The first quadrant is lava. The rest of the plane (including the x and y axes)...

 

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The first quadrant is lava. The rest of the plane (including the x and y axes) is safe. To traverse the lava, you can place a board, which is a line segment of length at most 1, between two safe endpoints. Once a board is placed, the line segment becomes safe, and points on it may be used as endpoints of a subsequent board. Your goal is to reach the point (2023, 2023). (a) Prove that one million boards aren't enough. (b) Give a specific number of boards with which you can reach (2023, 2023). (c) Prove that one billion boards aren't enough. The first quadrant is lava. The rest of the plane (including the x and y axes) is safe. To traverse the lava, you can place a board, which is a line segment of length at most 1, between two safe endpoints. Once a board is placed, the line segment becomes safe, and points on it may be used as endpoints of a subsequent board. Your goal is to reach the point (2023, 2023). (a) Prove that one million boards aren't enough. (b) Give a specific number of boards with which you can reach (2023, 2023). (c) Prove that one billion boards aren't enough.