Category Archives: Science

What is Love?

Sometimes love is only recognised when something leaves. And in the feeling of loss, the emotional tumult becomes manifest in some undefinable ache. Something in your chest that once was,  now  is not. Other times it is something that fills you, that breaks apart the worries of life and focuses everything into the space between heartbeats.

Is love just hormones? Or nerve receptors? Or is it something more?

“But that afternoon he asked himself, with his infinite capacity for illusion, if such pitiless indifference might not be a subterfuge for hiding the torments of love.” – Gabriel Marquez, Love in the Time of Cholera.

Love in the blood

It is a habit of science to reduce the complex into understandable pathways. By explaining something to a point of being irreducible we can better understand it. But for something as universal as love the idea may seem not just implausible, but somehow offensive.

After all, what does it all matter if we know our atoms, if we cannot feel love?

Scientists believe that love can be explained in terms of our physiology (i.e how our body works,) and this can be explained as way of surviving. From our earliest ancestors we needed something strong to hold us together, to cement a protective unit for our children.

Love seems to fit the call.

It all starts, they say, with lust. The feeling of attraction, physical in nature and overwhelming. Surges of hormones such as testosterone and oestrogen fill our bodies, and promote sense of wellbeing.

This is what the poets immortalise, the feeling of connection beyond space and time.

Next comes attraction, the impetus of familiarity. New hormones such as adrenaline, cortisol and dopamine begin to work. Dopamine itself is a hormone much concerned with addiction, our bodies literally pine for the subject of our love.

That ache, often good if longing, comes somewhere in this whirlwind. This is where the stark images of colour and beauty found in songs finds home.

Finally comes attachment, a feeling mediated by oxytocin, serotonin and vasopressin. These hormones work in different ways, but science shows they foster feelings of connection and comfort. In fact, studies have correlated blood levels of these substances with healthy relationships. This is where the image of old hands intertwined on a park bench lives.

In a small town taking my hand from the words into a promised land. How I wish for a thorn in my heart and deadly was the rose that I got. – Kite, Dance Again.

Love in our hearts

So you may be wondering, does learning all of this remove the special nature of love? Was the lifelong love of Florentino Ariza nothing more than an interplay of chemicals on a lonely mind? Or is there something poetic to it, something that transcends the biology?

We know little about the brain, and less about the mind. Sure, with functional MRI we can begin to tease apart the physicality of thought, and with psychiatry the cognition of life, but even with complex models it seems something is amiss.

Perhaps love is just a victory of evolution, a hormone driven delusion designed to bring us together. Perhaps it serves no function other than to provide a stable resource base for the young and a gateway to reproduction.

But that doesn’t full explain it, much like a painting can never fully capture a mountain.

There is a point, somewhere between our dreams and our reality that life finds comfort. And the complexity of the human mind has led to great poetry, art, literature and film that portrays everything from the spark of eyes meeting to the squalor of heartbreak.

Within our love and pain we have created great beauty far beyond the dance of molecules.

So, for me at least, even if love can be categorised and explained by hormones and biology, it may never be captured. It exists between the heartbeats, and in that silence is a secret that no instrument can reveal.

Welcome home. Ships are launching from my chest
Some have names but most do not. Radical Face, Welcome Home.

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5 Quotes that make Carl Sagan Unforgettable

sagan love quote universe

If you haven’t heard of Carl Sagan then you are missing out. Not only was the renowned astrophysicist a pioneering scientist, but a leader in the field of public science communication. And personally, one of the greatest people to have ever lived.

Without him, it is unlikely that many of us would know much about the universe beyond basic education. And without his television show Cosmosa generation of scientists may have never come to be.  But Carl’s greatest contribution to humanity was his unending patience, empathy and personal charge toward empowering people with knowledge.

sagan top 5

On his ‘ship of the imagination’, Sagan traversed the stars. Image courtesy of Flickr

Even now, his profound insights into human life ring true in arenas ranging from politics to social reform. So let’s count down our top list of his most enduring quotes, perhaps you will find something that you love.

1. On understanding and knowledge.

People are not stupid. They believe things for reasons. The last way for skeptics to get the attention of bright, curious, intelligent people is to belittle or condescend or to show arrogance toward their beliefs.

Throughout his career, Carl was persistent in his pursuit of public empowerment. By treating us all as friends, capable of the greatest feats, he established a paradigm of education by right.

sagan carl quote top 5 universe

Carl Sagan’s Universe was one we could all explore, and he tried to be the greatest guide. Image courtesy of Flickr.

