1RB1RC_1LC1LE_1RA1RD_0RF0RE_1LA0LB_---1RA

Unsolved problem:

Does this TM run forever?

1RB1RC_1LC1LE_1RA1RD_0RF0RE_1LA0LB_---1RA (bbch) is a BB(6) Cryptid found by @mxdys and shared on Discord on 27 Jul 2024. It's closely related to Hydra and Antihydra.

Xl643 Calls this the Ambihydra.

Analysis by @mxdys

antihydra variant:
1RB1RC_1LC1LE_1RA1RD_0RF0RE_1LA0LB_---1RA

(n,m) := 0^inf C> 01011 01^n 1 01^m 0^inf

start from (3,1)
(2n+1,m) --> (3n+3,m+2)
(2n,m+1) --> (3n+3,m)
(4n,0) --> (9n+6,1)
(4n+2,0) --> halt

The rules are proved in Coq.

Analysis by @dyuan01

Using B(a, b) = (3a-6, b), I've come up with these rules:

Start from B(3, 1)
B(2n, m+1) → B(3n, m)
B(2n+1, m) → B(3n+1, m+2)
B(4n+2, 0) → B(9n+4, 1)
B(4n, 0) → halt

This is actually just Hydra except you start at B(3, 1) instead of B(3, 0) and you don't necessarily halt if you reach -1. In fact, this can only halt if Hydra halts.