Imagine a hydra with a scaly body and a number of necks sprouting from it. At the end of each neck is either a head or another set of necks. This fearsome creature is covered with necks and heads –and you need to slay it.
Due to its ferocity, you can only cut off one head by severing one neck at a time. Unfortunately, this mystical beast has a magic power. As soon as you remove any head, the neck below the neck connecting this head regrows a copy of what is left above it (refer to the picture). If you cut a head directly from the body, it is gone forever and doesn’t regrow.
This hydra is the most dangerous of its species – a mathematical hydra. This means that each time you slice a head, it will increase the number of copies it makes from that lower neck. For example, when you remove the fifth head, it will regrow five copies of what remains on top of the lower neck. Dastardly!
Mere mortals would admit defeat; however, those tenacious enough to persevere will always win. It doesn’t matter the order in which you cut off the heads; as long as you keep on slicing, you’ll win.
Those tenacious enough to persevere will always win
There is some serious math behind your victory.
The reason you can win is that no matter how many heads the hydra grows, the level of the heads will continue to decrease. Eventually there will be just a layer of heads around the body that can be cut off and which won’t regrow.
Just how many heads do you need to cut to kill a given hydra that is just a line of necks with one head?
If the hydra has one neck, you’ll need three cuts. If it has two necks, you’ll need 37 cuts. So far, so good. But, if you have three necks, you’ll need at least a ‘Graham’s number’ of cuts. Graham’s number is a finite (not infinite) number so impossibly large that it can’t be output by a computer before the universe collapses in on itself.
In 1982, Laurie Kirby and Jeff Paris produced the formal proof that any given hydra can be defeated. In doing so, they assumed the existence of an infinite number – a number that is greater than any of the counting numbers. It can even be proven that you can’t prove their result without assuming the existence of an infinite number.
Kirby and Paris truly defeated the Hydra!
Sound Bites
• A tiling is a covering of the plane using one or more shapes without gaps and overlaps. A set of tiles is said to be aperiodic if there is no way to tile a plane that creates a repeating pattern. In 1967, mathematician Robert Berger discovered a set of 20,426 tiles that were aperiodic. Seven years later, physicist and Nobel Prize winner Roger Penrose reduced this number to two.
• In April 2023, a retired engineer called David Smith, together with mathematicians Craig Kaplan, Joseph Mayers and Chaim-Goodman Strauss found a single aperiodic tile. They named this the ‘einstein’ tile after the German phrase ‘one stone’. The einstein tile is a 13-sided shape that resembles a hat or shirt. Their work has not yet been peer-reviewed. You can find it here.
For more science news, listen to Radio Mocha on www.fb.com/RadioMochaMalta/.
DID YOU KNOW?
• Google named their company after the googol number: 10^100. There are fewer than googol atoms in the observable universe.
• Graham’s number makes a googol look small. Graham’s number is so big that it cannot be expressed in the common mathematical notation as a power or powers of powers. Perhaps Google should have been called Graham?
• Just like ChatGPT is replacing Google, Graham’s number is no longer the largest number used in serious mathematics. There is now the TREE(3) number that makes Graham’s number look tiny by comparison.
• What replaces TREE(3), the ChatGPT of numbers? A Mexican philosopher, Agustin ‘Mexican Multiplier’ Rayo, trounced Professor Adam ‘Dr. Evil’ Elga in a 2007 ‘Big Number’ duel. The Mexican Multiplier defined the Rayo number as the smallest number bigger than any number that can be defined in the language of first order set-theory with less than a googol symbols.
For more trivia, see: www.um.edu.mt/think.