The mathematics that can defeat monsters and cut cards.

Hércules and Minerva enfrentam Marte, deus da guerra, em pintura de Peter Paul Rubens

In Homer's great Greek classic "The Odyssey," after the Trojan War, the hero Odysseus embarks on a long journey back to his city, Ithaca. Ten years of adventures across the Mediterranean follow, adding to the ten years of the war. At home, Odysseus was presumed dead, and the beauty and wealth of his wife, Penelope, attract increasingly audacious suitors. Always believing in her husband's return, she devises a cunning pretext to avoid marrying any of them: first, she needs to finish weaving her father-in-law's shroud. But everything Penelope weaves during the day, she unravels at night. And so two decades pass.

This is an unusual situation. Normally, what we want is to finish tasks as soon as possible, and that requires effort. "Procrastinating" to avoid finishing is usually easier; in fact, some people are great at it. Like my son, when it comes to eating his soup… But there are situations where not finishing the task can be impossible.

Imagine a box containing a finite number of billiard balls, each numbered (1, 2, 3…). The task is to remove the balls from the box, one by one. You – remember that you don't want to finish the task! – can only replace each removed ball with any number of balls, as long as they all have lower numbers. For example, if you remove a ball numbered 5, you can replace it with any number of balls numbered 1, 2, 3, or 4. When you remove a ball numbered 1, you cannot replace it, since there are no balls with lower numbers.

Completing this task can take a long time: after all, you can put as many balls as you want in the box each time.

However, mathematics guarantees that, whatever you do, sooner or later the box will be emptied! Can you understand and explain why?

In ancient times, the demigod Hercules had to perform twelve tasks to prove his worth and be received into the abode of the gods. One of them was to kill the Hydra of Lerna, a monster with seven ferocious serpent heads. The problem was that every time he cut off one head, two new ones grew back! Hercules eventually won. But we recently discovered that this didn't solve the problem, because the Hydra left offspring! We need to find a new hero, a reader with a strong arm and a brave heart, to rid humanity of this new scourge.

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