Cubic covering
A blue cube has blue cubes glued on all of its faces. Yellow cubes are then glued onto all the visible blue facces. How many yellow cubes are needed?
Problem
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Take one blue unit cube and glue a further blue unit cube to each of its faces (to make a 3D cross).
If unit cubes coloured yellow are now glued face-to-face to all the spare faces of the blue cross, how many yellow unit cubes are required?
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If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
We first imagine the cross with the blue cubes only:
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We now glue the yellow faces to the up and bottom faces of the blue cross. We require 5 yellow cubes for wrapping each of the two blue cubes. (We used transparent yellow cubes)
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Now we need one yellow cube for each of the four corners:
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Finally, we have 4 faces to cover, so we need 4 more squares.
Therefore, we used $5+5+4+4 = 18$ yellow cubes.
Therefore, we used $5+5+4+4 = 18$ yellow cubes.