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Here’s what you’ll learn when you read this story:
The Busy Beaver number, or BB(n), represents a mathematical problem that tries to calculate the longest possible run-time of a Turing machine recording 1s and 0s on an infinitely long tape for various sets of instructions called states.
While the first three BB(n)s equal 1, 6, and 21, respectively, it takes 107 steps to get to BB(4), and BB(5) produces a staggering 17 trillion possible Turing machines with a 47,176,870 steps being the answer.
Now, as mathematicians attempt to wrap their minds around answer BB(6), they’re beginning to realize that even expressing the unfathomably large number, likely bigger than the number of atoms in the universe, is itself a problem.
For most people, the term “busy beaver” brings to mind a tireless worker or (for the biologists among us) an absolutely vital ecosystem engineer. However, for mathematicians, “busy beaver” takes on a similar-yet-unique meaning. True to the moniker’s original intent, the idea represents a lot of work, but it’s work pointed at a question. As computer scientists Christopher Moore puts it in a video for Quanta Magazine, “What’s the longest, most-complicated thing [a computer] can do, and then stop?” In other words, what’s the longest function a computer can run that does not just run forever, stuck in an infinite loop?
The solution to this question is called the Busy Beaver number, or BB(n), where the n represents a number of instructions called ‘states’ that a set of computers—specifically, a type of simple computer called a Turing machine—has to follow. Each state produces a certain number of programs, and each program gets its own Turing machine, so things get complicated fast. BB(1), which has just 1 state, necessitates the use of 25 Turing machines. For decades, many mathematicians believed that solving the Busy Beaver number to four states was the upper limit, but a group of experts managed to confirmed the BB(5) solution in 2024 (on Discord, of all places). Now, participants in that same Busy Beaver Challenge are learning fascinating truths about the next frontier—BB(6)—and how it just might represent the very edge of mathematics, according to a new report by New Scientist.
First, a brief explanation. The aforementioned BB(1), which is the simplest version of the BB(n) problem, uses just one set of rules and produces only two outcomes—infinitely moving across the tape, or stopping at the first number. Because 1 is the most amount of steps that any of the 25 Turing machines of BB(1) will complete before finishing its program (known as halting), the answer to BB(1) is 1. As the number of states increases, so do the steps and the number of Turing machines needed to run the programs, meaning that each subsequent BB(n) is exponentially more taxing to solve. BB(2) and BB(3) are 6 and 21 respectively, but BB(4) is 107 and takes seven billion different Turing machines to solve. Granted, many of these machines continue on indefinitely and can be discarded, but many do not.
The Busy Beaver number was first formulated by the Hungarian mathematician Tiber Radó in 1962, and 12 years passed before computer scientist Allen Brady determined that BB(4) runs for 107 steps before halting. For decades, this seemed like the absolute limit of what was discernible, but then mathematicians solved BB(5) in 2024 after sifting through 17 trillion (with a t) possible Turing machines. The answer? An astounding 47,176,870 steps. Quanta Magazine has an excellent explainer about how this was achieved.
But finally solving BB(5) presented the next obvious question: What about BB(6)? Of course, adding just one more rule makes the problem beyond super exponentially harder, as BB(6) is estimated to require 60 quadrillion Turing machines.
“The Busy Beaver problem gives you a very concrete scale for pondering the frontier of mathematical knowledge,” computer scientist Tristan Stérin, who helped start the Busy Beaver Challenge in 2022, told New Scientist.
In a new post, anonymous user “mxdys”—who was instrumental in finally confirming BB(5)—wrote that the answer to BB(6) is likely so unfathomably large that the number itself likely needs its own explanation, as it’s likely too big to describe via exponentiation. Instead, it relies on tetration (written as, say, yx, as opposed to exponentiation’s xy) in which where the exponent is also iterated, creating a tower of exponents. As Scott Aaronson, an American computer scientist who helped define BB(5), notes on his blog, that means 1510 can be thought of as “10 to the 10 to the 10 and so on 15 times.”
As mxdys notes, BB(6) is at least 2 tetrated to the 2 tetrated to the 2 tetrated to the 9. One mathematician speaking with New Scientist said that it’s likely that the number of all atoms in the universe would look “puny” by comparison. While these large numbers boggle the mind, they also tell mathematicians about the limitations of the foundation of modern mathematics—known as Zermelo–Fraenkel set theory (ZFC)—as well slippery mathematical concepts like the Collatz conjecture.
It’s unlikely that mathematicians will ever solve BB(6), but if the Busy Beaver Challenge is any evidence, that fact likely won’t stop them from trying.
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