Mind-Blowing Numbers: How Busy Beaver Hunters Surpass Ordinary Math Limits | Quanta Magazine

In 1962, mathematician Tibor Radó introduced a fascinating concept called the busy beaver game. It’s a game where you pick a number of rules, let’s call it n. The goal is to find a Turing machine that follows n rules and runs for as long as possible before it eventually stops. The total steps it takes before halting is known as the busy beaver number, BB(n).

To identify this busy beaver, you first list all possible Turing machines that follow your chosen number of rules. Then, you run a computer program to see how long each machine runs. Some machines never halt; they get caught in endless loops and can be disregarded. For the rest, you track how many steps they take until they stop. The machine with the longest runtime is your champion, the busy beaver.

However, this task isn’t as simple as it sounds. The number of potential machines grows exponentially as you add more rules. Analyzing each machine is impractical, so custom computer programs are essential. Some machines are easy to categorize; they either stop quickly or get stuck in infinite loops. Yet, others run for unexpectedly long times without clear patterns, adding complexity.

Shawn Ligocki, a software engineer and avid busy beaver explorer, highlights the challenges: “Technology improvements help, but they have limits.” As you increase the rules, you also need more computing power. Some machines may run for so long that a straightforward simulation isn’t feasible, so you require clever mathematical methods for calculations.

The quest for the busy beaver has been alive and thriving since the 1990s. Ligocki and his father, Terry, utilized high-powered computers to tackle the BB(6) problem. In 2007, they discovered a six-rule Turing machine that ran for an astonishing number of steps—almost 3,000 digits long. To put that into perspective, you could fit all those digits on a single sheet of paper!

By 2010, Pavel Kropitz, a computer science student, took on the BB(6) challenge for his senior thesis. Using a network of 30 computers, he found a new record: a machine that ran nearly 30,000 digits long, enough to fill about 10 pages. His record held for 12 years, but in May 2022, Ligocki found an even larger machine, sparking a flurry of activity in the busy beaver community. In just two weeks, Kropitz and Ligocki exchanged record after record, each trying to outdo the other.

The numbers involved in these machines can be incomprehensibly large. For example, consider the concept of tetration, a mathematical operation involving repeated exponentiation. A number raised to the power of itself multiple times grows rapidly. To illustrate, 10↓10 (ten tetrated by two) is 10 billion, and if you take it a step further, you enter realms of numbers that dwarf even the atoms in the universe.

Today, research into the busy beaver continues to captivate mathematicians and computer scientists alike. It not only challenges our understanding of computation but also sheds light on concepts of infinity. It’s a puzzle that blends curiosity with the limits of technology, and there’s still much to explore.

For more on Turing machines and their implications in computer science, check out this comprehensive study by the ACM.

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