50% chance that two will share a birthday. Seem unlikely? Let us explain.

Counting backwards

To demonstrate how this works, we’re going to need to scale things up for a second.

Instead of imagining your kitchen or classroom, think about a ballroom (those from The Shining or Titanic will do nicely) and instead of twenty three of your friends, I want you to invite three hundred and sixty six.



(That’s got to fit 367 people, surely?)

Once you’ve crammed all of these people into one room, you can be certain that two of them will share a birthday. How do we know this?

There are only 366 possible birthdays (including February 29th on a leap year), even if somehow you managed to find one person whose birthday was on a different day of the year for the first 366, if you were in the room, one of them would have to share a birthday with you (of course, in real life, some birthdays are more common than others).

In discrete mathematics, this is called the Pigeonhole principle. If you have x pigeons and y pigeonholes, where x > y, then at least two pigeons are going to have to buddy up.

This sounds pretty simple, but gets fairly odd when we introduce the idea of infinity and upwards, but that’s for another article.

Half A Chance

Going back to our party in the ballroom, there are either one of two probabilities; that someone in that room shares a birthday with another or nobody does. These are mutually exclusive; they cannot both be true.

If we calculate this probability, we see that it is in fact comprised of several more, one for each person. The probability that nobody in the room shares a birthday can be split down into the probability that Person A doesn’t share a birthday and then Person B and so on.

Probability isn’t as simple as just heads or tails, especially when we think about big numbers. Image courtesy of Flickr.

In probability, we can express mutually exclusive outcomes as:

Probability something happens + probability something doesn’t happen = 1

P + not P = 1 or P + ‘P = 1

In our paradox, either at least two people in that room share a birthday (B) or no-one does (‘B)

The probability that nobody shares a birthday then can be displayed as:

‘B = 366/366 x 365/366 x 364/365 …

And so on, for each of people in the room we are comparing to. Each comparison leads to a smaller and smaller value of ‘B (that nobody shares a birthday).

We know that the first person can have any birthday in the year without invalidating the probability that none of them do. The second may only have a birthday on all of those days minus the one which the first person has the birthday.

The next person has one less day and so on and so on. A simpler way of writing this is:

‘B = 366!/((366-n)!x366^n)

Where n is the number of people in the room. If the value is 367, then we may be certain that two people share a birthday (though at 70, the odds are 99.9%).

If the value is set at 23, the probability works out roughly as 0.5, which is half a chance.

You can work it out with different sizes here.


(Here’s a graphic representation from Cornell University)


The Birth of the Paradox

This paradox was featured as part of a maths puzzle in Scientific American in 1957. It had been separately been proposed by mathematicians Harold Davenport and Richard von Mises prior to this.

But why should we care about all of this? Why do we celebrate birthdays? As well as being a strange piece of probability, there is a real world application in hacking called the birthday attack.

It involves attacking the way data can be encrypted via a hash function – stealing data or passwords. If two people are sending data to each other, it will be encrypted with each at both ends able to translate the code via a hash function.

The birthday attack examines the way that two pieces of information may be encoded in the same way, much like two people sharing the same birthday.

The attack means all the hacker needs to do is find any matching pair before and after encryption rather than any single pair. This reduces the time taken to break into a system and is so much more important than it may seem at first!

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The opinions expressed in this article are those of George Aitch and Dr Janaway alone and may not represent those of their affiliates. Images courtesy of flickr.


  1. Su, Francis E., et al. “Pigeonhole Principle.” Math Fun Facts. <>

How Good Are You At Simple Maths? Maybe Not As Good As You Think.

Mathematics is not always as simple as two and two making four. Some particular problems are so counter-intuitive that they’ll make your head spin. And we aren’t talking about quantum physics or binomial equations, we are talking about simple fractions. Something you would have learned in Primary School. The most famous of these is the Monty Hall Problem, though there are plenty more. The mind does funny things when faced with problems of ‘chance’. and these brief thought experiments demonstrates how bizarre probability can be. So we will place a bet, if you get this wrong, give it a share.

