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 number of times Ivan and Adeline have the same number written on the board is 6.

Explanation:

Consider the procedure as follows:

• On each half of the board, the number 2 is written.
• On Ivan's teacher's 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.

Consider the numbers on each half of the board:

Ivan Adeline

2 2

2 × -2 = -4 2 × 2 = 4

-4 × -2 = 8 4 × 2 = 8

8 × -2 = -16 8 × 2 = 16

-16 × -2 = 32 16 × 2 = 32

32 × -2 = -64 32 × 2 = 64

-64 × -2 = 128 64 × 2 = 128

128 × -2 = -256 128 × 2 = 256

-256 × -2 = 512 256 × 2 = 512

512 × -2 = -1024 512 × 2 = 1024

-1024 × -2 = 2048 1024 × 2 = 2048

Thus, the number of times Ivan and Adeline have the same number written on the board is 6.