Question

Part A: The Painted Cube Problem A wooden cube is dipped in a can of paint, then removed and allowed to dry. It is then sliced up, as in the picture below. When the cube is taken apart and the resulting small cubes are examined, how many will have no paint on them? How many small cubes will have paint on exactly one face? How many will have paint on exactly two faces, on three faces, on four, five or six faces? Part B: The figure below shows a sequence of painted cube problems. In the 10th figure in this sequence, how many small cubes would have paint on 3 faces, 2 faces, 1 face, and no faces? 1 2 3 ... Part C: Find all generalized (closed) formulas for the number of faces painted for an nnn cube. You MUST USE pictures to justify your formulas.

          Part A:
The Painted Cube Problem
A wooden cube is dipped in a can of paint, then removed and allowed to
dry. It is then sliced up, as in the picture below. When the cube is taken
apart and the resulting small cubes are examined, how many will have no
paint on them? How many small cubes will have paint on exactly one face?
How many will have paint on exactly two faces, on three faces, on four, five
or six faces?
Part B:
The figure below shows a sequence of painted cube problems. In the 10th
figure in this sequence, how many small cubes would have paint on 3 faces,
2 faces, 1 face, and no faces?
1
2
3
...
Part C:
Find all generalized (closed) formulas for the number of faces painted for
an n*n*n cube. You MUST USE pictures to justify your formulas.

Part A:
The Painted Cube Problem
A wooden cube is dipped in a can of paint, then removed and allowed to
dry. It is then sliced up, as in the picture below. When the cube is taken
apart and the resulting small cubes are examined, how many will have no
paint on them? How many small cubes will have paint on exactly one face?
How many will have paint on exactly two faces, on three faces, on four, five
or six faces?
Part B:
The figure below shows a sequence of painted cube problems. In the 10th
figure in this sequence, how many small cubes would have paint on 3 faces,
2 faces, 1 face, and no faces?
1
2
3
...
Part C:
Find all generalized (closed) formulas for the number of faces painted for
an n*n*n cube. You MUST USE pictures to justify your formulas.

Added by Noelia G.

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