A Good Problem

The Monty Hall problem refers to a US 1960’s game show called ‘Let’s Make A Deal‘ whose host was Monty Hall. The show involved three doors. In this case, a goat, another goat, and a roadster. When we talk about the statistics problem, we can imagine that we are contestants on ‘Let’s Make A Deal‘. There are three doors with three prizes, and we want to guess the best door for the best prize. We could win one of two goats or a roadster. We want the roadster, so we wish to maximise our chances of making the ‘right guess’.

Your chance here of picking the roadster, based on the knowledge you have, is 1/3. One door in three. Pretty simple. But let’s make it a little harder. The host, Monty Hall, shows us the prizes and shuffles them behind doors number 1, 2 and 3. After some thought, we decide to pick the second door. We’re about to open the door, when the host (who knows where everything is), opens the third door to reveal one of the goats. He then asks us “Do you want to change your mind?” (i.e do you want to stay with door number 2, or switch to door number 1.)

So what do you do? Do you stay or switch? It seems like a 50/50 chance of winning the roadster. But actually, it isn’t. There is a hidden factor. But, you can make a decision here which increases your chances of winning. Figured it out yet? This is the Monty Hall problem. Is it to our advantage to switch our selection? If you haven’t heard of the problem before, think about it while I explain the history to you. The puzzle was first posed in an American mathematical magazine, with the answer appearing later down the line. The answer, being not what you’d expect, caused a media furore against Marilyn Vos Savant, the woman who’d explained the solution.

Are you ready for the answer?

A Simple Paradox

Believe it or not, the best move is to switch doors. You know door 3 has a goat behind it, so door 1 and 2 are left. So you switch to door one, and it is opened to reveal the luxury red roadster. But why was it beneficial to change your mind here?

In the beginning, the roadster has a 1/3 chance of being behind each door.

1      2      3

1/3    1/3    1/3

You pick 2, which has equal probability of being any of the prizes that point, a third chance. The host, then reveals to you a goat.

1        2         3

1/2    1/3    Goat

At this point, your original choice remains one in three whereas the other door has half of chance of being the roadster. The answer lies with the assumptions of the problem. As the host is aware of where each prize is hidden, he cannot open a door to reveal a roadster (he would lose the show money! He has all of the information and by revealing the goat to you, he imparts some information which alters the balance of probability. As one mathematician points out: “Probabilities are expressions of our ignorance about the world, and new information can change the extent of our ignorance.” This is the simpler version of the proof, but requires a little more heavy work to explain it fully.

goat monty hall maths paradox

Do you want a Goat, or a Roadster? You can cuddle either, and ride both. But one bites. Image courtesy of Tamsin Cooper

So bear with us.

The Best Solution

They key to solving the problem is the following assumptions:

  1. At the start you have a 1/3 chance of getting the roadster and a 2/3 chance of getting a goat. So you picked a door (door 2 in this case.)
  2. The host opened a door (door 3 in this case,) which had a goat behind it.
  3. The host will not help you win, so his behaviour would directly effect your chances of winning the roadster.
  4. So at this point instead of the remaining doors having a 50:50 chance, you must now account for another factor, the probability that the host may force you to lose.
  5. By multiplying together the probabilities, you can get a clear answer of the ‘best’ door to pick.
  6. Remember,  you don’t know what is behind your door (2), or the other one (3.)

Take a moment to read that again. You aren’t dealing with just a 1/3 chance anymore, you are dealing with a multiplicity of uncertainty where you must account for a combination of the hosts effect on outcome (1/2) and your original choice of winning (1/3.) So let’s see what happens when you apply the math in each case. Each fraction is here is your chance of winning if you switch in each case;

1: You picked a goat (2/3 chance of doing so). The host reveals the remaining goat behind door (1/2 of doing so), he cannot reveal the car. (2/3 x 1/2 = 1/3) Your chance of winning is 1/3.

2: You picked a car (1/3 chance of doing so). The host reveals a goat behind door 3 (1/2 chance of doing so). (1/3 x 1/2 = 1/6) Your chance of winning is 1/6.

3: You picked a car (1/3 chance of doing so). The host reveals a goat behind door 2 (1/2 chance of doing so). (1/3 x 1/2 = 1/6) Your chance of winning is 1/6.

4: You picked a the other goat (2/3). The host reveals the remaining goat behind door 2 or 3 (1/2) again, he cannot reveal the car. (2/3 x 1/2 = 1/3.) Your chance of winning is 1/3.

