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- May 25th 2007, 10:56 AM #1

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## Problem 23

Consider a tunnel (not necessarily straight) of length 101 units through a

sphere of radius 51 units.

Prove that the sphere can be cut into two congruent pieces one of which

contains the tunnel in its interior.

RonL

- May 25th 2007, 12:38 PM #2

- May 25th 2007, 01:31 PM #3

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- May 28th 2007, 09:14 AM #4

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Solution:

Let the entry and exit points be A and B. Then all posible paths of length

101 lie in an ellipsoid such that the sum of the distance from A and that from

B is less than 101 units. But the centre of the sphere cannot be inside this

ellipsoid as the sum of the distance from A to the centre and from B to the

centre is 102.

Now the plane through the centre normal to the plane containing the points

A, B and the centre does not intersect the ellipsoid, and so divides the sphere

(ignoring the hole) into two equal hemisphere.

RonL