Solution To Mensa 3D Geometry Circle Problem

 

Let O be the center of the cube after envisioning the two spheres centered at O (passing through the 2 circles) and sketched in two dimensions (as shown above).  Then the sphere that passes through the inscribed circle has radius:

s Ö (2)/ 2 

The larger sphere that passes through the inscribed circle has radius:

 s Ö (3)/ 2 

The  distance between the 2 circles cannot be smaller than the distance between the spheres. Thus, for any point P on the circumscribed circle and any point Q on the inscribed circle:  

PQ  >  (s/ 2) (Ö (3)  -  Ö (2) )

To show that this lower bound is attained, envisage a ray from O aimed toward the point where the inscribed circle is tangent to the face on which the other circle is circumscribed (as depicted in the revised 2D sketch).  Initially this ray passes through the interior of the circumscribed circle.

Now, visualize rotating the ray while keeping it tangent to the inscribed circle. Eventually, the ray will rotate far enough that it no longer passes through the interior of the circumscribed circle. By the principle of continuity, it follows that in some intermediate position the ray touches both circles.

Let P be the point where the ray touches the circumscribed circle and Q be the point where the ray touches the inscribed circle.  The points O, P and Q are co-linear and:

PQ = (OP - OQ) =  sÖ (3)/ 2)  -  sÖ (2)/ 2) 

= (s/ 2) {Ö (3)  -  Ö (2)}  

Which is the minimal distance between the two circles.