## USACO 2012 January Contest, Gold Division

## Problem 2. Cow Run

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Problem 2: Cow Run [Mark Gordon, 2011]
Farmer John and Bessie have devised a new exercise game for the cows. The
cows are running on a circular track of length M (2 <= M <= 1,000,000,000)
starting from the same position. The game proceeds in N (1 <= N <= 14)
rounds using a deck of 8N cards each with a number X_i (0 <= X_i < M)
written on it.
Each round FJ moves the top 8 cards into a separate pile and selects either
the top 4 or bottom 4 cards for Bessie to play with. Bessie then chooses
either the top 2 cards or bottom 2 cards of the 4 cards FJ selected. After
this FJ calls out the number on the top card, X_top, and the cows run a
distance of R * X_top, where R is the total distance the cows have run so
far. Bessie then calls out the number on the bottom card, X_bottom, and the
cows run a distance of X_bottom.
FJ is concerned that after the exercise the cows will be too tired to get
back to the beginning of the track if they end up too far away. He
believes if the cows end up more than a distance of K (0 <= K <=
floor(M/2)) from their starting position they won't be able to get back
home.
It is guaranteed that if FJ plays correctly, he will always be able to
ensure the cows can come home, irrespective of the moves made by Bessie!
For each round, your task is to determine which half of the cards FJ should
choose such that no matter what Bessie does from that point on, FJ can
always get the cows home. Bessie will then make the move provided in the
input and you can then continue onto the next round. Note that even though
Bessie's moves are provided to you in the input, you are to specify moves
for FJ that would have worked no matter what Bessie chooses (so it is
effectively as if FJ does not really know what Bessie will do during her
moves).
PROBLEM NAME: cowrun
INPUT FORMAT:
* Line 1: Three space-separated integers N, M, K
* Line 2: A string N characters. If the ith character is 'T' it means
Bessie will select the top 2 cards in the ith round. Otherwise
the ith character will be 'B' to indicate Bessie will select
the bottom 2 cards in the ith round.
* Lines 3..2+N: Each line contains eight integers representing the 8
cards to be used that round from top to bottom.
SAMPLE INPUT (file cowrun.in):
2 2 0
TT
1 0 0 0 0 0 0 1
0 1 1 1 0 0 1 0
OUTPUT FORMAT:
* Line 1: A string of N characters where the ith character is 'T' if
FJ should choose the top 4 cards or 'B' if FJ should choose
the bottom 4 cards in the ith round. If there are multiple
ways to get the cows home, choose the lexicographically first
(that is, the string that is alphabetically smallest).
SAMPLE OUTPUT (file cowrun.out):
TB
OUTPUT DETAILS:
The cows must end up exactly where they started to be able to come home.
Note that FJ is not aware of what choices Bessie is going to make
beforehand. If FJ did know, he could have chosen the bottom half each time.

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