When I was a kid I had a handheld version of the game "lights out" [1,2], and I
was addicted. The game is played on a 5-by-5
board of lights, and starts with some of the lights on and all the others off;
the goal is to turn off all the lights. You can
toggle the state of a light and its 4 neighbors (NSEW) by pressing on it.
For example, here is the classic starter board:
00000
00000
10101
00*00
00000
Where 0 is an off light, 1 is an on light, and * is a light I'm about to press
(which happens to be off, but you can't see that in
this notation). After I press that light, the board looks like this:
00000
00000
10001
0111*
00100
Where again the light I'm about to press happens to be off (i.e. there's a 0
under that *). Having pressed that, the board now
looks like this:
00000
00000
10000
01101
00101
and so on until I turn off all the lights.
I have an implementation of this in J that I wrote when I was first learning
the language (I frequently use this game as my
language-familiarization problem, following "Hello,world!" -- I have one in x86
assembler, somewhere). When I wrote it, one of
the things that bothered me was that I couldn't come up with an elegant,
array-ish way to toggle the lights, given the light the
user pressed.
So, given J's array mastery, can you? That is, given the inputs:
] BOARD0 =: 5 5 $ 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0
0 0 0
0 0 0 0 0
0 0 0 0 0
1 0 1 0 1
0 0 0 0 0
0 0 0 0 0
PUSH =: 3 2
can you write a dyad out such that:
] BOARD1 =: BOARD0 out PUSH
0 0 0 0 0
0 0 0 0 0
1 0 0 0 1
0 1 1 1 0
0 0 1 0 0
in an elegant, array-oriented way? I'm thinking about maybe an inner product
of some sort, as opposed to calculating the index
array and manually amending the board (e.g. BOARD0 4 : '(-. z { x ) (z=.<"1 y
<@+"1 _2 ]\ _1 0 0 _1 0 0 0 1 1 0)} x' PUSH). I'd
also like to see solutions which avoid handling the edge- and corner-cases
specially, especially if this can be done *without*
changing the representation of the as a boolean table (but if necessary, feel
free to select your own representations for BOARD or
PUSH).
-Dan
[1]: You can play the game online here:
http://www.whitman.edu/mathematics/lights_out/
and also see its Java source.
[2] Wikipedia nicely describes the mathematics of Lights Out:
http://en.wikipedia.org/wiki/Lights_Out_(game)
including spoilers on how to win (or write a program to determine if a
given board is winnable).
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