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Troy made the following problem (titled WA) for a programming contest:
There is a game with N levels numbered from 1 to N. There are two characters, both are initially at level 1. For i < j, it costs Ai,j coins to move a character from level i to level j. It is not allowed to move a character from level i to level j if i > j. To win the game, every level (except level 1) must be visited by exactly one character. What is the minimum number of coins needed to win?
JP is a contestant and submitted the following Python solution.
def Solve(N, A): # A[i][j] is cost of moving from level i to level j # N is the number of levels x, y, sx, sy = 1, 1, 0, 0 # Initialize x and y to 1, sx and sy to 0 for i in range(2, N + 1): # loop from 2 to N if sx + A[x][i] < sy + A[y][i]: sx += A[x][i] x = i else: sy += A[y][i] y = i return sx + sy
Troy is certain that JP’s solution is wrong. Suppose for an input to WA, JP’s solution returns X but the minimum number of coins needed is Y. To show how wrong JP’s solution is, help Troy find an input N and Ai,j such that X/Y is maximized.
There is no input.
Print an input to WA in the following format:
On the first line, print one integer N (2 ≤ N ≤ 100). Then print N−1 lines; the i-th line should contain N−i integers Ai,i+1, · · ·, Ai,N (1 ≤ Ai,j ≤ 100). If your output is not the correct format, it will get an incorrect verdict on the sample test in the grader and score 0 points.
Otherwise, suppose that for your input, JP’s solution returns X but the minimum number of coins needed is Y. Then you will receive X/(4Y) points.
5 1 2 3 4 10 9 8 7 6 5
The optimal way to win the game is for one character to visit level 2 and the other character to visit levels 3, 4 and 5. This costs (1) + (2 + 7 + 5) = 15 coins. JP’s solution returns 18. Thus X/(4Y) = 18/(4×15) = 0.3, so this output will receive 0.3 point.