Question 98
Daniel invented a game allowing only these operations:
- add $4$;
- multiply by $4$;
- square.
Starting from one number, Daniel challenges a friend to obtain another by successively applying any allowed operations.
For example, Daniel challenged Alan to obtain $152$ from $3$. Alan succeeded as follows:
(a) Daniel challenged Alan to obtain $340$ from $3$. Alan succeeded using
$$3\longrightarrow9\longrightarrow81\longrightarrow85\longrightarrow340.$$
Describe the operation at each step.
(b) Show Alan could also obtain $340$ starting from $5$.
(c) Suppose Alan starts from a number whose remainder on division by $4$ is $1$. Show that after any step the number obtained can have only remainder $1$ or $0$.
(d) Show it is possible to obtain $43$ from $3$, but not from $5$.
- add $4$;
- multiply by $4$;
- square.
Starting from one number, Daniel challenges a friend to obtain another by successively applying any allowed operations.
For example, Daniel challenged Alan to obtain $152$ from $3$. Alan succeeded as follows:
(a) Daniel challenged Alan to obtain $340$ from $3$. Alan succeeded using
$$3\longrightarrow9\longrightarrow81\longrightarrow85\longrightarrow340.$$
Describe the operation at each step.
(b) Show Alan could also obtain $340$ starting from $5$.
(c) Suppose Alan starts from a number whose remainder on division by $4$ is $1$. Show that after any step the number obtained can have only remainder $1$ or $0$.
(d) Show it is possible to obtain $43$ from $3$, but not from $5$.