So now we can add the probabilities together. If we talk options 1 and 4 (i.e you picked either goat,) your chance of winning by switching = 1/3 + 1/3 = 2/3.  If you picked a car, (options 2 and 3) then your chances of winning by switching are 1/6 +1/6 = 2/6 = 1/3.  So comparing the probabilities, if you pick a goat door, and the host doesn’t want you to win, then you best switch (2/3 chance of winning. If you picked the car door, and the host doesn’t want you to win, you best not switch (1 – 1/3 = 2/3.) Admittedly, and some of you may have spotted this already, if the host was not biased, or wanted you to win, the answer may be different. Let us know what you find in the comments below.

goat roadster monty hall problem paradox

Einstein may even have had trouble with this one.
Image courtesy of Tom Haex

If this still sounds strange to you, don’t worry; you’re not alone. After Vos Savant published her proof, many attacked her and claimed that she was wrong despite many simulations and proofs. Even the (arguably) greatest mathematician alive at that point Paul Erdős wasn’t convinced at first. This is an example of veredical paradox; that is, a situation or result which at first appears to be wrong but can be demonstrated to be true. How many of your friends do you feel would get it right first time?

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  • Getting your mind blown was just the beginning when you consider that we probably live in a simulation
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The opinions expressed in this article are those of Dr George Aitch and Dr Janaway alone and may not represent those of their affiliates. Images courtesy of flickr. Note from the Editor:  I had to write this out and work out the fractions myself before I was convinced (lights cigarette, stares blankly into the sea listening to soft circus music.)


In Reality Just A Dream? (You Will Need A Cup Of Coffee For This One!)

You can’t leave the park if you stay on the rides boy. Stop being a tourist and take a look behind the curtain.”

The idea that everything we know ‘is a lie’ and we have been dreaming all of this time pervades culture en masse; From Plato to the Wachowskis, the possibility that we are all collectively experiencing a simulated reality is a juicy subject for discussion. But there is something to this idea. If we are in a simulation, how would we know? How might we begin to prove this? Defining a hyperobject (or a hyper-hyperobject), such as reality itself, is difficult. We come up like the fish searching for water. It is everything to the fish,  so where do we even begin with being?

A Philosophical Dream

The human mind is not equipped to answer the big questions very well. In fact, our very logic is based on very restrictive parameters.  Our understanding of distances, time and flying things is limited to what we see day to day. This is why we are easily tricked by the massive or very small, our brains aren’t evolved to make sense of the information. Or indeed, there has been no demand to do so that restricts our survival as a species. And answering whether our Universe is in fact ‘real’ isn’t a question that would have vexed our ancestors, so its little wonder we have trouble with it. Today’s big questions confuse yesterdays brain.

Questioning the nature of reality is one of those big questions. Take optic illusions and hallucinations for example, or the auditory hysteresis as best demonstrated by ‘Laurel’ or ‘Yanny‘. We have a limited number of sensory cues which we can attach to our environment. When we try to cut corners, our brains attempt to fill the gap and make mistakes. Our brain will attempt to make sense of ambiguity by pushing previous experience on to it. VSauce has a great video explaining how and why this can happen, so take a look. So knowing this, where do we know where objective reality stops, and our own shortcuts begin? What is truly real outside of our own interpretation?

simulation reality descartes science physics philosophy dream

Rene Descartes – Philosopher and Pioneer 

This idea a, that reality is not ‘real’ is not so foreign to us as it may seem. The first consideration of this with which most people are familiar is perhaps the cogito ergo sum of René Descartes: ‘I think, therefore I am‘. This simple statement was the basic building block Descartes used to establish his metaphysical philosophy. He reasoned that, as we know the senses can be misleading, everything which he perceives may be the illusion of a clever and malign demon. If this is the case, he would have difficulty in establishing which percepts were real and which were not, as each one might be designed to fool him. Although this touches on the idea of a ‘false reality’, it appears to appeal to some higher power ‘tricking us.’

Although a powerful idea, it doesn’t answer the question objectively but actually throws another layer of faith on the issue.