But with some controversy, he took what was privy only to a select few in academia and made it not just palatable, but wondrous to the rest of us. For Carl, you were not just deserving of the universe but enriched for understanding it.

And as an avowed skeptic of common wisdom and conspiracy, he approached each subject with evidence, understanding, and compassion. Simply, he forgave people human mistakes, where others would simply dismiss them.

We should do the same.

2. On the transience of human life and the immortality of words.

One glance at a book and you hear the voice of another person, perhaps someone dead for 1,000 years. To read is to voyage through time.

During one episode of his poetic homage to humanity, Cosmos, Sagan visited the ancient library of Alexandria. It was here, for a short while, that the world’s greatest minds came together in a shared mission of understanding.

sagan quote books

Books are nothing less than a voyage of discovery, be they history. science or fiction. Our words stay behind when we leave. Image courtesy of Flickr.

And although much of Alexandria’s history was lost, small amounts remain in collected writings. An enduring legacy of another time. But for Carl words were more than just communication between friends and colleagues, but a version of immortality.

Through the written word we learned to overcome death, share the wisdom of our time with those who would come after. The ‘information-organism’ of humanity finds feet in ink over millennia.

3. On the fragility of understanding and the wonder of creation

If you wish to make an apple pie from scratch, you must first invent the universe.

In this short epithet, Sagan reflects on the vast gulf between human knowledge and the nature of the universe. And depending on your interpretation, he is either jocularly revealing a comedy of nature, or providing a deep insight into the linearity of thought.

supernova sagan star

All that we are was formed in ancient furnaces. Including apple pie. Image courtesy of Flickr.

To make an apple pie requires the ingredients created from the formation of our universe. All that is once was in the bellies of ancient stars, cast into our universe and eventually mealtimes by cosmic forces and eons of time.

But to understand the world we must first invent a way of understanding, and for that the best we have is science. It is through a skeptic and imaginative mind that we may create our universe.

4. On the humility of human life in an infinite universe (see video.)

In his famous soliloquy, Sagan reduces human accomplishment, greatness, cruelty and misunderstanding to the tiny significance it has in the greater universe.

Within his poetic testament he not only shows us just how small we are, but hints at how pointless our self destruction is.

And at the same time he conveys a message of hope disguised in a eulogy. He warns us that our future is down to us, and that hopefully by realising what we have, as tiny as it is, that we may create a better future.

Earth is our only home.

5. On the saving grace of companionship.

For small creatures such as we the vastness is bearable only through love.

For all his poetry, lessons and foretelling, Sagan hits on something truly profound. Upon recognising the inescapable truths that are our mortality and ineffectual existence beyond a pale blue dot, he returns to what unites us all.

love sagan universe

In the infinite abyss of a dark universe hides rare moments of light and love. Image courtesy of Flickr.

By embracing love we can overcome any distance. And in that the paucity of meaning is rendered mute, bearable and even empowering, as through love we can find meaning in an ocean of irrelevance. Where science can bring humility, love can bring back purpose.

Throughout his career it appears to me that Sagan’s underlying driving force must have been a deep and powerful love for the universe and his fellow man. To continuously fight for public empowerment, against governments, critics and even himself, Sagan had a heart much greater than even his ‘ship of the imagination’ could explore

But to encapsulate Sagan in five quotes is impossible, so we encourage you to explore his work further.

So which was your favourite? What have we missed and what did you take away? Let us know in the comments below. And if you found yourself touched, please help us reach out  by sharing.

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The opinions expressed in this article are those of Dr Janaway alone and may not represent those of his affiliates. Featured image courtesy of Flickr.

R.I.P Carl.

Ten Incredible Pearls of Wisdom From Great Minds

ten wisdom dawkins

The world can be a confusing place, but academics, authors, artists, poets and philosophers are just some of the many who have tried to explain things. In my wanderings through scientific texts, popular non-fiction and and documentaries, there have been a few stand out comments.

Whether it be Malcolm Gladwell’s keen insight into human nature, or Carl Sagan’s prophetic view of the world, all have resonated in some way beyond comprehension. They seem intrinsically correct, and universally true.

So here are ten incredible pearls of wisdom from the great minds.

1. On The Origins And Nature Of Human Behaviour

‘Let us understand what our own selfish genes are up to because we may then at least have the chance to upset their designs.’ – Richard Dawkins, ethologist, evolutionary biologist and popular-science writer.

