Here is a version of one of Zeno’s famous paradoxes. Achilles will run eight metres. He can cover four metres in one second. But this means that in the first second he is half way there, in the next half of a second he covers two more metres, then one metre in the next quarter of a second, and so on. But does this mean that Achilles takes forever to reach the end?

The solution involves summing up the terms 1 + 1/2 + 1/4 + 1/8 + … .

Although this sum contains an infinite number of terms, its value is two. And, running eight metres at four metres per second does take exactly two seconds. The ancient Greeks’ strong intuition made them wary of summing an infinite number of terms.  It was only in the 18th-19th century that mathematicians developed the right tools to show rigorously that the sum of an infinite number of terms can be finite – equal to two  in this case.

But now consider a modern variant of this paradox, called Thomson’s Lamp. An idealised being, Joviana, has an idealised lamp with a toggle switch which turns it on/off. Joviana turns the lamp on, then after one second she flips the switch to off, then flips it on again after one quarter of a second, and so on. Using the same summation as above, the whole procedure takes two seconds. But now, after two seconds, is the lamp on or off? This question is equivalent to finding the value of the sum of Grandi’s series: 1 – 1 + 1 – 1 + … .

The terms ‘1’ and ‘-1’ in this series correspond to switching the lamp on or off, respectively.  Even great mathematicians disagreed about the value of this sum. If you group the terms in pairs starting from the first two, you get the answer 0. This would mean that Thomson’s lamp is off at the end of two seconds. If you group them in pairs starting from the second term you get the answer one, meaning that the lamp ends up on! One answer which seems to be the most mathematically correct is 1/2. But then the mystery deepens because starting from this value of 1/2 one can arrive at the conclusion that 1 + 2 + 3 + 4 + … = -1/12.

An infinite sum of positive integers adding up to a negative fraction! This calculation can be found in one of the notebooks left by Srinivasa Ramanujan, the protagonist of the recent film The Man Who Knew Infinity.

And if this sounds too distant from reality, then let us end by noting that this last formula appears in that branch of modern physics called string theory.

Did you know?

• Rubik’s Cube was invented by Erno Rubik, a Hungarian professor of architecture, in 1974. It is widely considered to be the world’s best-selling toy.

• General solutions to the Rubik’s Cube were discovered independently. David Singmaster gave one in his book Notes on Rubik’s ‘Magic Cube’, published in 1981. His solution is based on solving a layer completely before solving another one. In the same year, at 13 years of age, Patrick Bossert gave another one in his book You Can Do The Cube, which became a bestseller.

• The Rubik’s Cube has 43,252,003,274,489,856,000 different configurations. However, it can be solved in 20 moves or less. This was proved by a team of researchers working with Google. The number 20 is optimal, meaning that there are configurations which cannot be solved in less than 20 moves.

For more trivia see: www.um.edu.mt/think

Sound bites

• Do mathematics and maps make you nervous? A new study by researchers from King’s College London shows that genes play a significant role in how anxious a person feels when faced with mathematical tasks or spatial tasks, such as map reading or navigation. The study is based on a sample of 2,928 twins (1,464 pairs) whose ages range from 19 to 21. It was published in the following paper: M. Malanchini, K. Rimfeld, N.G. Shakeshaft, M. Rodic, K. Schofield, S. Selzam, P.S. Dale, S.A. Petrill and Y. Kovas, ‘The genetic and environmental aetiology of spatial, mathematics and general anxiety’, Scientific Reports 7 (2017), article 42218.

• What does a robot do after it fails the entrance exam to a university? Find a job in industry, of course. Artificial Intelligence researchers in Japan have given up on the challenge to build a robot that will, by 2021, gain entry into the prestigious University of Tokyo after Torobo-kun, the robot they built, failed the exam for the fourth consecutive year. This exam consists of eight tests in five different subjects. Torobo-kun actually got an average of 51.7 per cent, which would have gained it entry into several of Japan’s universities, but still far off from the 80 per cent required for entry into ‘Todai’. It did very badly in English, average in Maths, but much better in World History because it can store in its memory vasts amounts of data and use its artificial intelligence to make connections and produce good answers in the test. Its creators have now decided to focus on this skill and improve on it so that Torobo-kun can apply it to problems in industry.

For more science news, listen to Radio Mocha on Radju Malta 2 every Monday at 1pm and every Friday at 6pm.