Descartes’ response to this unusual problem was to throw the whole thing out; he only knew that he was thinking. Thus, Descartes knew that he existed but about the rest, he could not sure. This was a logical move, as he realized that objective realities would be consistent regardless of who perceived them, only the inferred reality (a very personal one,) would be his alone. Obviously, we can all infer the same when seeing an apple (and tend to, its red, hard, tasty,) so there is something consistent. But even then, the ‘essence’ of the object considered may be inferred differently by everyone, and you would never know quite how (i.e is my red your red?)

This was termed methodic doubt or Cartesian scepticism. The take-home message is that seeing is not believing. The extension of this, Solipsism, is the belief that you are the only aware rational agent (agent meaning one capable of observing and influencing.) From a simulation perspective, it means that you are the only ‘real’ person. Of course, our video games are populated by Non-Player Characters (Cortana in Halo, Navi in Zelda.) If we are in a simulation, it is more likely that you are not ‘real’. Why would a simulation be built for us alone?

Of course, this is a basis for a line of thought, not an encouragement to live your life in this way. People still look both ways before crossing the street. An NPC is not benevolent and doesn’t exist to help you by nature (i.e any character who attacks you in a game.) Solipsism, as understood by Karl Popper, is not a falsifiable hypothesis. Traditional scientific method seeks to disprove ideas via a null hypothesis (the chance that the association between X and Y is due to chance). Solipsism cannot meaningfully be disproved in this way (the death of the main agent ends the argument, one way or the other). This doesn’t mean that it isn’t true, but that solipsism is in the hands of philosophy over science.

Which is an uncomfortable position to be in. If you can’t objectively prove it, or at least reliably disprove it, nothing can be concluded. Popper himself is aware of this and forms the basis of his work.

We can approach this problem from the other direction; that is to say, by considering the ethics of simulation after the fact. As software becomes more advanced and hardware becomes more capable, our simulations or the possibility of any simulation becomes more sophisticated. The simulated minds we might develop could be more complex and we have every reason to suppose that we might pursue this. The map might start to look more and more like the territory.

science popper simulation descartes dream

Karl Popper – Father of Falsificationism and proponent of reasoning

These sim people (sims?) would have behavior like ours, they might even have thoughts like ours. At some point, they might become indistinguishable from us and there are ethical considerations to running this. We do not consider the ethics of running a sim, thus any advanced civilization is unlikely to do this either. The economist Robin Hanson recommended that anyone living in a simulation better be as entertaining as they can, otherwise they might get switched off. An uncomfortable thought. So if we are simply lines of code, it makes sense for that code to be useful. Although we can see that ‘bad humans’ (Hitler as a prime example,) seemed to operate for years before ‘termination.’

Clearly, either this isn’t true, or Hitler’s suicide was a programmed termination carried out as volitional. We couldn’t be certain either way. Popper once again becomes very relevant, as we have no way of proving any hypothesis of even this one act.

These sim people would be ‘p-zombies’ or philosophical zombies. A p-zombie is not a horror movie villain. They look like people (or sims) and we cannot tell them apart, even from their reactions. If you tickle them, they laugh and if you pinch them, they would cry. However, they do not feel that indescribable sensation (‘qualia’). At some point, surely this becomes indistinguishable also? A simulant human such as found in Blade Runner was virtually human, and Robin William’s Bicentennial Man was actually declared human as ‘he’ became ethically synonymous with his organic peers.

bicentennial man williams science dream plato descartes

Robin William’s Bicentennial Man achieved human status through consciousness.

So we have established a reasonable proposal that these simulations are possible (although not provable only within philosophy.) We have now a frame of understanding with which to appreciate this issue. Next, we must turn theory into practice. How do we find the proof?


A ‘Physical’ Dream

The best way to analyse the problem of our potential simulation is to look at how we would do it. We need to examine how we build simulations and models. What limits do we put on them and how does that map onto what we have observed in the universe? After all, we have built simulations to model economic or anthropological behavior and VR goggles encourage us to leap into cyberworlds, is it that unrealistic that these might become more sophisticated and take on lives of their own? And what would reassure us that we weren’t indeed sentient ‘code’? Are we virtual reality convinced of physicality because of that same programming?

