Teacher Raquel invented the following game: write a number on the board, if it is odd, add 3 units to the number, and if it is even, divide the number by 2.

This operation can be performed multiple times. The teacher is interested in ending up with the number 1 and asked the class: How to get the number 1 after 3 operations? And after 4 operations? And after 5 operations?
(a) The number 1 can only be obtained from $2 \leadsto 1=2 \div 2$, and the number 2 can only be obtained from $4 \leadsto 2=4 \div 2$, but the number 4 can be obtained from $1 \leadsto 1+3=4$ or $8 \sim 4=8 \div 2$.

Therefore, we have 2 ways to obtain 1, starting from 1 and 8 after 3 operations:

\[
\left\{\begin{array}{l}
1 \leadsto 4 \leadsto 2 \leadsto 1 \\
8 \leadsto 4 \leadsto 2 \leadsto 1
\end{array}\right.
\].

(b) For one more operation, we see that the number 8 can be obtained from $5 \sim 8=5+3$ or $16 \sim 8=16 \div 2$. Therefore, we have 3 ways to obtain 1 starting from

![](https://cdn.mathpix.com/cropped/2024_05_01_dbce7bc2a200e234c3e2g-36.jpg?height=176&width=873&top_left_y=2010&top_left_x=412)

(c) Similarly, we see that for 5 operations, we have the numbers: $4,10,13$

![](https://cdn.mathpix.com/cropped/2024_05_01_dbce7bc2a200e234c3e2g-36.jpg?height=217&width=914&top_left_y=2231&top_left_x=413)