Usaco Jan09 Bronze

Part of USACO Jan09

                           BRONZE PROBLEMS
                  Three problems numbered 11 through 13

Problem 11: Scrabble [Traditional, 2009]

Bessie is playing Bovine Scrabble which is just like regular Scrabble
except that a player's tray has T (3 <= T <= 20) letters instead
of always having 7 letters as in the normal game.

For those who haven't played English Scrabble, it is a word game
where each player has a set of T letters and uses them to form a
word to place on a gameboard which ends up looking a little bit
like a crossword puzzle. For this task, Bessie has the first play
and cares only to make a word from one or more of her letters
(without worrying how to fit it onto the board).

Each letter has a value (see the table below). A word's value, for
this task's purposes, is the sum of values of each of the letters.
Thus, the word "TAX" has three letters, each with a value: "T"
with 1 point, "A" with 1 point, and "X" with 8 point -- total: 10
points. No other bonus points are considered in this task. Bessie
might use one, some, or all of her letters to make the highest
scoring word.

This game also includes blank letters (which are represented as "#"
in the input). Blank letters can be used to represent any other
letter but receives 0 points no matter which letter is chosen. Only
three of the test data inputs will contain one or more blanks. Each
blank letter can, if desired, represent a different actual letter
for purposes of forming words.

Given T and a set of T letters, read the dictionary from file
dict.txt (which is alphabetized and has fewer than 25,000 words;
each word is no longer than 20 characters) and determine which is
the most valuable word that can be made from Bessie's letters. If
two words have the same value, choose the one that comes first

The actual dictionary used in this task is available at . Bessie can always find
some word in that dictionary to form with her letters.

Letter values:
     0 points:  # (blank)
     1 point:   A, E, I, L, N, O, R, S, T, U
     2 points:  D, G
     3 points:  B, C, M, P
     4 points:  F, H, V, W, Y
     5 points:  K
     8 points:  J, X
     10 points: Q, Z



* Line 1: A single integer: T

* Lines 2..T+1: Each line has a single character that appears in
        Bessie's letter tray




* Line 1: A single line with a single word (the word -- no '#' blanks)
        that is the highest-scoring word Bessie can make.

SAMPLE OUTPUT (file scrab.out):



Problem 12: Back to the Barn [Neal Wu, 2008]

As every cow does occasionally, Bessie has lost herself in the
woods! She desperately needs to get back to the barn but has no
idea where to go.

The woods can be thought of as an R x C grid (1 <= R <= 5; 1 <= C
<= 5). Bessie is located in the lower left corner at row 1, column
1; the barn is located in the upper right corner at in row R, column
C.  Each square in the grid is either empty (denoted by '.') or
blocked by a tree (denoted by 'T'). Bessie's square and the barn
will always be empty.

From a given square, Bessie may move to any adjacent square (one
that shares a long edge with Bessie's current square) as long as
it is empty. As Bessie is quite a smart cow, she never visits a
square twice on the way back to the barn.

Determine the number of different ways Bessie can take that lead
her from her initial position back to the barn while visiting at
most K different squares (1 <= K <= R * C).

By way of example, consider this forest:


Bessie has a number of ways to get to the upper right corner, here
are all seven of them and their path lengths (the number of squares

         cdef  ...f  ..ef  cdeh  cdej  ...f  
         bT..  .T.e  .Td.  .Tfe  bTfg  bTfi  .Tde  
         a...  abcd  abc.  abcd  a...  abc.  
Length:    6     6     6     8     8    10    6

PROBLEM NAME: backbarn


* Line 1: Three space-separated integers: R, C, and K

* Lines 2..R+1: Line i+1 contains the C characters representing row
        R+1-i of the forest (with no spaces)


3 4 6


The forest is a 3 x 4 grid. A single tree sits almost in the middle,
as shown. No more than six steps can be taken.


* Line 1: A single integer that is the number of different paths,
        visiting at most K squares, that Bessie can take to get back
        to the barn. Note that this number will always fit into a
        signed 32-bit integer.

SAMPLE OUTPUT (file backbarn.out):



Of the several evenays to complete the traversal, only four of them touch
no more than six squares along the route.


Problem 13: Music Notes [Neal Wu, 2008]

FJ is going to teach his cows how to play a song. The song consists
of N (1 <= N <= 100) notes, and the i-th note lasts for B_i (1 <=
B_i <= 100) beats. The cows will begin playing the song at time 0;
thus, they will play note 1 from time 0 to time B_1 - 1, note 2
fromtime B_1 to time B_1 + B_2 - 1, etc.

The cows have lost interest in the song, as they feel that it is
long and boring. Thus, to make sure his cows are paying attention,
FJ asks them Q (1 <= Q <= 1,000) questions of the form, "During the
beat at time T_i (0 <= T_i < length of song), which note should you be
playing?" The cows need your help to answer these questions.

By way of example, consider a song with these specifications: note
1 for length 2, note 2 for length 1, and note 3 for length 3:

NOTES    1   1   2   3   3   3
TIME     0   1   2   3   4   5



* Line 1: Two space-separated integers: N and Q

* Lines 2..N+1: Line i+1 contains a single integer: B_i

* Lines N+2..N+Q+1: Line N+i+1 contains a single integer: T_i


3 5


The song has 3 notes, with lengths 2, 1, and 3. FJ has 5 queries.


* Lines 1..Q: Line i contains the a single integer that is the note
        the cows should be playing at time T_i

SAMPLE OUTPUT (file mnoteb.out):



As per the chart above.

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