Darts#1254
After a long week of work at the ICPC Headquarters, Bill and his friends usually go to a small pub on Friday evenings to have a couple of beers and play darts. All of them know that their ability at darts decreases at the same rate as the amount of beer left in their mugs.
They always play 501, one of the easiest games. Players start with a score of (typically, N = 501, hence the name) and take turns throwing darts. The score of each player decreases by the value of the section hit by the dart, unless the score becomes negative, in which case it remains unchanged. The first player to reach a score of wins. The figure below shows the dartboard used in the game.
As midnight approaches and the beer runs out, everyone wonders: is it worth trying to aim the dart at a specific section? Or is it better to just throw the dart at a random section on the dartboard? You are asked to determine what would happen if two players (A and B), each applying one of these two strategies, play against each other:
• Player A throws darts randomly, so they land with equal probability in each of the sections on the dartboard. • If Player B aims at a certain section, the dart lands in the correct section or one of its two neighboring sections (to the left and right) with equal probability. Furthermore, he is fully aware of his skills and will aim at the section that maximizes his chance of winning.
Given the initial score for both players, can you determine the probability that the first player wins? Of course, being the one to throw the first dart may be advantageous, so the answer depends on who plays first.
The input consists of a series of lines, each containing an integer (1 \le N \le 501), the initial score for both players. The case N = 0 marks the end of the input and must not be processed.
For each number in the input, your program should output a line containing two real numbers: the probability that A wins if A throws the first dart, and the probability that B wins if B throws the first dart. Your answers should be accurate within an absolute or relative error of 10^{-8}.
Example 1