BB(2,5)

The 2-state, 5-symbol Busy Beaver problem BB(2,5) is unsolved. With the discovery of the Cryptid machine Hydra by Daniel Yuan in April 2024, we now know that we must solve a Collatz-like problem in order to solve BB(2,5) and thus BB(2,5) is Hard.

The current BB(2,5) champion is 1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ (bbch), also discovered by Daniel Yuan in June 2024. It is a halting shift overflow counter, and is in fact the only known champion machine that exhibits Counter behavior. It provides the lower bound:

$$S(2,5)>\Sigma (2,5)>10^{10^{10^{3\,314\,360}}}>10\uparrow \uparrow 4$$

Cryptids

Known BB(2,5) Cryptids

• 1RB3RB---3LA1RA_2LA3RA4LB0LB0LA (bbch), known as Hydra.
• 1RB3RB---3LA1RA_2LA3RA4LB0LB1LB (bbch), known as the bonus cryptid.

Holdouts

In April 2024, Shawn Ligocki publicly released a list of 23,411 undecided BB(2,5) machines. Justin Blanchard then made substantial progress over the course of the next month, reducing the list to 499 holdouts by late May 2024. In June 2024, @mxdys cut down the list to 273 using halting and inductive deciders, and again to 217 using CTL. This list of holdouts can be found here.

In February 2025, @mxdys ran a decider pipeline in Coq that resulted in only 173 holdouts. This list has some overlap with previous lists due to 20 machines previously solved by Justin Blanchard not yet being implemented in Coq. Therefore, the list of holdouts as of February 2025 is as low as 153.