This prospect is not that unrealistic according to Hans Moravec, an Austrian futurist. Eventually, a civilization of some sort or another will become highly technologically advanced. This civilisation will be able to mass produce self-contained virtual simulations. They might do this for entertainment purposes or to model certain situations, as we do. These widespread simulated realities may become so numerous that any thinking entity has a greater chance of being inside one than out. Simply put, if the code can perceive and experience, how would it know if it was real or code? And if most of the ‘entities’ in a given universe are code, statistically you are more likely to be one of them.

Nobel Prize-winning physicist, George Smoot, encourages us to examine the basic physical constants which govern the universe. In his opinion, the fact that our environment is quantized (‘fuzzy on a small scale’, think of it as pixellation) so that physics works differently on a large scale compared to a small one may be a way of saving space an computational power. This discrete-ness is our binary. Basically, the way the Universe works, the rules it plays by, aren’t there by chance. They are created by a programmer, and that the base levels of ‘reality’ such as quantum physics, are an example of this.

dream plato science simulation

In physics our universe is quantised. 10 points to Gryffindor if you get the joke

Its just data, and since the small doesn’t reliably approximate to the big (i.e no one has developed a Unified Theory of quantum vs classical physics, it might be because a programmer has made a subroutine to relate the two to save data.) He also points to entangled states as another ‘simulation memory’-saving device. Other people take the computer science element a little further and examine Planck lengths, absolute zero and the speed of light. These unbending limitations could also better enable such a simulation to run smoothly.

So what we know about writing code, the concessions we make for ‘functionality’ may be present in the Universe itself. This is disconcerting because it speaks of ‘design.’ And we can see it. Its like Halo’s Master Chief realizing that the loading screens are actually real.

Tying The Physical To The Philosophical – A Dream Becomes Real

Back to philosophy again with the anthropic principle; the idea that the universe is meant for conscious minds to inhabit and observe it. There are two variants to this idea: the weak and the strong. The weak anthropic principle posits that we are only able to observe our universe because of the presets producing its formation. If the big bang never happened, or the earth was too far away from the sun, our civilization would never have arisen. Thus it is easy to say ‘of course the universe was made for us’, if it wasn’t, we wouldn’t be here to make that observation. A million other universes with different laws of physics or other presets might exist, but we’d never know because we are unable to observe them.

The strong variant of this argument goes similarly. It states that the scale of time and place in the universe is such that life must arise within it somewhere. Given how many billions of years and how large it is, there is a strong probability that intelligent life will come about and begin asking questions. However, this is a circular argument, suggesting that the proof in the pudding is that since we can question, the universe exists for it to be so questioned. Once again we are visited by the idea of a simulation.

IYou can consider further what the anthropic principles might mean for our position in the grand scheme of things. At this point we might speculate that if simulations are powerful and advanced enough, we could have sims running simulations and circles within circles. I don’t want to linger on who or what would do this; that takes the frame of this discussion from the strange into solipsism and mental illness. But if we are to entertain the philosophical argument for simulation, and note that physics may give it strength, we are met with an uncomfortable ‘reality’.

Or at least, we may be programmed to.

What’s Next?

  • Even if you are in a simulation, it doesn’t matter because the universe is out to get you.
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The opinions expressed in this article are those of Dr George Aitch and Dr Janaway alone and may not represent those of their affiliates.  Article written by Dr Aitch and embellished and edited by Dr Janaway (But the vast majority goes to Dr Aitch!!) Images courtesy of flickr.


  1. Hyperobjects by Timothy Morton (2013) University of Minnesota Press
  2. The Meditations on First Philosophy by René Descartes (1641)
  3. Mind Children by Hans Moravec (1995) Harvard University Press
  4. You are a Simulation & Physics Can Prove It: George Smoot at TEDxSalford (watch here:
  6. Image of Rene Descartes
  7. Image of Karl Popper
  8. Image of Robin Williams
  9. Image of Halloween costume (Walter White.)

Voices from the Distant Stars – The Story of a Pulsar

In space no one can hear you scream, or can they? Actually, we hear a lot from space.  We just need the right set of ears. And for a long time we have listened to the stars, and every once in a while, we hear something that might just be something special. A regular pattern that suggests an alien intelligence, or shadows on a sun suggesting a giant structure. Here are some of the great space oddities that have given us pause to reconsider our place in the Universe. One such ‘oddity’ is a Pulsar.

