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:
- If x=0, delete x and append nothing. If x=1, delete x and append rule pᵢ’s string to the end of b
- Advance to rule (i mod k)+1
- 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:
| n | Value | Champion |
|---|---|---|
| 1 | 1 | 0 |
| 2 | 5 | 10 |
| 3 | 9 | 101 |
| 4 | 13 | 1001 |
| 5 | 15 | 10101 |
| 6 | 334 | 010111 |
| 7 | 404 | 1010111 |
| 8 | 670 | 11100101 |
| 9 | 12584 | 001101110 |
| 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.