Here is a wheel with 10 numbers:

wheel

Suppose that the wheel begins in the position above. We are given an integer N ≥ 0. We will spin the wheel three times. On each of the three spins, the wheel will rotate counterclockwise by a number of positions randomly chosen (uniformly) from the integers 0 .. N. For example, if N = 3, then on each spin there is

a 1/4 chance that it will not rotate
a 1/4 chance that it will rotate by 1 position
a 1/4 chance that it will rotate by 2 positions
a 1/4 chance that it will rotate by 3 positions

What is the exact probability that after the 3 spins the wheel will end up in the same position in which it started, i.e. pointing to number 1?

Write a program that can answer this question. The program should read the integer N on a single line of standard input, and write the answer as a fraction in reduced form.

Sample input:

Output:

1/64

Explanation: When N = 3, the wheel cannot possibly make a complete rotation in 3 spins. So it will end up at the starting position only if it never moves. On each spin, the probability that it won't move is 1/4, so the probability that it never moves is

(1/4)(1/4)(1/4) = 1/64

Sample input #2:

Output:

1/10

Explanation:

When N = 9, the wheel will be in a (uniformly) random position after every spin. Therefore after 3 spins (or any number of spins, for that matter), the probability that it will be pointing to the number 1 is 1/10.

Hint:

Consider every possible sequence of spins.

MJay_Lee

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