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.

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 1563

127 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**