Most people don’t think about math when they hear “busy beavers.” But these animals represent a fascinating concept in the world of mathematics: not everything can be calculated, no matter how hard you try. The busy beaver function is an example of a noncalculable mathematical expression. It refers to the largest number of steps a computer program can take before stopping, based on the complexity of the problem.
For over 40 years, experts believed that the value of BB(5) was beyond the limits of computability and therefore unattainable. However, an international project called the Busy Beaver Challenge has successfully determined the value of BB(5) to be 47,176,870. This means that a program with five states can take a maximum of 47,176,870 steps before halting or running indefinitely. This achievement marks a significant breakthrough in the field of mathematics.
Mathematicians have long grappled with the concept of unprovable statements in mathematics, as demonstrated by Kurt Gödel in 1931. One such problem is the halting problem, which deals with the execution of algorithms. Alan Turing showed that there is no algorithm that can predict whether a computer program will run forever or eventually stop. This led to the development of the busy beaver game in 1962, where mathematicians sought to find the most diligent Turing machine of a certain size.
The search for busy beavers has led to the discovery of the longest-running programs for machines with different numbers of states. The latest achievement in determining the value of BB(5) highlights the complexity and significance of these calculations. The Busy Beaver Challenge project has provided a platform for mathematicians and computer scientists to collaborate and verify results in this challenging field.
Moving forward, the search for busy beavers with more states continues, with the record holder for Turing machines with six states performing an astronomical number of arithmetic steps. The quest to calculate BB(6) faces new challenges, with indications that it may be beyond reach. Despite the complexity and uncertainties in this field, mathematicians remain determined to push the boundaries of what is calculable and continue to explore the fascinating world of busy beavers.
