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Lights Out! (2) (Posted on 2022-10-31) Difficulty: 3 of 5
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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Cleaned-up code for the extension and an observation | Comment 11 of 24 |
The code below does not require recursion as each row's switches are completely determined by the previous row's lights. The results of the first found solution through n=21 follow the listing.

An observation is that most of the solutions  could, in one rotation or another, be done by marking off a corner by a diagonal of flipped switches forming a triangle of non-flipped switches.

I was going to say that 13 is an exception, but in this case, the diagonal is of length of only 2, with the degenerate triangle being a single non-flipped switch (zero). It's symmetric so it appears in all four corners and the solution is therefore completely unique.

 clearvars,clc 
 
 for sz=3:2:21
   disp(' ')
   fprintf('%s %3d\n','size',sz)
   
 rowChoice={}; found=false;
 for k=0:sz
   c=nchoosek(2:sz+1,k);
   for i=1:size(c,1)
      rowChoice{end+1}=c(i,:);
   end
 end
  
 middle=ceil(sz/2);
 
 lowest=999;
 
 for whichSwitches=1:length(rowChoice)
   lights=zeros(sz+2); lights(middle+1,middle+1)=1;
   switches=zeros(sz+2);
   
    which = rowChoice{whichSwitches};
    switches(2,:)=zeros(1,sz+2);
    for i=1:length(which)
       switches(2,which(i))=1;
       for dr=-1:1
         for dc=-1:1
           if dr==0 || dc==0
             lights(2+dr,which(i)+dc)=1-lights(2+dr,which(i)+dc);
           end
         end
       end
    end
    for row=3:sz+1
       switches(row,2:sz+1)=lights(row-1,2:sz+1);
       for col=2:sz+1
         if switches(row,col)
           for dr=-1:1
             for dc=-1:1
               if dr==0 || dc==0
                 lights(row+dr,col+dc)=1-lights(row+dr,col+dc);
               end
             end
           end
         end
       end
    end
    if isequal(lights(sz+1,2:sz+1),zeros(1,sz))
       fprintf([repmat('%2d',1,sz) '\r'],switches(2:sz+1,2:sz+1))
       disp(' ')
       found=true;
       break
 %       disp(lights(2:sz+1,2:sz+1))
 %       disp(' ')
 %       disp(' ')
    end
 end
  
end

 
size   3
 0 1 0
 1 1 1
 0 1 0
 
 
size   5
 1 0 1 1 0
 1 0 0 0 1
 0 1 1 0 1
 0 0 1 0 0
 0 0 0 1 1
 
 
size   7
 0 0 0 1 0 0 0
 0 0 1 1 1 0 0
 0 1 0 0 0 1 0
 1 1 0 1 0 1 1
 0 1 0 0 0 1 0
 0 0 1 1 1 0 0
 0 0 0 1 0 0 0
 
 
size   9
 1 1 1 0 0 0 1 1 0
 0 1 0 1 0 1 0 0 1
 0 0 1 1 0 1 0 0 1
 0 0 0 1 0 0 1 1 0
 0 0 0 0 1 0 0 0 0
 0 0 0 0 0 1 1 1 0
 0 0 0 0 0 0 1 0 1
 0 0 0 0 0 0 0 1 1
 0 0 0 0 0 0 0 0 1
 
 
size  11
 0 1 0 0 0 1 0 1 0 0 0
 1 1 1 0 1 1 0 1 1 0 0
 0 0 0 0 0 1 0 1 0 1 0
 1 1 1 0 0 0 0 0 1 1 1
 0 1 0 1 0 1 0 0 0 0 0
 0 0 1 1 0 1 1 0 1 1 1
 0 0 0 1 0 0 0 0 0 1 0
 0 0 0 0 1 1 1 0 0 0 0
 0 0 0 0 0 1 0 1 0 1 0
 0 0 0 0 0 0 1 1 0 1 1
 0 0 0 0 0 0 0 1 0 1 0
 
