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A 3 cm cube is painted on all its faces.It is then cut into 1 cm cubes.How many cubes will have paint on exactly two faces ?

  • math -

    The middle edge cubes on each side will have two sides painted. That is 4 on each side and 2 more each on the front and back.

    4 + 4 + 2 + 2 = ?

  • Mathematics -

    There will be 12 cubes with paint on exactly two faces.

    There will be 27 smaller cubes in total, 9 on the 'top', 9 in the 'middle' and 9 on the 'bottom. The top and bottom are equivalent.

    Consider the top 9 cubes. The cube in the center only has one painted side.

    The corners each have three painted sides.

    This leaves 4 cubes left, which are around the perimeter of the top, and are in the middle (in between each corner).

    Since the top and bottom are equivalent, the bottom must also have 4 cubes with exactly two sides that are painted.

    This leaves the middle section. The center cube has zero painted faces. The four corners each have two painted faces, while the cubes in between each corner only have one painted face.

    So there are four cubes on each the 'top,' 'middle' and 'bottom' that have exactly two painted sides.

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