Breaking Chocolate

Alright, new puzzle time. I'm a bit surprised I hadn't posted this puzzle before, as I thought of it many years ago. Anyway, I sort of came up with it myself, but I may have had some influence from Carl Michal. Also, this game appears exactly in Winning Ways For Your Mathematical Plays, so I cannot claim that that book didn't give me inspiration. Anyway, on to the puzzle:

Consider a two player game in which the players take turns breaking up an NxM block of chocolate. On a players turn, they select one existing block of chololate and break it in a straight line along one of its break lines making two smaller blocks of chocolate. The players alternate turns, selecting a single block and breaking it into two smaller blocks. If on a players turn they have no legal moves (because there are only 1x1 blocks left), they lose. Determine the optimal strategy and who will be the winning player from a starting NxM position.

To clarify, a legal move is to select an available NxM block, then select a positive integer k less than N or a positive integer i less and M and turn that NxM block into either a (N-k)xM block plus a kxM block, or into a Nx(M-i) block plus a Nxi block. To give some examples, in the 1x1 case the first player loses (having no legal moves), and in the 1x2 case the first player wins (by making the only legal move).

Trinary Answer

Two posts in one month? Unheard of. Anyway, I found a new puzzle recently, so now I have two puzzles I haven't posted and I want to waste them as soon as possible (also, post them before I forget what they are).

Alright, solution to last times light bulb puzzle:

You turn on one switch and leave it on for a reasonable amount of time, then turn it off and turn on a different switch. Go to the other building and check to see if the light bulb is on, warm, or cold, each of these results points to a different switch.

Not much to it, there are proably other answers, but its hard to come up with one as reasonable as this one. If you have any, I would love to hear them.

A Silly Puzzle

New puzzle time? I am probably out of puzzles by this point, time to dig up the old bad ones from years ago. Actually, I also thought of a somewhat decent one, but I want to do this bad one first. I first heard this puzzle from Carl Michal:

There is a building with a lightbulb in it, and in the building next door there are three switches, one of which can turn the lightbulb on or off. You are in the building with the switches, and may manipulate them as you like, when you are done you may go to the building with the light bulb. You may not return to the bulding with the switches once you leave. You must determine which switch operates the light bulb.

The solution is somewhat silly, in that I mean to say that if you converted this problem into "math" then there is no solution (this isn't hard to prove). You can make use of the "reality" of the situation in your solution. The actual solution is quite reasonable, not actually bad at all, just not mathematical. It is a practical solution one could impliment and does not require anything unstated from the puzzle (such a calling a friend to come help, or being able to see the light from the next building over).