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There may be a 1000 faces on the outside of the cube, …

1000 are not just small cubes on the outside layer, they are ALL small cubes that make the wooden cube.

When you study math, start with simple objects and move on to more complicated ones. Imagine that we have just 1 row of 10 small cubes lying on a table.

How many cubes are there? 1 × 10 = 10

Then imagine we place 10 rows and 10 columns of small cubes on a table.

How many cubes are there? 10 × 10 = 100

Now imagine we add another level on top the previous one.

How many cubes are there? 2 × 10 × 10 = 200

Now you should fully understand why there are 10 × 10 × 10 small cubes in the original wooden cube:

And also why there are 8 × 8 × 8 = 512 not-painted small cubes:

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…, but the cubes at the corners account for 3 of those faces each and the cubes at the edges account for 2 of those faces each.

This is a good observation. But we use it not when counting the whole number of small cubes that compose the wooden cube, but if we count the number of small cubes that make the surface.

In this case the corner cubes (marked green) will be counted 3 times (because each one belongs to three sides) and the edge cubes (marked blue) will be counted 2 times (because each one belongs to two sides). While we need to count all the surface cubes only once!

There are 10 × 10 = 100 small cubes on each side and there are 6 sides. Considering that there are 8 corner cubes and 8 × 12 = 96 edge cubes, the correct counting of the painted cubes is:

(100 × 6) – (8 × 2) – (96 × 1) = 488