questioner wrote:

**Why will the number of cubes painted at least on one side ie the cubes on the outside not be 600, 10 × 10 on each side × 6 sides?**

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).

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

It's really easy to make a mistake counting the number of painted cubes, while counting the number of unpainted cubes is much simpler: 8 × 8 × 8 = 512.