Locked in a room with your worst enemy. On a table in the centre of the room is a bar of chocolate, divided into 23x45 squares in the usual way. One square of the chocolate is painted with a bright green paint that contains a deadly nerve poison. You and your enemy take it in turns to break off one or more squares from the remaining chocolate (along a straight line) and eat them. Whoever is left with the green square must eat it and die in agony. You may look at the bar of chocolate and then decide whether to go first or second. Describe your strategy in the following 3 cases: a) The poisonous square is in the corner b) The poisonous square is the middle square c) The poisonous square is adjacent to the middle (consider more cases if needed).
a) In the first case, I will play first and cut the chocolate along a line so that I leave my enemy with a square piece 23x23 having the deadly piece in one of its four corners. This is obviously a losing position for him.
b) In the second case, I don't see any winning strategy for me, if I play first. Apparently, I can't leave my enemy with a square piece of chocolate!
c) Well, I would start with eating the part that contains the middle square and then leaving my enemy with a rectangular piece that contains the poisonous square in the middle of its first 1x45 column. Any ideas on how to continue?
Hint: Imagine that instead of playing on the whole bar of chocolate, we start by throwing away all the squares that don't share a row or column with the poisoned square (so the game board now looks like a possibly lopsided cross). Convince yourself that this modified game is equivalent to the original game. Now find a way to interpret the modified game as Nim, and use what you presumably know about winning strategies in Nim.