 
size  13
 0 1 1 1 0 1 1 1 0 1 1 1 0
 1 0 1 0 0 0 1 0 0 0 1 0 1
 1 1 0 0 0 0 0 0 0 0 0 1 1
 1 0 0 0 0 0 1 0 0 0 0 0 1
 0 0 0 0 0 1 1 1 0 0 0 0 0
 1 0 0 0 1 0 0 0 1 0 0 0 1
 1 1 0 1 1 0 1 0 1 1 0 1 1
 1 0 0 0 1 0 0 0 1 0 0 0 1
 0 0 0 0 0 1 1 1 0 0 0 0 0
 1 0 0 0 0 0 1 0 0 0 0 0 1
 1 1 0 0 0 0 0 0 0 0 0 1 1
 1 0 1 0 0 0 1 0 0 0 1 0 1
 0 1 1 1 0 1 1 1 0 1 1 1 0
 
 
size  15
 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0
 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0
 0 0 0 0 1 1 0 1 0 1 1 0 0 0 0
 0 0 0 1 0 1 0 1 0 1 0 1 0 0 0
 0 0 1 1 1 0 0 0 0 0 1 1 1 0 0
 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0
 1 1 0 1 1 0 1 1 1 0 1 1 0 1 1
 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0
 0 0 1 1 1 0 0 0 0 0 1 1 1 0 0
 0 0 0 1 0 1 0 1 0 1 0 1 0 0 0
 0 0 0 0 1 1 0 1 0 1 1 0 0 0 0
 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0
 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0
 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
 
 
size  17
 1 0 1 0 1 1 0 0 0 1 1 1 0 1 1 1 0
 1 0 1 0 0 0 1 0 1 0 1 0 0 0 1 0 1
 0 0 0 1 1 0 1 0 1 1 0 0 0 0 0 1 1
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
 1 1 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0
 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 1 1
 1 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 1
 1 0 1 0 1 0 1 0 1 1 0 0 0 1 1 1 0
 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0
 1 0 1 1 0 0 0 1 1 0 1 0 1 0 1 1 0
 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
 0 1 1 1 0 0 0 1 1 0 1 0 1 0 1 0 1
 0 0 1 0 1 0 1 0 0 0 1 0 1 0 1 0 0
 0 0 0 1 1 0 1 0 1 1 0 0 0 0 0 1 1
 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0
 0 0 0 0 0 1 1 0 1 1 0 1 1 1 0 1 1
 
 
size  19
 0 1 0 1 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0
 1 1 0 1 0 1 1 0 0 0 0 0 1 1 0 1 1 0 0
 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 1 0 1 0
 0 0 1 1 1 0 1 1 1 0 1 1 1 0 0 0 1 1 1
 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0
 0 0 0 0 1 1 0 1 1 0 1 1 1 0 0 0 1 1 1
 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 1 0 1 0
 0 0 0 0 0 0 1 1 1 0 0 0 1 1 0 1 1 0 0
 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0
 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0
 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 0 0 0
 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 1 0 0
 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0
 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 1
 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 1 1 0 1 1
 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0
 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1
 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0
 
 
size  21
 0 0 0 0 0 0 0 0 0 1 1 1 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 0 0 0 0 0 1 1 1 0 1 1 1 0 0 0 0 0 0 0
 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0
 0 0 0 0 0 1 1 0 1 0 1 0 1 0 1 1 0 0 0 0 0
 0 0 0 0 1 0 1 0 1 0 0 0 1 0 1 0 1 0 0 0 0
 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0
 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0
 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0
 1 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 1
 1 1 0 1 1 0 1 1 0 1 1 1 0 1 1 0 1 1 0 1 1
 1 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 1
 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0
 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0
 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0
 0 0 0 0 1 0 1 0 1 0 0 0 1 0 1 0 1 0 0 0 0
 0 0 0 0 0 1 1 0 1 0 1 0 1 0 1 1 0 0 0 0 0
 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0
 0 0 0 0 0 0 0 1 1 1 0 1 1 1 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 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0


  Posted by Charlie on 2022-11-13 11:32:26
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