Question 9
A strange calculator has only digit keys $0$ to $9$ and two special keys $A,B$. Pressing $A$ squares the displayed number; pressing $B$ adds $3$. One can obtain $22$ from $1$ by pressing $BABB$:
$$1\xrightarrow B4\xrightarrow A16\xrightarrow B19\xrightarrow B22.$$
a) Starting with $3$, what number appears after pressing $BBAB$?
b) Show how to obtain $55$ from $1$ using $A,B$.
c) Explain why $54$ cannot be obtained from $2$ using $A,B$.
$$1\xrightarrow B4\xrightarrow A16\xrightarrow B19\xrightarrow B22.$$
a) Starting with $3$, what number appears after pressing $BBAB$?
b) Show how to obtain $55$ from $1$ using $A,B$.
c) Explain why $54$ cannot be obtained from $2$ using $A,B$.