A wooden cube 4 units on a side is dipped in paint so that every outer face is covered, then sawn into 64 unit cubes. You reach in and draw one small cube at random. What is the chance that it shows painted paint on **exactly two** of its faces?

\dfrac{24}{64} = \dfrac{3}{8}, so a probability of 0.375.

Dipping a Cube in Paint

Issue 4 Warm-up Published 15 Aug 2026 By The Chalkboard Desk

The problem

A wooden cube units on a side is dipped in paint so that every outer face is covered, then sawn into unit cubes. You reach in and draw one small cube at random. What is the chance that it shows painted paint on **exactly two** of its faces?

All unit cubes are equally likely to be drawn.
A face counts as painted only if it was part of the original outer surface.

Take a minute

Reveal hint 1

Classify the little cubes by how many of their faces were on the outside: corners, edge cubes, face centres, and the hidden interior.

Reveal hint 2

An edge cube is one that sits on an edge of the big cube but is not a corner.

Solution

Answer, so a probability of .

Derivation

Count the cubes with three painted faces.
Count the cubes with two painted faces.
Confirm the general formula.
Divide by the total.
Cross-check the whole partition.