1RB0LD_0RC1RB_0RD0RA_1LE0RD_1LF---_0LA1LA

1RB0LD_0RC1RB_0RD0RA_1LE0RD_1LF---_0LA1LA (bbch) (and its equivalent 1RB0LD_0RC0RF_0RD0RA_1LE0RD_1LF---_0LA1LA (bbch)) is an unsolved BB(6) TM. It is the 8th Beaver Math Olympiad problem.

Analysis by mxdys

https://discord.com/channels/960643023006490684/1239205785913790465/1402211374565953609

1RB0LD_0RC0RF_0RD0RA_1LE0RD_1LF---_0LA1LA

(n+a,b,2n+c) --> (a,n+b,c)

(0,b,c) --> (5+3b,0,c)

(1,10+b,0) --> halt
(1,10+b,1) --> halt

(2,b,0) --> (2,0,8+3b)
(2,b,1) --> (2,0,11+3b)

(3+a,b,0) --> (a,0,6+3b)
(3+a,b,1) --> (a,0,9+3b)

start: (10,0,12)

this machine is weird
it does't halt in 8.9e12 rules (the first rule doesn't count)
for small (a,0,c), none of them halts between 1e5 and 1e8 rules (use the simplified model that (1,b,0) and (1,b,1) halts)
for each visited (a,0,c), log(c/a) seems to have a good distribution, which means that it doesn't fall into a fixpoint quickly
and the (2,b,0),(2,b,1) rules are used for several times:
T=462: 2 673 1
T=41231: 2 59502 0
T=6056795: 2 8748186 1
T=45863796: 2 66242895 0
T=2012212661: 2 2906495658 1
T=10335020887: 2 14928329409 1
T=468871071868: 2 677258129269 1

Analysis by dyuan

https://discord.com/channels/960643023006490684/1468065303278911613

1RB0LD_0RC1RB_0RD0RA_1LE0RD_1LF---_0LA1LA 

A(a, b, c) = 1^4 0^a <A 1^3b 0 1^c 0 1^2
A(a+1, b, c+2) → A(a, b+1, c)
A(0, b, c) → A(3b+3, 0, c+2)
A(a+2, b, 0) → A(a, 0, 3b+5)
A(a+2, b, 1) → A(a, 0, 3b+8)
A(1, b+5, 0) → Halt (after some time)
A(1, b+4, 1) → Halt (after some time)

Start: A(4, 0, 11)


B(a, b) = A(a, 0, b)

B(a+b+2, 2b+1) → B(a, 3b+8)
B(b+1, 2b+1) → Halt
B(a+b+2, 2b) → B(a, 3b+5)
B(b+1, 2b) → Halt
B(a, 2a+b) → B(3a+3, b+2)

Start: B(4, 11)