As the current face of evolutionary theory, Dawkins’ is no stranger to controversy. And although his work has revolutionised our understanding of the natural world, his opinions about how we must overcome our nature have caused the most conversation.

ten wisdom dawkins

The nature of life is conservative, selfish and driven by unconscious forces. Dawkins’ understands that to ‘be good’ you must understand our basic urges. Image courtesy of Flickr.

Dawkins’ is very aware how our evolutionary history, and how the selective pressures of the environment and each other, have shaped our behaviour. And in his book ‘The Selfish Gene‘, gives us to pause to consider the true morality of nature.

What is ‘natural’ isn’t inherently ‘moral’, and what we consider ‘moral’ is not inherently survival.  So Dawkins’ asks us to understand our primal natures if we are to best them.

2. On The Power Of Words

‘To acquire immunity to eloquence is of the utmost importance to the citizens of a democracy. ‘ – Bertrand Russell, Philosopher, Logician and Nobel Prize winner.

Although Russell is best known as a philosopher, his life of work reached deeper into the shared mind of society than we realise. By studying and writing on the academic disciplines of logic, mathematics and epistemology (the study of knowledge,) he became a strong advocate for peaceful societal reform.

A man’s words may make beautiful the macabre. Russell relied on logic to unify humanity toward a common good. Image courtesy of Flickr.

And most noteworthy are his observations of how people can be manipulated by words. The use of eloquent language, a flowery vocabulary or poetic arrangement can make the terrible seem empowering.  You need only read the words of Nazi spokesperson Joseph Goebbels to see how language can betray human decency.

We must understand a man’s motivation, and place it in the context of the sociopolitical climate, to truly understand what may be hiding behind the words.

3. On The Risks Of Virtue

‘Whoever fights monsters should see to it that in the process he does not become a monster. And if you gaze long enough into an abyss, the abyss will gaze back into you.’ – Friedrich Nietzsche, Philosopher, Author and culture commentator.

As the central reference to ‘Nihilism‘, Nietzsche examined the purpose of life without purpose. His infamous quote ‘God is dead‘ instilled the idea that the concept and role of any God is limited, and that through pain and suffering we may choose a virtuous path.

wisdom nietzsche

Nietzsche warned us not to become monsters in the pursuit of greatness. Image courtesy of Flickr.

Whilst an avid skeptic of religion in general, his infamous parable ‘Thus Spoke Zarathustra‘ presented, in humbling terms, the true insanity of zealotry. He warns us not to lose ourselves in purpose, and to recognise how a belief in achieving ‘the good’ can lead to evil.

And in a world where popular influencers claim a moral authority, his words could not carry greater weight.

4. On The Importance Of Responsibility

‘Man is condemned to be free; because once thrown into the world, he is responsible for everything he does.’ – Jean Paul Satre, Philosopher.

Regarded by many as the father of ‘Existentialism‘, Satre believed that existence precedes essence, and that we must find our own way in a meaningless universe. But in accepting this freedom in action, we cannot ignore our role in what comes next.

wisdom star ten star

Satre  believed that with action comes responsibility, and with freedom comes the same. Image courtesy of Pixabay.

Many existentialists reject the concept of ‘good’ or ‘bad’ behaviour being an inherent natural motivation, and instead suggest that we make our decisions based on complex (although often incorrect,) contextual interpretations.

Not only does this mean we must be aware of our own limits, but take responsibility when they are exposed.

5. On The Stoicism Of Knowledge

The older I get, the more I understand that the only way to say valuable things is to lose your fear of being correct. – Malcolm Gladwell, Author and Journalist.

Famed author of ‘David And Goliath’, ‘Outliers‘ and ‘The Tipping Point‘, Gladwell explores the interconnectedness of humanity with the world around it. In his poignant prose he unravels what may seem miraculous, often challenging widely held beliefs.

wisdom pearls gladwell quote

Gladwell has exposed and explained the hidden reasons behind cultural success and individual power. To question convention is as useful as it is risky. Image courtesy of Flickr.

His work tells us not only to dig deeper to explain the world, but that explanations may exist beyond the obvious. He also extols the value of expressing new ideas, fearlessly with no regard for your own ego.

We must challenge convention to find the truth, even if it risks our reputations.

6. On The Insights Given By Friendship

‘You can learn something about a person by the company she keeps.’ – Sam Harris, Philosopher and Neuroscientist.

Although more likely to be a figure of repute for his views on religion, Harris is a distinguished author and surveyor of the interface between neuroscience, morality and the world at large.

