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I was reading about the the four turtles/bugs math puzzle Four bugs are at the four corners of a square of side length D. They start walking at constant speed in an anticlockwise direction at all times directly towards the bug ahead of them. How far does each bug walk before they meet with each other?

and the a-ha answer given is that the turtles are always in the shape of a square, and since each one is always walking perpendicular to the direction of his pursuer's approach, he neither hinders nor helps the pursuer in reaching him. Therefore the time to the center is the same as the time it takes one turtle to walk across a side of the square.

Since each creature is running directly to its target, the paths of a chaser and its chasee are always at right angles. That is, the distance between the chaser and its chasee depends only on the movement of the chaser, not its chasee. Therefore, the distance that the chaser runs is the same as if the chasee did not run at all.

Could you please explain how this happens?

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Suppose you are a mile directly north of me. At exactly the same time, I take one step towards you (north) and you take one step west. By how much did the distance between us decrease? Basically by one step. Not quite one step, actually (work it out, Pythagorean theorem). But in the limit of taking very small steps (like the bugs that are continually tracking each other) we would be getting closer by one step's distance. Similarly, the rate of decrease of the inter-bug distance is equal to the bugs' speed. – Greg P Jun 7 '12 at 20:47

The four creatures spiral toward the center of the square as they follow eachother. Eventually they meet at the center after travelling a certain distance along the spiral, which apparently is equal to the distance betwene the creatures initially. Therefore if two creatures at opposite corners did not move, the other two creatures would reach them after the same distance as if they all chased eachother in the spiral fashion.

The reason they spiral toward the center is the following: The creatures travel perpendicular to eachother but not perpendicular to a radial line from the center of the square. Perpendicular to each other (90 degrees) is 45 degrees from a line towards the center = spiraling at that angle (45 degrees).

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Thank very much for the responses. The information was useful and made me to see thinks clearer... – kostis Jun 7 '12 at 21:01
@kostis: So maybe accept the answer? – Colin K Jul 7 '12 at 20:45

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