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?
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![Cubic Covering Cubic Covering](/sites/default/files/styles/large/public/thumbnails/content-id-4955-cube.gif?itok=G4K5S86b)
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![Cubic Covering Cubic Covering](/sites/default/files/styles/large/public/thumbnails/content-id-4955-cube.gif?itok=G4K5S86b)
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![Cubic Covering Cubic Covering](/sites/default/files/styles/large/public/thumbnails/content-id-4955-cube.gif?itok=G4K5S86b)
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![Cubic Covering Cubic Covering](/sites/default/files/styles/large/public/thumbnails/content-id-4955-cube.gif?itok=G4K5S86b)
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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![Cubic Covering Cubic Covering](/sites/default/files/styles/large/public/thumbnails/content-id-4955-ycube.gif?itok=bf-mPLm9)
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![Cubic Covering Cubic Covering](/sites/default/files/styles/large/public/thumbnails/content-id-4955-ycube.gif?itok=bf-mPLm9)
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![Cubic Covering Cubic Covering](/sites/default/files/styles/large/public/thumbnails/content-id-4955-ycube.gif?itok=bf-mPLm9)
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![Cubic Covering Cubic Covering](/sites/default/files/styles/large/public/thumbnails/content-id-4955-ycube.gif?itok=bf-mPLm9)
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
We first imagine the cross with the blue cubes only:
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![Cubic Covering Cubic Covering](/sites/default/files/styles/large/public/thumbnails/content-id-4955-Cub1.png?itok=uh5EIQZw)
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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![Cubic Covering Cubic Covering](/sites/default/files/styles/large/public/thumbnails/content-id-4955-Cub2.png?itok=sVGDmYL7)
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![Cubic Covering Cubic Covering](/sites/default/files/styles/large/public/thumbnails/content-id-4955-Cub3.png?itok=5omCJ7c6)
Now we need one yellow cube for each of the four corners:
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![Cubic Covering Cubic Covering](/sites/default/files/styles/large/public/thumbnails/content-id-4955-Cub4.png?itok=-kFYzNih)
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.