Tiny Tag

On December 2nd, 2025, Discord User Jack created a variant of Cyclic Tag that operates on a singular known binary string. 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 “largest 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

Trivially, there are 2ⁿ total binary strings of length n.