A fierce critic of authoritarian dogma, Harris asks us to take responsibility for building our knowledge toward creating a better world.

wisdom harris quote ten

Sam Harris is a vocal critic of authoritarian regimes and their numerous abuses. Image courtesy of Flickr.

He also asserts that morality itself exists independently of religious doctrine, and empowers a human approach toward a coalescence of society. And as a neuroscientist, he is all to aware of how our behaviour may make us, or betray our intentions.

So if  you want the measure of a man, consider who they value as friends.

7. On The Illusion Of Simplicity.

‘I think you’ll find it’s a bit more complicated than that.’ – Ben Goldacre, Physician and Author.

The namesake of his popular book tells us a lot about Ben Goldacre. An academic and scourge of pseudoscience and ‘folk wisdom’, Ben uses evidence to expose the lies many are sold by the few to the many. He also tells that what is made simple, or appears so, may not be.

wisdom goldacre bad science

Ben Goldcare makes it his mission to challenge misleading beliefs, expose bad science and explain the misunderstood. Image courtesy of Flickr.

What is claimed to simple may be complex, and what lies between may be inaccurate, underhanded and deliberate.

And with that, we should try to understand the motives behind simplification, and why it is so easy for us to be sold a lie. Amongst his many targets is Homeopathy and the risks involved in being misled.

8. On the Arrogance Of The Human Mind.

‘See that the imagination of nature is far, far greater than the imagination of man.’ – Richard Feynman, Theoretical Physicist, Nobel Prize winner and  Science Communicator.

Whilst it may seem odd that a theoretical physicist is so humble about uncertainty, Feynman shows us just how wonderful the universe is.

Although a pioneer in our understanding of the nature of our reality, he recognises that there is simply more we don’t know.

feynman physics quote wisdom

Feynman studied and revealed some of the most hidden secrets of our universe, but in doing so realised that was is unknown is our greatest teacher. Image courtesy of Flickr.

The intricacies of our reality, currently hazy between the infinitesimally small and unimaginably large, appear to us through rigorous questioning and often teach us that our presuppositions are not just wrong,  revealing a drastic flaw in human understanding.

We claim to be intelligent, and yet this intelligence often blinds us to our own folly. We must revel in the wonder of whats left to wonder about, and not be afraid to look stupid doing so.

9. On The Value Of Choice And Humility

‘I was never born to write. I was taught to write. And I am still being taught to write.’ – Atul Gawande, Surgeon, Research and Author.

If you are a doctor, you no doubt are aware of Gawande. Whilst a strong advocate of evidence and comprehensive approaches, Gawande has also ventured into a philosophical musing of the human condition.

gawande life wisdom

As a surgeon, Gawande not only saves lives, but has taken it upon himself to understand what the true value of life is. Image courtesy of Flickr.

In his best-selling book, ‘Being Mortal’, Gawande examines the true value of human life, and what we lose and gain as we age. And true to his nature, he treats himself with a level of skepticism coherent with his humble world view.

We are born with a choice in a difficult world, expertise is only a measure of dedication tempered by self criticism, and arrogance undermines greatness.

10. On The Size Of Our Influence

‘The Earth is a very small stage in a vast cosmic arena.’ – Carl Sagan,  Astronomer, Astrophycist and Pulitzer Prize winner.

To reduce the works of Sagan to one sentence would vastly sell him short. Not only did he lead the way in popularising science, but housed a mind so in tune with the human condition that his loss is truly universal.

Having inspired legions of scientists, including protege Neil deGrasse Tyson, Sagan’s holistic approach to science and its delicate implications toward society rings as true today as it did years ago.

sagan quote wisdom science

Carl Sagan was not just a pioneer of knowledge, but arguably one of humanity’s greatest teachers. Image courtesy of Flickr.

He was exceptionally kind, humble and patient, expressing the very tenets he postulated as a universal ideal.

Sagan reminds us that the true beauty of the universe is not just in its nature, or its creation, but in our pursuit of explanation and the inherent ability to use this knowledge to better ourselves and future generations.

And that perspective matters, for the universe is much greater than such complex molecular machines as we. We are so very small, but in that there is much to be learned, gained and valued.

So Much Left To Learn

Ten quotes simply isn’t enough to even scratch the surface of the grand insights accumulated in the wealth of human knowledge, or beyond it. And with each quote, you may have taken your own interpretation of meaning and purpose.

Perhaps you disagree with some, or worry that they are incongruent with each other. But I am willing to contend the opposite, that each shares a unity in placing the pursuit of knowledge through humility, truth and beneficence as a true virtue.

