Paint on a big cube
A cube is built from 1000 unit cubes and dipped in paint, so that its whole outer surface is painted.
How many of the unit cubes have no paint at all?
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
Which unit cubes touch the surface?
Remove the outer layer. What is left?
h
s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. A unit cube is unpainted exactly when it does not touch the surface, that is when it lies in the inner block left after peeling off one layer on every side.
Step 1. Peeling one layer from each of the six faces removes one unit from both ends of each edge, leaving a block.
Step 2.
Follow-ups. 488 cubes have some paint. The 8 corner cubes have 3 painted faces, the edge cubes have 2, and the face cubes have 1: .