A 3\times 3\times 3 cube is made of 1\times 1\times 1 cubes glued together. What is the maximal number of small cubes one can remove so the remaining solid has the following features:
1) Projection of this solid on each face of the original cube is a 3×3 square,
2) The resulting solid remains face-connected (from each small cube one can reach any other small cube along a chain of consecutive cubes with common faces). geometry3D geometrycubeprojectioncombinatoricscombinatorial geometry