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.