1RB1LD 1RC1RB 1LC1LA 0RC0RD

1RB1LD_1RC1RB_1LC1LA_0RC0RD (bbch)

Blanking Beaver BLB(4,2) champion which creates a blank tape after 32,779,477 steps. It was discovered and reported by Nick Drozd in 2021.

Analysis by Shawn Ligocki

Let $$D(a,b)=0^{\mathrm{\infty }}{1}^a{0}^b\text{<mrow data-mjx-texclass="ORD">D<mo>></mo></mrow>}{0}^{\mathrm{\infty }}$$then:$$\begin{array}{lccc}D(a+3,& b)& \to & D(a,b+5)\\ D(0,& b)& =& \text{<mrow data-mjx-texclass="ORD">Blank</mrow>}\\ D(1,& b)& \to & D(b+2,4)\\ D(2,& b)& \to & D(b+3,4)\\ \end{array}$$let $D(a)=D(a,4)$, then we can simplify to:

$$\begin{array}{lcc}D(3k)& \to & \text{<mrow data-mjx-texclass="ORD">Blank</mrow>}\\ D(3k+1)& \to & D(5k+6)\\ D(3k+2)& \to & D(5k+7)\\ \end{array}$$Starting from $D(2)$ (at step 19) we get the trajectory:

$$\begin{array}{lllllllllll}D(2)& \to & D(7)& \to & D(16)& \to & D(31)& \to & D(56)& \to & D(97)\\ & \to & D(166)& \to & D(281)& \to & D(472)& \to & D(791)& \to & D(1322)\\ & \to & D(2207)& \to & D(3682)& \to & D(6141)& \to & \text{<mrow data-mjx-texclass="ORD">Blank</mrow>}\\ \end{array}$$which has the remarkable luck of applying this Collatz-like map 14 times before reaching the blanking config (expected # of applications before halting is 3).

The map and trajectory are equivalent to that of the BB(5) champion. For all $k$, let $f(k)$ be the number such that $D(k)\to D(f(k))$, or $\text{HALT}$ if $D(k)\to \text{<mrow data-mjx-texclass="ORD">Blank</mrow>}$, and let $g$ be the map simulated by the BB(5) champion. Then:

$$\begin{array}{lcccccc}g(2(3k)+2)& =& g(6k+2)& =& \text{HALT}& =& f(3k)\\ g(2(3k+1)+2)& =& g(6k+4)& =& 10k+14& =& 2f(3k+1)+2\\ g(2(3k+2)+2)& =& g(6k+6)& =& 10k+16& =& 2f(3k+2)+2\\ \end{array}$$

So the size of this machine's BLB output is tied to the size of the BB(5) champion's output.