Probability One

Number guessing is a popular game among elementary school children. Teachers encourage students to play this game as it develops their arithmetic skills, logical thinking, and ability to follow simple procedures. You will probably master it within a few minutes. Here is an example of how to play this game: Ask a friend to think of a number, let's call it n_0.

Then:

1. Ask your friend to compute n_1 = 3 \cdot n_0 and tell you if n_1 is even or odd.

2. If n_1 is even, ask your friend to compute n_2 = n_1 / 2. Otherwise, if n_1 is odd, ask your friend to compute n_2 = (n_1 + 1) / 2.

3. Now ask your friend to calculate n_3 = 3 \cdot n_2.

4. Ask your friend to tell you the result of n_4 = n_3 / 9. (n_4 is the quotient of the division. In computer terminology, "/" is the integer division operator.)

5. Now you can simply reveal the original number by calculating n_0 = 2 \cdot n_4 if n_1 was even, or n_0 = 2 \cdot n_4 + 1 otherwise.

Here is an example: If n_0 = 37, then n_1 = 111 which is odd. Now we compute n_2 = 56, n_3 = 168, and n_4 = 18, which is what your friend will tell you. Calculating 2 \cdot n_4 + 1 = 37 reveals n_0.