Tiny Tag

On December 2nd, 2025, Discord User Jack created a variant of Cyclic Tag that operates under a singular binary string known. It is known as "Tiny Tag" and is similar to Self-Bitwise Cyclic Tag (SBCT). Its strength is not known.

Definitive Statements

b is an initial binary string,

x is the leftmost bit of b,

i=(1,2,…,k) is an instruction counter (initially 1),

p₁,p₂,…,pₖ are the runs of b (also known as: the “rules”).

(NOTE: The “runs of b” are the maximal consecutive sequence of the same bit. Ex. b=10011101 has rules/runs 1,00,111,0,1)

Instructions

While b is not empty (b≠∅), and the current rule is pᵢ:

(1) If x=0, delete x and append nothing. If x=1, delete x and append rule pᵢ’s string to the end of b,

(2) Advance to rule (i mod k)+1,

(3) Repeat.

Example

b=010111 (therefore, p=0,1,0,111), results in the following:

010111 (Initial b)

10111

01111

1111

111111

111110

111101

111010

11010111

10101110

01011101

1011101

011101111

11101111

11011111

10111110

0111110111

111110111

111101111

111011110

11011110111

…

…

...

Reaches ∅ in 334 total steps (rule applications)!

Relation to BB(n)

From here, we can define a Busy Beaver function as the “longest finite number of steps required for a binary string of length n to reach empty.” Call this function f(n).

here is a list of known values followed by their initial string b:

f(1)=1 (0)

f(2)=5 (10)

f(3)=9 (101)

f(4)=13 (1001)

f(5)=15 (10101)

f(6)=334 (010111)

f(7)=404 (1010111

f(8)=670 (11100101)

f(9)=12584 (001101110)

f(10)=2180995 (0100011110)

Values for n>10 are not currently known. However, an extremely weak lower bound of f(48)>>971038 (with an initial b of 010111...010111 (with 8 total 010111's)) was given on the date the definition of Tiny Tag was originally published.

Number of "Programs"

This is trivial. there are 2ⁿ total binary strings of length n.