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 Lights Out! (2) (Posted on 2022-10-31)
Imagine there is a 7x7 grid of lights, and only the middle in the grid is on.

The lights are wired such that when you flip the switch for one light (from on to off or off to on) the others next to it (not diagonally) flip as well.

Using this weird wiring of lights, what is the fewest number of switch changes it takes to turn all the lights off, and which lights should you switch?

Note: Assume all the switches work in the manner explained, and there is one switch for each of the lights.

 See The Solution Submitted by K Sengupta Rating: 5.0000 (1 votes)

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 continuing the fun - edited | Comment 16 of 24 |
`I experimented a bit more. I was able to find all solutions`
`for grids from side length 3 to 27. For a range of lengths from  3 to 127, I also tried potential solutions of the form that begin with only one button push in the center of the top row, and the pushes for all other rows following from that. These solution worked only for the squares n = (3, 7, 8, 9, 15, 19, 31, 39, 63, 79, and 127) Can you see the two interleaved sequences?   The "one push" patterns are shown here. The next ones in the sequence will be side lengths 159 and 255.... because the sequence seems to be n=(2^i-1) and n=(10 * 2^j -1), so far, at least. I tested one-push for n=1023 and that works as well - displayed here. Likewise, n=2047 and n=2559 were found to work, as expected.   For the grids with all solutions found, n= 3 to 27, I counted total number of different patterns that work (some are likely not completely symmetric and so present as multiple rotations) I also counted the number of solutions that had the same maximum number of pushes and those that had the same minimum number of pushes.  There appears to be one min solution (minimum number of pushes) and one max solution for each square size. (When there are four counted, these are likely four rotations of one asymmetric pattern.)  These min and max pushes grids are displayed here. Table for n sides:   n  solutions points points cnt   cnt               (min)  (max) (min) (max)----------------------------------------      3     1          5     5    1     1  5     4         11    11    4     4  7     1         17    17    1     1  9   256 (2^8)   25    51    4     4 11    64 (2^4)   43    67    4     4 13     1         65    65    1     1 15     1         61    61    1     1 17     4        103   103    4     4 19 65536 (2^16)  93   219    4     4 21     1        125   125    1     1 23 16384 (2^14) 151   311    4     4 25     1        209   209    1     1 27     1        229   229    1     1  As mentioned, the results for the one-push top-row patterns are displayed here. Only n=8 (which is not from either sequence) shows an asymmetrical pattern).  Table for squares solved with the "top row one-push" pattern:  n    i  j   points---------------------     3    2  -       5  7    3  -      17  8    -  - ?    25  9    -  0      35 15    4  -      61 19    -  1     123 31    5  -     217 39    -  2     439 63    6  -     773 79    -  3    1563127    7  -    2753  So square sizes 19 and 23 have enormously more solution than other size squares. I am not sure why. Patterns need to cancel themselves out - perhaps these (19, 23) size scales are the easiest for this stoplight pattern to self-cancel. I show here for the case n=19 each 1000th solution (65 in total) here.  I see no pattern to these n=19 configurations. Perhaps the expanding and contracting push patterns have a useful self-cancelation column height commensurate with these side lengths. The code is here.  Misc. References:I find these patterns reminiscent of various other endeavors into cellular automatons. The cross pattern for lighting lights is a rule much like on John Conway's Game of Life, which one may play.  Likewise the notion that the algorithms that yield changing patterns are somehow fundamental in nature - as in Wolfram's immodest tome "A New Kind of Science."  These "lights-out" puzzles show up often in Bart Bonte's  games - like Green here. (Check out his "Sugar, Sugar" there too, just for the fun of it.)`

Edited on November 18, 2022, 8:38 pm
 Posted by Steven Lord on 2022-11-18 19:33:47

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