Consider an infinite chessboard. Each square contains either a 1 or an X in some pattern. (X can be any real number but for a given board, all the X's are the same.)
Each square with an X on it has weight equal to zero.
Each square with a 1 on it has a weight of 1 + N*X where N is the total number of X's on the 8 surrounding squares.
For a given value of X, find a way of tiling the board with the highest average weight per square.
Inspired by various Tower Defense games.
I want each '1' to be surrounded by as many 'X' as possible, but to also have them shared with at least another '1' if possible. The best sharing of a particular cell is 4 but that is not optimum for this problem.
While my logic is slightly flawed, in each case below I am allowing extrapolation of one cell beyond the grid in recognition that the plane is infinite while I my thoughts are within finite limits.
I am allowing myself a 5x5 grid in which I distribute the '1's. I am also considering the 7x7 grid in which that resides.
a) b) c)
1 . 1 . 1 1 . 1 . 1 1 . . . 1
. . . . . . 1 . 1 . . . . . .
1 . 1 . 1 1 . 1 . 1 . 1 . . .
. . . . . . 1 . 1 . . . . . .
1 . 1 . 1 1 . 1 . 1 1 . . . 1
X=8 X=4 X=8
d) e) f)
1 . . . 1 1 . . . 1 1 . . 1
. 1 . . . . . 1 . . . 1 . . 1
. . 1 . . 1 . . . 1 . . 1 . .
. . . 1 . . . 1 . . . . . 1 .
1 . . . 1 1 . . . 1 . 1 . . 1
X=6 X=8 X=6
g) h)
1 . . . 1 1 . . 1 .
. . . . . . . . . .
. . . . . . . . . .
1 . . . 1 . 1 . . 1
. . . . . . . . . .
x=8 x=8
Pattern No.1 X val Grid Wt Grid Wt/25 Grid Wt/49
a) 9 8 9 + 9*8 81/25 81/49
b) 13 4 13 +13*4 65/25 65/49
c) 5 8 5 + 5*8 45/25 45/49
d) 7 6 7 + 7*6 49/25 45/49
e) 8 8 8 + 8*8 72/25 72/49*
f) 9 6 9 + 9*6 63/25 63/49
g) 4 8 4 + 4*8 36/25 36/49
h) 4 8 4 + 4*8 36/25 36/49
Now, if my analysis of this situation is correct
e) is the solution although
72/49 will not be the exact average.
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Posted by brianjn
on 2012-02-08 05:51:45 |
Shaky ground?
