
Football 
Time Limit: 1000ms, Special Time Limit:2500ms, Memory Limit:32768KB 
Total submit users: 35, Accepted users: 32 
Problem 10045 :
No special judgement

Problem description 
Consider a singleelimination football tournament involving 2n teams, denoted 1, 2, . . . , 2n. In each round of
the tournament, all teams still in the tournament are placed in a list in order of increasing index. Then, the
first team in the list plays the second team, the third team plays the fourth team, etc. The winners of these
matches advance to the next round, and the losers are eliminated. After n rounds, only one team remains
undefeated; this team is declared the winner.
Given a matrix P = [pij] such that pij is the probability that team i will beat team j in a match determine
which team is most likely to win the tournament.

Input 
The input test file will contain multiple test cases. Each test case will begin with a single line containing n
(1 ¡Ü n ¡Ü 7). The next 2^n lines each contain 2^n values; here, the jth value on the ith line represents pij . The
matrix P will satisfy the constraints that pij = 1.0 − pji for all i!=j, and pii = 0.0 for all i. The endoffile
is denoted by a single line containing the number 1.
Note that each of the matrix entries in this problem is given as a floatingpoint value. To avoid precision
problems, make sure that you use either the double data type instead of float.

Output 
The output file should contain a single line for each test case indicating the number of the team most likely
to win. To prevent floatingpoint precision issues, it is guaranteed that the difference in win probability for
the top two teams will be at least 0.01.

Sample Input 
2
0.0 0.1 0.2 0.3
0.9 0.0 0.4 0.5
0.8 0.6 0.0 0.6
0.7 0.5 0.4 0.0
1 
Sample Output 
2 
Judge Tips 
In the test case above, teams 1 and 2 and teams 3 and 4 play against each other in the first round; the
winners of each match then play to determine the winner of the tournament. The probability that team 2
wins the tournament in this case is:
P(2 wins) = P(2 beats 1)P(3 beats 4)P(2 beats 3) + P(2 beats 1)P(4 beats 3)P(2 beats 4)
= p21*p34*p23 + p21*p43*p24 = 0.9*0.6*0.4 + 0.9*0.4*0.5
= 0.396.
The next most likely team to win is team 3, with a 0.372 probability of winning the tournament.

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