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?
Age
14 to 16
Challenge level
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.