Analysis of Flip by Gary Watson

This Lights Out variant was invented by Gary Watson (around 1985 - note this is before Tiger's patent!). It is played on a 5x5 grid, and the idea is to turn all the lights on.The game randomly choses five lights to turn on at the start of play. The twist is that you are not allowed to press any lit buttons. For more about this visit The Lights Out Fan Club and click on the link to Flip.

The obvious question to ask about this game, is whether all the puzzles that would be solvable in normal Lights Out are solvable here. Of course, changing from Lights Out to Lights On makes no difference whatsoever - and in fact from now on, I'm going to talk about turning the lights out, and not pressing any unlit buttons. This is clearly the same problem. I'm also going to ignore the fact that only five buttons are lit at the start, and just assume that the game is chosen completely randomly.

It turns out that every solvable lights out puzzle is also a solvable Flip puzzle. The solutions will inevitably be longer (in general) - and they certainly require a bit more work - but I think that makes it a good challenge. Also, since Flip choses it's configuration completely at random, only 25% of the puzzles can be solved (see this page)

Incidentally - the condition that you are only allowed to press lit buttons is also made in some play modes of Lights Out Deluxe and Orbix. I imagine that, again, this makes no difference to the solvability of either of these games, although I've not checked. (Actually, I don't know about Orbix...)

Light Chasing

Before you read the solution to Flip, you should read the solution to Lights Out. There we introduce the Light chasing algorothm. With only slight modifications we can use exactly the same method here.

In the light chasing algorithm, we turn the lights out row by row. We can do exactly the same here. Ordinarily we just push any button that is underneath a lit button. Now, however, we have the problem that these buttons may not be lit. This isn't actually a probelm at all. To turn out a light above an unlit button, press the lit button (i.e. the one you want to turn out) first. This will turn this light off, and the light underneath on. Now press the underneath button, and finally press the other button again. This has the net effect of just pressing the button under the lit one. Here's an example.

Using this trick, we can chase the lights all the way to the bottom, just as before. Now, we know from our Lights Out solution, that this will leave us with 7 possible combinations of lights on the bottom row. Now consult this table. (cf. table in Lights Out solution)

we use this table slightly differently for Flip. The second column shows the buttons we press on the bottom row. So if we've chased the lights down to the bottom row, we consult this table, and then press the appropriate buttons in the bottom row. Notice that you can always do this easily (for example, suppose you get to the position in the second row. If you press button 2 first, then you will be able to press buttons 1 and 3).

The next step is to chase the lights up! That is, use the usual light chasing algorithm, only upside-down. The puzzle will then be solved!

Comments

You can of course play Flip on a normal Lights Out just by being careful only to press lit buttons. (I may add a mode to my Javascript Lights Out that enforces the rule).

I have no idea how to work out the shortest solutions, and suspect this may be pretty difficult, but at least the above method always works!