AMC 2003 Cut the cube into pieces.

Question

For each vertex of a cube a plane is constructed through the three vertices which are neighbors of that vertex. Into how many parts do these eight planes dissect the cube?
(A) 9 (B) 13 (C) 21 (D) 27 (E) 24

My answer is 21. I think about each corner. for example , the green corner(with green point), it is cut by the green triangular plane, with the green point, it forms a tetrahedron. This tetrahedron is also cut by other 3 planes. This dissect it into 3+1=4 parts (3 tetrahedron corners and 1 tetrahedron center).
It seems that all the 8 cube corners are cut into 3x8+1x8=32 parts. But each tetrahedron corner part has been counted twice. So there are 3x8/2+1x8=20  parts. Do not forget, after all the dissections, there is 1 part in the cube center.
Therefore, total parts are 20+1=21 parts.
But the official answer is NOT 21 ?!

Oziter · Answer

My answer is 21. I think about each corner. for example , the green corner(with green point), it is cut by the green triangular plane, with the green point, it forms a tetrahedron. This tetrahedron is also cut by other 3 planes. This dissect it into 3+1=4 parts (3 tetrahedron corners and 1 tetrahedron center).
It seems that all the 8 cube corners are cut into 3x8+1x8=32 parts. But each tetrahedron corner part has been counted twice. So there are 3x8/2+1x8=20  parts. Do not forget, after all the dissections, there is 1 part(octahedron) in the cube center.
Therefore, total parts are 20+1=21 parts.

AMC 2003 Cut the cube into pieces.

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