The Busy Beaver Problem
A NEW MILLENNIUM ATTACK
Christmas Tree Non-Halters
In a general sense, a christmas tree non-halter sweeps back and forth across the tape in a repeatable manner. More specifically, at some point during machine execution, the tape has the following structure: 0*[U][Vs]0* (see the discussion on simple loops for an explanation of this particular tape represenation). Then at some later point, the tape has the structure: 0*[U][X][Vs]0*. This does not complete the proof however as it must be shown that the machine progresses through a series of specific transformations such that additional [X]'s are repeatedly added to the middle of the structure ad infinitum. See Owen's non-halter presentation for details on the christmas tree detection strategy.
References
Brady. "The Determination of the Value of Rado's Noncomputable Function Sigma(k) for Four-State Turing Machines" Mathematics of Computation Volume 40, Number 162. April 1983, pp 647-665

Machlin and Stout. "The Complex Behavior of Simple Machines" Physica D 42 (1990). pp. 85-98
Non Halt Detection
Project Components
Current Champions
n B(n) b(n) O(n) o(n)
1 1T 1T 1T 1T
2 2T 3T 2T 3T
3 3? 13R 3O 13R
4 5? 31R 8O 37R
5 11? 57R 15O 111O
6 25M 255M 239R 41606R
7 196M 13682M

8 672M 198339M


n P(n) p(n) R(n) r(n)
1 1T 2T 1T 2T
2 2T 4T 2T 4T
3 4R 14R 4R 14R
4 7R 32R 8R 38R
5 16R 112R 16R 112R
6 163R 27174R 240R 41607R

RSet by present RPI effort
MSet by Machado and Pereira
OSet by Oberschelp, et al.
TTrivial records
?Unknown origin

Also note: A solid yellow background indicates records have been explicitly confirmed by the present effort. A faded yellow indicates relative confidence but not yet an explicit proof.
Busy Beaver Research Team
Bram van Heuveln
Selmer Bringsjord
Boleslaw Szymanski
Carlos Varela
Kyle Ross
Owen Kellett
Shailesh Kelkar