Inn dimensions, where a point is ann-tuple of coordinate values { x_{1}, x_{2,} x_{3,} ... , _{,}x_{n} }, the unit sphere is the locus of points for which

In one dimension this is a pair of points that encloses a volume (length) of 2 and comprises a surface area of 0. In two dimensions it's a circle that encloses a volume (area) of pi and comprises a surface area (circumference) of 2pi.

So the enclosed volume and surface area both start out, at least, as increasing functions of n.

What happens as n continues to increase?

Puzzle: Is there a value of n for which the volume reaches a maximum?

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## bonanova 85

is inspired by the seventh of Rocdocmac's Difficult Sequences:This puzzleIn

ndimensions, where a point is ann-tuple of coordinate values {x_{1},x_{2,}x_{3,}... ,_{,}x_{n}},the unit sphere is the locus of points for which

x_{1}^{2}+ x_{2}^{2}+x_{3}^{2}+ ... +x_{n}^{2}= 1.In one dimension this is a pair of points that encloses a volume (length) of 2 and comprises a surface area of 0. In two dimensions it's a circle that encloses a volume (area) of

piand comprises a surface area (circumference) of 2pi.So the enclosed volume and surface area both start out, at least, as increasing functions of

n.What happens as

ncontinues to increase?: Is there a value ofPuzzlenfor which the volume reaches a maximum?: What about the surface area?Bonus question## Link to post

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