A quick and dirty solver for Inverter http://gorried.github.io/inverter/game.html

invert.rb

$size = ARGV[0].to_i

if $size <= 0
  puts "Supply numeric level (grid size) as an argument"
  exit
end

def show(placement)
  0.upto($size - 1) do |x|
    0.upto($size - 1) do |y|
      if placement.include?([x, y])
        print "[*]"
      else
        print "[ ]"
      end
    end
    puts
  end
end

$placings = []

0.upto($size - 1) do |x|
  $placings[x] = []
  0.upto($size - 1) do |y|
    $placings[x][y] = (1 << (y * $size + x)) +
    (y > 0 ? (1 << ((y - 1) * $size + x)) : 0) +
    (x > 0 ? (1 << (y * $size + x - 1)) : 0) +
    (x < $size - 1 ? (1 << (y * $size + x + 1)) : 0) +
    (y < $size - 1 ? (1 << ((y + 1) * $size + x)) : 0)
  end
end

$solution = (1 << $size * $size) - 1

def check(startindex, bits, cur_placings)
  
  x = startindex % $size
  y = startindex / $size
    
  v = $placings[x][y]
  
  new_bits = bits ^ v
  if new_bits == $solution
    return [cur_placings + [[x,y]], true]
  end
  
  return [cur_placings, false] if startindex == $size * $size - 1
  
  above_bit = (y == 0 ? 0 : (1 << (y - 1) * $size + x))

  if y == 0 || bits & above_bit > 0
    (placement, solved) = check(startindex + 1, bits, cur_placings)
    return [placement, true] if solved
  end
  if y == 0 || bits & above_bit == 0
    (placement, solved) = check(startindex + 1, bits ^ v, cur_placings + [[x,y]])
    return [placement, true] if solved
  end
  return [placement, false]
end

(placement, solved) = check(0, 0, [])

if solved
  show(placement)
else
  puts "No solution"
end

Improved. It will solve up to level seventeen under 10 secs on my machine. Algorithm is still O(2^n) though.