Two players play a game on the Cartesian plane. The game starts by placing a token at a lattice point in the first quadrant. The players alternate turns, with player one going first. On her turn, player 1 can move the token 2 units to the left or 1 unit down. On his turn, player 2 can move the token 1 unit to the left or 2 units down. A player loses the game if s/he makes either co-ordinate of the token negative. The starting position of the token is determined by randomly choosing an x∈{1,…,30} and a y∈{1,…,30}. Of the 900 different possible starting positions for the token, how many positions result in a guaranteed winning strategy for the first player?
To determine the number of positions that result in a guaranteed winning strategy for the first player, we need to find all the winning positions for player one.
Let's consider the possible moves for each player:
- Player one can move the token 2 units to the left or 1 unit down.
- Player two can move the token 1 unit to the left or 2 units down.
We can approach this problem by looking at the positions and analyzing the possible moves for each player. We will start by analyzing each position on the Cartesian plane and determine if it leads to a guaranteed win for player one or not.
1. Start by considering the bottom-left corner (where x = 1 and y = 1). For this position, player one can move the token 1 unit down to (1, 0), making it a losing position. Hence, this position does not result in a guaranteed winning strategy for player one.
2. Now, let's analyze the positions along the bottom row (where y = 1 and 1 < x ≤ 30). For any position (x, 1) in this row, player one has two options:
a) Move 1 unit down to (x, 0).
b) Move 2 units to the left to (x - 2, 1).
In both cases, player two will be able to move the token to a losing position in the next turn. Therefore, all positions in the bottom row result in a guaranteed winning strategy for player one.
3. Next, let's analyze the positions along the leftmost column (where x = 1 and 1 < y ≤ 30). For any position (1, y) in this column, player one has two options:
a) Move 2 units to the left to (-1, y).
b) Move 1 unit down to (1, y - 1).
In both cases, player one will make the x-coordinate negative, resulting in a loss for them. Therefore, all positions in the leftmost column do not result in a guaranteed winning strategy for player one.
4. Now, let's analyze the positions in the remaining quadrant (where 1 < x ≤ 30 and 1 < y ≤ 30). For any position (x, y) in this quadrant, player one has two options:
a) Move 2 units to the left to (x - 2, y).
b) Move 1 unit down to (x, y - 1).
In both cases, player one will force the token into the positions in row x and column y, where player two will have a guaranteed winning strategy (as analyzed in steps 2 and 3). Therefore, all positions in this quadrant do not result in a guaranteed winning strategy for player one.
5. Finally, we can sum up the number of positions that result in a guaranteed winning strategy for player one by considering the bottom row (29 positions) and subtracting the positions in the leftmost column (29 positions). This gives us a total of 29 positions.
Thus, there are 29 different starting positions for the token that result in a guaranteed winning strategy for the first player.