Question 83
Let $N_1 = 12$. The goal in this problem is to obtain the number $N_k = 21$ after performing a finite number of steps according to the following rules. In each step, the numbers $N_i$ and $N_{i+1}$ need to be in *Shamrock Relation*. Two numbers are said to be in *Shamrock Relation* if and only if
(a) one number is obtained from the other by replacing any block $X$ of consecutive digits by its triple repetition $XXX$; or
(b) one number is obtained from the other by replacing any connected block of the form $XXY$ with $YYY$, where $X$ and $Y$ represent blocks of consecutive digits.
Is the goal of obtaining 21 achievable?
(a) one number is obtained from the other by replacing any block $X$ of consecutive digits by its triple repetition $XXX$; or
(b) one number is obtained from the other by replacing any connected block of the form $XXY$ with $YYY$, where $X$ and $Y$ represent blocks of consecutive digits.
Is the goal of obtaining 21 achievable?