where a bulb would "trap" another unlit cell, making it impossible to light it up without breaking the rules. More advanced techniques tend to focus on different combinations of clues. Two 3s that are one space apart, for example, with nothing between them or to the other two sides of the cell in between, must have a lightbulb in that space, and the two spaces next to the two threes, on the line joining them. If not, then one would have two lightbulbs illuminating each other. Also, from this deduction, the remaining four cells surrounding the threes must contain two lightbulbs. Note that as the four spaces are arranged in two rows with nothing in between, one must have one lightbulb to each row, so one can mark all other spaces in those rows as empty. Another fairly common pattern is a 1 diagonally adjacent to a 2, with one of the spaces next to the 2 but not adjacent to the 1 either empty or walled off. At most one lightbulb can be placed in the two cells common to the two clues, so the last lightbulb must go in the last space around the 2. Now, it is known that there is exactly one lightbulb in those cells, so the other cells next to the 1 must both be empty. Computational Complexity Determining whether a given Light Up puzzle is solvable is NP-complete. This is proved by a polynomial-time reduction from Circuit-SAT, which is known to be NP-complete, to Light Up puzzles. Variations on the original Light Up puzzle in which there are walls without numbers plus walls with one certain number, so either 0, 1, 2, 3 or 4 (we call these variations Akari-), have also been researched in complexity. It is shown, by a polynomial-time reduction from Circuit-SAT, that Akari-1, Akari-2 and Akari-3 are NP-complete; for Akari-4 and puzzles without any numbers it is shown that these are in P; Akari-0 is thus far uncategorized. See also List of Nikoli puzzle types :Category:Logic puzzles References External links Nikoli's English page on Light Up Logic puzzles NP-complete problems