Voices of Little Green Men

In 1967, astrophysicist Jocelyn Bell Burnell was astonished to observe regular radio pulses of 1.33 seconds in the Vulpecula constellation. The Cambridge researchers were so taken with the idea that this was some sort of interstellar beacon, a voice from distant stars, that they nicknamed the mysterious source LGM-1; “little green men 1”. In fact, the radio bursts they were receiving came from the first pulsar which was given the much more cumbersome designation ‘PSR B1919+21’. So what is a pulsar, and why do they sound like aliens?

Pulsars are neutron stars, superdense structures which may only be a few miles in diameter but pack enough mass to rival our own sun. As a result of this amount of matter existing in such a compact space, some pretty strange things start to happen. For a start, the entire star is rapidly spinning (sometimes at hundreds of times per second). This combined with its two beams of energy produces its pulse as the spinning beams move in and out of view, like a lighthouse. And we see this information as regular ‘pulses’, enough to trick us into thinking a being is looking to talk to us. But how did we discover them?

Pulsars were discovered through interplanetary scintillation (IPS), a technique in radio astronomy, which was implemented in the sixties to scan the skies for quasars: large interstellar objects emitting vast amounts of energy across a variety of frequencies. As these signals pass through space, they encounter obstacles and diffract. This effect produces a characteristic twinkle, or scintillation which can be analysed and reveal the position of such objects as pulsars. The discovery would lead to the first Nobel Prize in physics to be given to an astronomer. So even though we didn’t find aliens, we learned a lot.

What we know about Pulsars

We now understand that neutron stars are the finished product of dead stars, when all that’s left has burnt out. The end of a star is a hostile environment; these neutron stars are incredibly hot, incredibly radioactive and as I said before, they are incredibly dense. The surface would be hard and smooth, though it is unlikely that any of us would be able to set foot on one. As the dying star collapses, its rotational speed increases. Imagine that you’re spinning on an office chair and you draw in your arms and legs. You speed up, so would the pulsar. The pulses which these star remnants emit are how we’ve discovered most neutron stars.

Since the initial discovery, we’ve found over two thousand pulsars. Merely collecting them isn’t their sole reason of interest. Recently we’ve been able to confirm the existence of another type of cosmic phenomenon; gravitational waves. As the universe expands, it does not does so uniformly. Ripples in space occur as part of this hodgepodge process as well as due to galaxies colliding. As these colossal clusters approach each other, the black holes in the centre release gravitational waves. Pulsars play a part in this, due to their regularity, which is altered by the distortion of merging black holes.

The Future of Pulsars

Pulsars were originally mistaken as the beacons or lighthouses of little green men. This is the reputation which history has afforded them, though that’s not to rule out their being put to a similar use in the future. Although pulsar signals do eventually slow down as they lose spin, this takes a long time and they are relatively stable points. Future deep space travellers will have the potential to navigate via triangulation of the pulsars in 3-D space.

Present space travellers are even able to do this. In 1972, the Pioneer 10 spacecraft was launched to explore Jupiter and beyond. In 1983 it left the solar system to enter deep space. Finally, in 2003 we lost its signal. As well as numerous scientific instruments on board, the spacecraft carried an engraving depicting humans, earth and the solar system. The latter featured as a sort of map, showing its location between the relative periods of 14 pulsars to enable any visitors from beyond the stars to say hello. Alongside the Arecibo message, the Pioneer engraving will give any civilisations out there a clue that there is something else trying to make contact.

We have come full circle, as many pulsars do. From being mistaken for communications beyond the stars, we have come to use them as our own messages to what might be out there. Pulsars are certainly one of most bizarre interstellar objects which we’ve detected. Certainly, we have a lot to learn from their structure and exotic matter. In the matter of aliens versus pulsars, perhaps the truth is stranger than fiction.

What’s Next?

  • Learn more about our imminent cosmic death.
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The opinions expressed in this article are those of Dr George Aitch and Dr Janaway alone and may not represent those of their affiliates. Images courtesy of flickr.


  1. Burnell, S. Jocelyn Bell “Little Green Men, White Dwarfs or Pulsars?” Cosmic Search Magazine. (1977)