Expert: Josh - 3/24/2004

Question
Hi, my question is: One thousand unit cubes are fastened together to form a large cube with edge length 10 units, each face of which is painted. If the large cube is seperated back into unit cubes, how may of these cubes will have paint on at least one face?
Thanks.

Answer
Hi Stella,

There are six sides to a volume.
We can look at the cube, as though that it consists of 10 vertical layers, each measuring 10 unit x 10 unit x 1 unit (height-by-width-by-depth).

Step1:
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If we look at the outer most layer from the side elevation, (say, looking into the easterly direction) clearly, it has 10x10 cubes painted. On the opposite end, (say, looking into the westerly direction), the same thing is observed.

Again, we have 10x10 painted cubes.

There are 8 vertical layers embedded between these two layers, agree? Before we get on to the next bit, remove the two 10x10 vertical layers from the "East" and "West".

Step2:
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Now, if we look at what remains at the "top" or "bottom" layer, the face measures 8 unit x 10 unit.

Next, remove these top 8x10 slices from the top and bottom face of the volume.

Step3:
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Finally, if we look at the remaining faces from the "south" or the "north", both measure "8x8, agree?

Adding all these different contributions, we get
2(10x10)+2x(8x10)+2x(8x8). This is the number of cubes with one or more painted face.

Okay?