Subsection 8.14.1 The Daisy Petal Game
First let’s think about whether you want to move first or second. You want to move second. One reason that this seems likely to be the right answer is that if both players are going to get the same number of moves, you want to move last (thus taking the last petal). That’s of course not a proof that moving second is the correct thing to do. We’ll get to that. But here’s another way to think of it: Before anyone moves, you have the possibility of winning (i.e., there are still petals for you to remove.) If you let your opponent move first, you get to choose your response in such a way as to preserve that possibility. (We’ll soon see how to do that.)
Now let’s look at a couple of small cases:
Imagine that the petals are numbered 1 – 4:
If your opponent moves first and takes petal 1, what should you do?
If your opponent moves first and takes petals 1 and 2, what should you do?
If opponent takes just 1, you should take 3. Then opponent must take one of the remaining petals and you can take the other. Opponent cannot take 2 because there aren’t 2 adjacent ones to take: you split them up. If opponent takes 2, you take the remaining 2 and win.
Now suppose that there are initially 6 petals, numbered 1 – 6:
If your opponent moves first and takes petal 1, what should you do?
If your opponent moves first and takes petals 1 and 2, what should you do?
If opponent takes just 1, you should take 4. Again you’ve taken the same number and you’ve taken the one directly opposite the one that got taken first. If opponent takes 2, you should take 4 and 5.
Now let’s solve the general case:
There is a winning strategy for the player who plays second. The idea is to preserve the following invariant:
There is a move that can be made (i.e. it’s not true that all petals have already been removed). You lose only if there is no move that you can make, so if this claim is always true when it’s your turn, you cannot lose.
Specifically, one such move is to mirror the opponent’s last move by removing the same number of petals and, more specifically, those that are exactly opposite the ones the opponent just took. (By opposite we mean the petals that you get by following a straight line from the opponent’s petals through the middle of the flower.) The following pictures illustrate the idea. If your opponent moves as in (A), you should move as in (B). If your opponent takes two petals, as in (C), you should also take two, as in (D).
(A) (B) (C) (D)
What about the claim that the invariant always holds for the second player (in, other words, player 2 hasn’t lost yet)? Must it be true? It is true right after the first time that player 1 moves since the number of initial petals is even (and thus there are opposite petals). And it can be maintained by player 2 throughout the game. Since after each of player 1’s moves, player 2 takes the opposite petal(s), player 1 is forced next to take other petal(s) whose opposite(s) are still available to player 2.