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(100a + 10b + c) – (100c + 10b + a)

The two b terms cancel each other out, leaving an intermediary result of:

99ac, or

99(a – c)

At a basic level algebra doesn’t involve any special insight, but rather the application of certain rules. The aim is to apply these rules until the expression is as simple as possible.

The term 99(a – c) is as neatly arranged as it can be.

Since the first and last digits in abc differ by at least 2, then a – c is either 2, 3, 4, 5, 6, 7 or 8.

So, 99(a – c) is one of the following: 198, 297, 396, 495, 594, 693 or 792. Whatever three-figure number we started with, once we have subtracted it from its reverse, we have an intermediary result that is one of the above eight numbers.

The final stage is to add this intermediary number to its reverse.

Let’s repeat what we did before and apply it to the intermediary number. We’ll call our intermediary number def, which is 100d + 10e + f. We want to add def to fed, its reverse. Looking closely at the list of possible intermediary numbers above, we see that the middle number, e, is always 9. And also that the first and third numbers always add up to 9, in other words d + f = 9. So, def + fed is:

100d + 10e + f + 100f + 10e + d

Or:

100(d + f ) + 20e + d + f

Which is:

(100 × 9) + (20 × 9) + 9

Or:

900 + 180 + 9

Hey presto! The total is 1089, and the riddle is laid bare.

The surprise of the 1089 trick is that from a randomly chosen number we can always produce a fixed number. Algebra lets us see beyond the legerdemain, providing a way to go from the concrete to the abstract – from tracking the behaviour of a specific number to tracking the behaviour of any number. It is an indispensable tool, and not just for maths. The rest of science also relies on the language of equations.

In 1621, a Latin translation of Diophantus’s masterpiece Arithmetica was published in France. The new edition rekindled interest in ancient problem-solving techniques, which, combined with better numerical and symbolic notation, ushered in a new era of mathematical thought. Less convoluted notation allowed greater clarity in descrig problems. Pierre de Fermat, a civil servant and judge living in Toulouse, was an enthusiastic amateur mathematician who filled his own copy of Arithmetica with numerical musings. Next to a section dealing with Pythagorean triples – any set of natural numbers a, b and c such that a2 + b2 = c2, for example 3, 4 and 5 – Fermat scribbled some notes in the margin. He had noticed that it was impossible to find values for a, b and c such that a3 + b3 = c3. He was also unable to find values for a, b and c such that a4 + b4 = c4. Fermat wrote in his Arithmetica that for any number n greater than 2, there were no possible values a, b and c that satisfied the equation an + bn = cn. ‘I have a truly marvellous demonstration of this proposition which this margin is too narrow to contain,’ he wrote.

Fermat never produced a proof – marvellous or otherwise – of his proposition even when unconstrained by narrow margins. His jottings in Arithmetica may have been an indication that he had a proof, or he may have believed he had a proof, or he may have been trying to be provocative. In any case, his cheeky sentence was fantastic bait to generations of mathematicians. The proposition became known as Fermat’s Last Theorem and was the most famous unsolved problem in maths until the Briton Andrew Wiles cracked it in 1995. Algebra can be very humbling in this way – ease in stating a problem has no correlation with ease in solving it. Wiles’s proof is so complicated that it is probably understood by no more than a couple of hundred people.

Improvements in mathematical notation enabled the discovery of new concepts. The logarithm was a massively important invention in the early seventeenth century, thought up by the Scottish mathematician John Napier, the Laird of Merchiston, who was, in fact, much more famous in his lifetime for his work on theology. Napier wrote a best-selling