The drawing below shows a building block 16 m×16 m×16 m Black and White surfaces are cubes visible on all four vertical surfaces and top surface, except the base. Calculate the total volume of black cubes that are visible from all sides except the base. Find the correct answer.
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The block is a 4 x 4 x 4 arrangement of 4 m cubes coloured like a 3D chessboard. Count the black cubes layer by layer, remembering that only the outer ring of each lower layer is visible, then multiply by the cube volume 64 cu.m.
Each small cube has side 4 m, so the block is a \(4\times4\times4\) arrangement of cubes, each of volume \(4^3=64\) cu.m. The figure shows a chessboard colouring, so every horizontal layer is a 4 x 4 chessboard with 8 black cubes and the colours alternate from one layer to the next. Top layer: all 16 cubes are visible from above, so 8 black cubes are visible. Each of the three lower layers: the cubes on the outer ring (12 cubes) are visible on a vertical face; the inner 2 x 2 block is hidden. In a chessboard layer the inner 2 x 2 block always has 2 black cubes, so the visible black cubes are \(8-2=6\) in each lower layer. Total visible black cubes = \(8+6+6+6=26\). Volume = \(26\times 64=1664\) cu.m. Answer: (3) 1664 cu.m.
