MHB What is the maximum distance between two points in a square of side length 1?

[Ackbach]

Here is this week's POTW:

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Show that if $5$ points are all in, or on, a square of side length $1$, then some pair of them will be no further than $\dfrac{\sqrt{2}}{2}$ apart.

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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!

 

[Ackbach]

Congratulations to greg1313 and Fallen Angel for their correct solutions. greg1313's solution follows:

Spoiler

Construct four quarter circles with radius $\dfrac{\sqrt2}{2}$, each with its centre on a vertex of the square. Notice that there is no area on the square that is not contained by, or on, a quarter circle. Hence one cannot place more that four points in or on the square without such a point being at most $\dfrac{\sqrt2}{2}$ from another point.