Topping Tetrahedral Ten

Geometry Level 4

Ten identical spheres are packed as above. You can either encase the entire stack in a tightly fitting tetrahedron, or you can balance one additional sphere on top. Which will be higher off the ground, the top vertex of the tetrahedron or the top of the additional sphere?


  • Each of the spheres in the top two layers of the original arrangement is supported by three spheres which are all tangent to one another.

  • The entire arrangement sits on a level ground.

  • Each face of the encasing tetrahedron is tangent to six of the spheres in the stack.

  • The additional sphere is identical to the others, is supported only by a single sphere below it to which it is tangent, and its center is directly above the center of the sphere below it.

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