In that case, it seems a bit of over-kill to use 2^24,
In your 3x8 grid, you can fit exactly 4 2x2 tiles in a row.
(It's surely equivalent to placing 4 2x1 tiles in a 3x4 grid.)
To make the graphics easier, I'll consider 4 1x2 tiles in a vertical 4x3
grid.
Each tile "[]" can then be "left" or "right", so there are 2^4
patterns: (best in fixed width!)
[m =: 4#,:'.[]' NB. Each tile is [], spaces are ....
.[]
.[]
.[]
.[]
0 1 0 1 |."0 1 m NB. two patterns out of the 16
.[]
[].
.[]
[].
0 1 1 1 |."0 1 m
.[]
[].
[].
[].
So #: i. 16 will generate all patterns.
Or have I missed something?
Mike
On 26/05/2020 22:14, Skip Cave wrote:
Ooops. You are right. I mis-copied the last dimension. :-(
Sorry for the confusion.
The problem I was working on, was to try to find all the ways to place 2"x
2" tiles non-overlapping on a 3"x 8" grid.
I know there's a technique called linear difference equations that can
solve this problem, but I wanted to try to
model it by brute force. So I generated all 16777216 possible 3x8 grids of
1s & 0s.
odo=:#: i.@(*/)
$n=.odo 24#2
16777216 24
$ns=.16777216 3 8$,n
16777216 3 8
_3{.ns
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 0 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
That fixed my problem, with the right dimensions. Now comes the hard part.
The problem now is to find all the 3x8 grids that only
have 2x2 blocks of non-overlapping 1s in them, and no isolated 1s or
non-2x2 groups of 1s.
Skip
On Tue, May 26, 2020 at 12:34 PM Skip Cave <[email protected]> wrote:
odo=:#: i.@(*/)
$n=.odo 24#2
16777216 24
_3{. n
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
_48{. ,n
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1
$ns=.16777224 3 8$,n
16777224 3 8
_3{.ns
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 1 0 1
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 1 1 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 1 1 1
Shouldn't the last few 3x8 arrays in ns be almost all ones, and the last
array definitely all ones?
Skip Cave
Cave Consulting LLC
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