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You are placing 626 points into a square with side length 3

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You are placing 626 points into a square with side length 3. You want to find the maximum distance possible that separates the three points closest to one another when placing the points in the square.

 

What is the largest number of squares you could use to subdivide the figure to induce the pigeonhole principle? 

 

Does this subdivision allow you to calculate the largest possible distance between the three points nearest one another? 

 

What is the largest number of squares you could use to subdivide the figure to induce the strong form of the pigeonhole principle? 

 

What is the least number of squares you could use to subdivide the region so that you can calculate the distance separating the three points closest to one another? 

 

Calculate the distance (guaranteed by the pigeonhole principle) within which the three points nearest one another must be situated.