Another Version of the Truth
Influence is a board game; while it can be played on almost any layout,
an interesting layout is a hexagonal NxN grid, such that the layout resembles
An example 9x9 board is as follows:
1 2 3 4 5 6 7 8 9
\ \ \ \ \ \ \ \ \
* * * * * * * * * — A
* * * * * * * * * — B
* * * * * * * * * — C
* * * * * * * * * — D
* * * * * * * * * — E
* * * * * * * * * — F
* * * * * * * * * — G
* * * * * * * * * — H
* * * * * * * * * — I
On the above grid, F5 is adjacent to F4, F6, E5, E6, G4, and G5. (These
coordinates are not used in the problem, but are useful for understanding
the underlying adjacencies.)
The rules of Influence are simple; the pertinent details are as follows:
- Players take turns placing Manipulators on the board. Manipulators
occupy a single location, and there may be at most one Manipulator at a
location. If there is no empty location on the board to place a piece, the
player must pass their turn.
- Each player has a certain amount of Influence. A player has a single point
of Influence for every location on the board that is strictly closer to
one of their Manipulators than a Manipulator of any other player. This is not
"straight-line" distance, but the number of cells in a minimal path to a
Manipulator. On the above grid, the cell F5 is 2 steps away from G6, 1 step
away from G5, and 0 steps away from itself.
- The player with the most Influence at the end of the game wins.
In the sample input below, the first player (represented by "!" marks) has
only two Influence, that provided by the locations that his Manipulators are
on; the second player (represented by "@" signs) has ten Influence, and the
third player (represented by "#" marks) has four Influence. There are nine
locations that provide Influence to no player, as they are equally distant
from two or more of the players.
Given a particular board layout, answer this question: what would the
resulting Influence be for each player's optimal move if they were making the
last move in the game?
Moves are to be considered independently; that is, the
maximum score for the second player should be calculated based on the original
board layout, not the one after the first player's best move.
Input to this problem will begin with a line containing a single integer
N (1 ≤ N ≤ 100) indicating the number of data sets.
Each data set consists of the following components:
- A line containing a single integer P (2 ≤ P ≤ 4)
indicating the number of players in the game;
- A line containing a single integer D (1 ≤ D ≤ 26)
indicating the board's dimension (9 would represent the 9x9 board above); and
- A series of D lines, each representing a row on the board from
top to bottom. Each location on the row is represented by one of the following
characters, separated by spaces:
Note that there may be extra whitespace on these lines (and only these
lines). This is to make the input resemble the layout shown above.
- . — An empty location;
- ! — A piece for the first player;
- @ — A piece for the second player;
- # — A piece for the third player (if playing); or
- $ — A piece for the fourth player (if playing).
For each data set in the input, output the heading
"DATA SET #K" where K is 1 for the first
data set, 2 for the second, etc. Then print P lines, each representing
the maximum score possible for, in order, the first, second, third (if playing),
and fourth (if playing) player if they were to make a single last move.
! . . # .
. @ . . !
. . . . #
. . @ . .
. . . . .
DATA SET #1