TMBR: December 2025

Prev: November 2025

This Month in Beaver Research

Next: January 2026

This edition of TMBR is in progress and has not yet been released. Please add any notes you think may be relevant (including in the form a of a TODO with a link to any relevant Discord discussion).

This is the last edition of TMBR this year. 2025 was a very productive year for BBChallenge: about 60% of the next domain, BB(6), was solved. Furthermore, new champions were discovered for BB(6), BB(7) and BB(4,3). Many models of computation other than Turing Machines were also explored - most notably Fractran and Instruction-Limited Busy Beaver. Some new methods were developed, such as mxdys's new version of FAR.

This Year in Beaver Research _{(TYBR - "Thank You Beaver Researchers!")}

Holdouts Reductions.

• BB(6) - Reduced from 3571 to 1416 holdouts. Hence, 2155 machines were solved this year. This is a 60% reduction.
• BB(7) - Enumeration was completed, the number of holdouts was reduced from an initial 85,853,789 to 20,405,295 machines, a 76.23% reduction.
• BB(4,3) - Reduced from 460,916,384 to 9,401,447 holdouts, a 97.96% reduction.
• BB(3,4) - Reduced from 434,787,751 to 15,136,283 holdouts, a 96.52% reduction.
• BB(2,5) - Reduced from 217 to 75, a 65.43% reduction.
• BB(2,7) - Enumeration started, 50K of the 1M subtasks have been enumerated (5%).

Champions.

• BB(6) - On 16 June 2025, mxdys discovered 1RB1LC_1LA1RE_0RD0LA_1RZ1LB_1LD0RF_0RD1RB (bbch), running for 10 ↑↑ 11010000 steps. This was surpassed on 25 June when mxdys discovered 1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE (bbch), a TM which runs for $10\uparrow \uparrow 10\uparrow \uparrow 10\uparrow \uparrow 8.10237$ steps.
• BB(7) - Within three days of the start of the enumeration of BB(7), three champions were discovered. The first two were discovered by Shawn Ligocki: 1RB0RF_1LC0RE_1RD1LB_1LA1LD_0RA0LE_1RG0LB_1RZ1RB (bbch) with a sigma score of about 10 ↑↑ 22 and 1RB1RA_1RC0LC_0LD1LG_1LF0LE_1RZ1LF_0LA1LD_1RA1LC (bbch) with a sigma score of about 10 ↑↑ 35. This was followed by the discovery of 1RB0LG_1RC0RF_1LD1RZ_1LF0LE_1RA1LD_1LG1RE_0LB0LB (bbch), achieving a sigma score of about 10 ↑↑ 46, by Terry Ligocki. On 10 May 2025, Pavel Kropitz discovered 1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF (bbch), a TM which runs for over $2{\uparrow }^{11}2{\uparrow }^{11}3$ steps.
• BB(4,3) - Polygon identified a new BB(4,3) champion with a score of over $10{\uparrow }^{4}4$ (1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD (bbch)). This TM was first proven to halt by Pavel Kropitz in May 2024, but its runtime was not known at the time.

New Methods.

TODO

Meta.

TODO

BB Adjacent.

• Instruction-Limited Busy Beaver was introduced and calculated up to BBi(7).
• Reversible Turing Machine Busy Beaver values were calculated up to BB_rev(5).
• Terminating Turmites (Relative Movement Turing Machines) were introduced.
• John Tromp introduced the $BB{\lambda }_1(n)$ function for Busy Beaver for lambda calculus with an oracle and computed it up to $BB{\lambda }_1(22)$.
• Instruction-Limited Greedy Busy Beaver gBBi(n) and an Instruction-Limited variant of the Blanking Busy Beaver (BLBi(n)) were introduced. gBBi(n) was computed up to n = 13 and BLBi(n) was computed up to n = 7.
• @savask shared the Bug Game (and fast-growing $Bug(H,W)$ function).
• Busy Beaver for Fractan (BBf) was introduced on 1 Nov by Jason Yuen.^[1] Exact values have been proven up to BBf(19) = 370 and exhaustive enumeration has been run up to size 21 (with BBf(21) ≥ 31,957,632 and 587 holdouts).
• Cyclic Tree Busy Beaver (CTBB) was introduced by @Jack on 14 Nov.^[2] The exact value is known for CTBB(2) = 5 and lower bounds have been found up to size 7 with CTBB(7) > 4↑↑↑↑(4↑↑↑3).

TODO

In the News.

• 1 July 2025. The Quanta Podcast. How Amateurs Solved a Major Computer Science Puzzle.
• 2 July 2025. Manon Bischoff. Spektrum. Wie der sechste Fleißige Biber die Mathematik an ihre Grenzen bringt.
• 7 July 2025. Karmela Padavic-Callaghan. New Scientist. Mathematicians are chasing a number that may reveal the edge of maths. (Paywalled)
• 9 July 2025. David Roberts. BB(5)=47,176,870: BB(6) is … astronomically larger.
• 11 July 2025. New Scientist podcast episode 311. Discusses mxdys's BB(6) pentation result "We’re brushing up against the edge of mathematics".
• 11 July 2025. Darren Orf. Popular Mechanics. Mathematicians Say There’s a Number So Big, It’s Literally the Edge of Human Knowledge.
• 18 July 2025 https://francis.naukas.com/2025/07/18/espeluznante-nueva-cota-inferior-para-la-funcion-castor-afanoso-bb6/
• 22 Aug 2025. Ben Brubaker. Quanta Magazine. Busy Beaver Hunters Reach Numbers That Overwhelm Ordinary Math.
• 14 Sep 2025. Ben Brubaker. Wired. The Quest to Find the Longest-Running Simple Computer Program. (Reprint of Quanta article from last month).
• 17 Sep 2025. Hacker News. Determination of the fifth Busy Beaver value.
• 18 Sep 2025. Tuomas Kangasniemi. Tekniikkatalous. Iso matematiikan ongelma ratkesi 63 v jälkeen (Finnish) (English: A big math problem solved after 63 years).

TODO

BB Adjacent

TODO. Register machines, General Recursive Functions.

Holdouts

• BB(6):
  • There are 14 holdouts left to simulate up to 1e12 steps, and 312 to simulate up to 1e13 steps^[1]. The two lists can be found here.
• BB(3,4):
  • TODO: Phase 2 stage 8.
• BB(2,7):
  • Terry Ligocki enumerated 10K more subtasks, increasing the number of holdouts to 150,662,006 and making 50K of the 1 million subtasks enumerated or 5%.