All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars
 perplexus dot info

 Boosted board (Posted on 2012-02-07)
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.

 See The Solution Submitted by Jer Rating: 4.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
 Shaky ground? | Comment 1 of 11
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.
 Posted by brianjn on 2012-02-08 05:51:45

 Search: Search body:
Forums (0)