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

Certified progress

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. In February 2025, @mxdys ran a decider pipeline in Coq that resulted in only 173 holdouts. Since then, additional machines have been proven in Coq using both deciders and individual proofs.

On 29 Mar 2025, @mxdys published a list of 83 holdouts that withstood state-of-the-art Coq deciders. Some of these machines were already decided before.

Holdouts

This section is based on @mxdys's March 2025 holdout list of 83 TMs.

Cryptids

• 1RB3RB---3LA1RA_2LA3RA4LB0LB0LA Hydra
• 1RB3RB---3LA1RA_2LA3RA4LB0LB1LB Bonus cryptid

Unsolved

• 1RB---4LB1RA4RA_2LB2LA3RA4LB0RB
• 1RB---4LB0LA4RA_2LB2LA3RA4LB0RB
• 1RB---0RB0LA2RA_2LB2LA3RA4LB0LB Shift overflow counter, potential cryptid
• 1RB3LA1LA1RA3RA_2LB2RA---4RB1LB Potential cryptid
• 1RB3LA1LA1RA1RA_2LB2RA---4RB1LB Potential cryptid
• 1RB3LA1LA4LA2RA_2LB2RA---0RA0RB
• 1RB4RA1LA4RB2LA_2LB3LA1RB2RA---
• 1RB---3RA2LA2RB_2LB3LA4LB4RA0RA
• 1RB4LA1RA1RB1LA_2LB3LA---4RA2RB BMO 3 variant?
• 1RB---4RB2RB4LA_2LB3LA3LB0RA0RB Bouncer + chaotic counter
• 1RB2RB3LA4LA1LA_2LB3RA---4RA1RB
• 1RB3RB3LA4LA2RB_2LB3RA---1RA1LA
• 1RB2LA4LA1RA1LA_2LB3RB4RB---2RA
• 1RB4RB4RA1LA3LA_1LB2LA3RB2RB---
• 1RB3RA3RB4LA1LA_1LB2LA1LA---1RB Potentially halt
• 1RB2RA4LA1RB4RB_1LB2LA3RA---0LB
• 1RB3RB1LA2LA3RA_1LB2RA4RB0LA---
• 1RB3LA1LA2RB2LB_1LB2RA4RA0RB---
• 1RB2LA0RB0LB3LB_2LA4RB3RA0RA---
• 1RB2RA3LB---2LB_2LA0LA4RB0RB1LA 30% chance of beating current champion
• 1RB2RA3LB4LA---_2LA0RB1LA2RB0RA Analysis by @dyuan01 and @Legion
• 1RB2RA3LA---2LB_2LA4RA4RB0RB0LA Spaghetti analysis by @nerdyjoe
• 1RB2RA3LA4LA2RB_2LA---3LB1RA3RA Bouncer + chaotic counter
• 1RB2RA3LA4LA2RB_2LA3RB---0RA1LA Longitudinal analysis suggests chaotic
• 1RB3LA1RA4LA2RA_2LA---1LA0RA3RB Longitudinal analysis suggests chaotic
• 1RB3LA3LB0RB0LA_2LA4RB1LB1RA---
• 1RB3LA1LA2RB2RA_2LA4RA3LB1RA---
• 1RB3LA3LA0RB2LB_2LA4LA4RA2RA---
• 1RB3RA2LB1LB1RB_2LA2RA4LA1LA— Skelet 17-like
• 1RB3RA4LB2RA2RB_2LA---3LA0LB1LA
• 1RB0LB2LA4LB3LA_2LA---3RA4RB2RB
• 1RB2LB3LA0RA1LB_2LA4RA3RB3LA--- Analysis by @nerdyjoe
• 1RB2RB---0LB3LA_2LA2LB3RB4RB1LB Longitudinal analysis suggests chaotic
