A correct solution was received from Shaun Lewis.  Partial solutions were received from Bill Phelps, Mike Dewitt, Paul Leisher.  Additional complete solutions were submitted by Robert T McQuaid, Al Zimmermann, Cyril Terakopiantz William Webb, Philippe Fondanaiche, Tim Kelly, Brian P. Glover. An additional partial solution was received from Burkart Venzke.

Refer to the sketches above for notation.  The larger sphere is centered at A and the smaller at B.  We may assume the larger sphere has radius 1.  The lengths of the two legs of the red triangle are then 1 and Ö2 with the hypotenuse having length Ö3.  Let r denote the radius of the smaller sphere.  By similarity, BO = rÖ3 and SO = r(Ö3 - 1).  Thus, r(Ö3 - 1) + r + r + 1 = Ö3, and r = (Ö3 - 1)/(Ö3 + 1).

In the case of a small cube with side length s, the diagonal of the cube would be TO = r(Ö3 + 1) = Ö3 - 1 = sÖ3, so s = (3 - Ö3)/3.

The volume of the small cube is s³ = .0755..., and the volume of the small sphere is 4pr³/3 = .0806..., so the sphere is about 6.7% larger in volume.

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Page last updated 18 October 1999.