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Since ancient times, humankind has mulled over paradoxes, with many left scratching their heads at the seemingly contradictory and logic defying statements.

A paradox can be defined as a statement or situation that may be true, but seems impossible to understand because it contains two opposite facts or characteristics. They can span all manner of field, from philosophy, to science or mathematics.

����ý Bitesize takes a look at some of the classical and modern paradoxes that have blown our minds across the centuries.

Achilles paradox

First argued by the 5th Century BC Greek philosopher Zeno, the Achilles paradox involves a race between the legendary swift-footed warrior Achilles and a slow-moving tortoise.

The paradox argues that if the tortoise is given a slight head start and continues to race ahead, Achilles can run at any speed and will never be able to catch-up.

For Achilles to win, he would first need to cover the distance to the point where the tortoise started the race. However, by the time Achilles would reach that point, the tortoise would have moved and logically this process would continue indefinitely.

The paradox indicates that to cover a certain distance, you need to cover a never-ending amount of smaller distances in between, until motion itself is negated. Zeno's concept is now considered to have foretold modern measure theory, which states that something finite can be divided an infinite number of times.

While the story is still one of the best-known paradoxes of all time, it has been widely explained and refuted by scholars, including Zeno's fellow Greek philosopher, Aristotle, in his work The Physics. So thankfully it doesn't look like you'll be getting caught in an endless race on sports day any time soon!

The birthday paradox

Ignoring leap years and assuming that each date was equally likely, how many people need to be in the same room for there to be a 50-50 chance that two have the same birthday?

Bizarrely, the birthday paradox, otherwise known as the birthday problem, states that you would only need a randomly selected group of 23 people in order for there to be a 50.73% chance. But why 23? It might be simpler than you think.

Oliver Johnson, Professor of Information Theory at the University of Bristol, explained the problem to ����ý Bitesize: “Imagine a football squad being picked at random, one by one, hoping to avoid a shared birthday. There’s a very good chance that the second person chosen has a different birthday to the first, since there is only one date to avoid.

“But the third selection now must avoid two dates, and so on, until the 23rd player needs to avoid twenty-two different dates. The more players that join, the harder it becomes to avoid a coincidence. As a result, the chance of a shared birthday jumps up very rapidly – from just over 10% with ten people, to nearly 90% with forty in the group.”

In 2022, two players from England’s 23-woman Lioness squad, Beth Mead and Ellen White, were found to have the same birthday. What are the chances?!

The twin paradox

The twin paradox invites you to imagine a pair of identical twins who decide to take part in an experiment. They decide that twin A is going to remain on Earth, while twin B, an astronaut, will board a high-speed rocket travelling almost at the speed of light towards a distant star.

Once the voyage has been completed, twin B returns to Earth and is reunited with twin A. But to their surprise, stay-at-home twin A has aged more than space-bound twin B.

Having embarked on their separate journeys over the same amount of time, how is this possible? The paradox contradicts our common sense and everyday experience that time is constant. But there is plenty of evidence to back it up.

Scientists believe that when travelling at speeds approaching the speed of light, there might actually result in a noticeable difference in aging. This is called ‘time dilation’ and it is one of the things that is explained by Einstein’s famous theory of special relativity.

This was put to the test in 1971, when physicist Joseph C. Hafele and astronomer Richard E. Keating took atomic clocks on board commercial aeroplanes which flew twice round the world, both eastwards and westwards. When the clocks were compared with a third identical clock which had remained in a lab on the ground, the clocks showed different measurements of time, which were consistent with the predictions from Einstein’s theory.

This article was published in November 2022