• 1RB2RB4LA2RA1LA_2LA4RA3LA---3RA Longitudinal analysis suggests chaotic
• 1RB3LB---4LA1RB_2LA4LA4LB3RB1RA Longitudinal analysis suggests chaotic
• 1RB3LB0RB---2LB_2LA3RA4RB2RB0LA
• 1RB3LB4LA0LB---_2LA0LA1RB0RA3RA
• 1RB3RB1LB---2RB_2LA1RA4LB2LA2RA Skelet 17-like
• 1RB3RB---4RA2RA_2LA2RA3LB4LB1LB
• 1RB3RB---0RA2RB_2LA4RA3LB1LB1LA
• 1RB3RB1LB2RA---_2LA2RB1LA4LB0RA
• 1RB2LA1RA---1LA_1LA4RB3LB0RB2RB Fractal?
• 1RB2LA3LA4RA0LA_1LA3RB1RB1LB--- Fractal?
• 1RB2LA0RB1LA3LB_1LA3LB1RA4RA---
• 1RB2LA0RB4LB1RA_1LA3RA1RA---0LA
• 1RB2LA0RB1LB---_1LA3RA1RA4RB0LB
• 1RB2LA0RB1LB---_1LA3RA1RA4LB2RB
• 1RB2LA0RB1LB---_1LA3RA1RA4LB2LB
• 1RB2LA0RB1LB---_1LA3RA1RA4LB1RB
• 1RB2LA0RB1LB---_1LA3RA1RA4LB1LB
• 1RB2LA0RB1LB---_1LA3RA1RA4LB0RB
• 1RB2LA0RB1LB---_1LA3RA1RA4LB0LB
• 1RB2LA0RB1LB---_1LA3RA1RA4LB3RA
• 1RB2LA0RB1LB---_1LA3RA1RA4LB2RA
• 1RB2LA0RB1LB---_1LA3RA1RA4LB2LA
• 1RB2LA0RB1LB---_1LA3RA1RA4LB1RA
• 1RB2LA0RB1LB---_1LA3RA1RA4LB1LA
• 1RB2LA0RB1LB---_1LA3RA1RA4LB0RA
• 1RB2LA0RB1LB---_1LA3RA1RA4LB0LA
• 1RB2LA0RB1LB0LB_1LA3RA1RA4RA---
• 1RB2LA0RB---4LA_1LA3LA1RA4RA1LB
• 1RB2LA0RB4LB0LA_1LA3LA1RA4RA---
• 1RB2LA4RA1LA3LA_0LA2RB3RB2LB--- 1D CA-like
• 1RB2LA4RA1LA3LA_0LA3RB3LB2RB--- 1D CA-like
• 1RB2LA1LA4RA2LA_0LA3RB3LB2RB--- 1D CA-like
• 1RB2LA3LA4RA1LA_0LA3LB3RB1RB--- 1D CA-like
• 1RB2LA0LB1LA2RA_0LA3RA1RA4LB— Potential bouncer
• 1RB2LA3LB4LB---_0LA4LB3RA4LA0RB Fractal?

Solved with moderate rigor

• 1RB1RB3LA4LA2RA_2LB3RA---3RA4RB BMO problem 3
• 1RB0RB3LA4LA2RA_2LB3RA---3RA4RB BMO problem 3
• 1RB4LA1LB2LA0RB_2LB3RB4LA---1RA Nonhalting argument by @dyuan
• 1RB3LA4RB0RB2LA_1LB2LA3LA1RA--- Current champion
• 1RB2RA3LA4RB---_2LA3RB3RA1LB3LB Counter that halts in at most 56452325275 steps
• 1RB2RA3LA4LA2RB_2LA---1LA1RA3RA Longitudinal analysis by @Legion implies halting
• 1RB3LA4LA1LA2RA_2LA4RB---0RA0LA Longitudinal analysis by @Legion implies halting
• 1RB3LA4LA2RB1LA_2LA4RB---3RA3LA Longitudinal analysis by @Legion implies halting
• 1RB2LB---4LB0RB_1LA3RB4RB4RA1LB Nonhalting argument by @racheline

Formally proven

• 1RB2RA3LA4LA2RB_2LA0RA---0RA1LA  Coq-decided by @mxdys. Longitudinal analysis by @Legion implies nonhalting