So what are your favourite quotes? What and who has changed your life? Let us know in the comments. And if you believe, like I do, that knowledge is best shared, then help us by sharing this article with your friends and family.

What’s Next?

The opinions expressed in this article are those of  Dr Janaway alone and may not represent those of his affiliates. Featured image courtesy of Flickr.

Note from the Author: Upon writing this article I became very aware of just how much I don’t know, and how much I can learn. I feel it only right to follow up on this article with more information about the works and lessons of the persons featured. There are many greats not featured on this list, but don’t worry, I will find ways to include them. I do not value my opinion of what is great  above any others, I only wish to signpost what is already there.

‘Did Mars Once Hold Life?’ Discovery Of Organic Molecules May Hold Vital Clues.

An exciting discovery by NASA’s Curiosity rover has strengthened the idea that Mars may have once been suitable for life. The chance finding of organic material in an ancient lake bed suggests Mars once held the foodstuffs necessary for life.

Although not conclusive, these findings add to the growing evidence for previous life on Mars, with seasonal methane and liquid water providing cause for excitement.  Mars is inhospitable for now, but may not have always been.

And with that, the tantalising hope that life may exist beyond earth.

mars life organic rover

Mars is considered a cold, dead world. But did life once flourish? Image courtesy of Flickr.

Curiosity’s Discovery Of Organic Molecules

The discovery of these ‘organic molecules’ asks many questions. We cannot tell exactly where they came from, perhaps remnants of long dead organisms, something crash landed from space or indeed simply ancient foodstuffs.

But what we can say is that these molecules, formed of carbon, oxygen and other elements fundamental to life as we know it, could very much  suggest something significant.  Whilst it is important to note that these structures can be created without life, life can’t exist without them.

“With these new findings, Mars is telling us to stay the course and keep searching for evidence of life. ” – Thomas Zurbuchen, NASA Science Directorate.

life organic simple mars

Life is a complex array of molecules and processes. But at its core, the foundational elements may coalesce with relative ease. Image courtesy of Flickr.

 

Food For Thought

Although direct evidence of life on Mar’s is yet to be discovered, if at all, these new findings tell us that the environment may have supported it. Biological creatures like we subsist  on organic compounds for energy, and this is consistent down the food chain.

There is little reason to thing extra-terrestrial life would be that different. In fact, many believe that the ingredients for simple life are abundant enough that finding alien organisms is almost a certainty. And the chances are that, on a fundamental level at least, we will all be built of the same stuff.

And with that, makes a strong case for a similar ecological energy source. Jennifer Eigenbrode, NASA biogeochemist says of the finding;

“It is not telling us that life was there, but it is saying that everything organisms really needed to live in that kind of environment, all of that was there.”

earth life mars organic

All life as we know it evolved from the same ingredients. And these ingredients grow universally. Image courtesy of Flickr.

A New Direction In Ancient Footsteps

Although direct evidence of extra-terrestrial life eludes us, the chances are good. Although the paucity of interstellar craft and signals presents some concerning questions, it may be that life is abundant between the stars.

And you have  to look no further than your own back yard to find out why. It is likely made of common stuff.

Whilst we imagine aliens as, well, very alien, very basic life may be a natural inevitability. Current theories of the origins of life on earth centre around the unconscious replication of favourable molecules, building more complex structures over time.

Eventually, these would become us and everything else that breathes, grows and dies. And the more we learn, the more beautiful and interconnected it all becomes.

life organic mars

The size of our universe is beyond comprehension. But can we really say that it is ours? Image courtesy of Flickr.

Given the vast numbers at hand, i.e trillions of stars, innumerable planets and billions of years, chances are that Curiosity’s discovery may be just the first of a truly cosmic collection.

So what do you think, is there life out there? What does it mean for us? Let us know in the comments below.

What’s Next?

The opinions expressed in this article are those of  Dr Janaway alone and may not represent those of his affiliates. Featured image courtesy of Flickr.

 

 

 

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.

 

16

(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.

image11

(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.

Sources

  1. Su, Francis E., et al. “Pigeonhole Principle.” Math Fun Facts. <http://www.math.hmc.edu/funfacts>
  2. http://mathforum.org/dr.math/faq/faq.birthdayprob.html
  3. https://www.geeksforgeeks.org/birthday-paradox/
  4. http://www.math.cornell.edu/~mec/2008-2009/TianyiZheng/Birthday.html
  5. http://mste.illinois.edu/activity/birthday/
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