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

Justin Blanchard generated a list of 499 holdouts in May 2024, which was cut down to 273 by an application of deciders written for BB(6), then again to 217 using CTL by the end of June 2024. This list of holdouts can be found here. Since then, between 50 and 100 machines have been claimed to have been solved, but no new reduced holdout list has been generated.