Arithmetic problem Ivan and Adeline

Question

Ivan and Adeline are in a classroom with a chalkboard. They are standing on different halves of the board, and on each half, the number $2$ is written. When Ivan's teacher gives a signal, Ivan multiplies the number on his side of the board by $-2$ and writes the answer on the board, erasing the number he started with. Adeline does the same on each signal, except that she multiplies by $2$. The teacher gives 10 signals in total. How many times (including the initial number) do Ivan and Adeline have the same number written on the board (including at the beginning)?

Answer 1

The pattern is

0) $2 \quad2$

1) $-4 \quad 4$

2) $8 \quad8$

...

10) $2^{11} \quad 2^{11}$

thus 6 times in total (including the start).

In general after the $k^{th}$ signal we have on the board:

$$(-1)^{k}2^{k+1} \quad2^{k+1}$$

and thus they are equals if and only if for $k=0$ or even.