A cube with all its faces coloured is divided into one…
A cube with all its faces coloured is divided into one thousand cubes of equal size. The cubes obtained in this way are well mixed. Determine the probability that a randomly drawn cube will have exactly two coloured faces.
Explanation of Method
$1000=10^3$, so each edge of the big cube is cut into $10$ pieces. A small cube has two coloured faces exactly when it lies on an edge of the big cube but not at a corner. We count these cubes (the favourable outcomes) and divide by $1000$, the number of all equally likely outcomes. As a check we count all four types of small cubes and verify that they add up to $1000$.
Step 1: How the cube is cut
$1000 = 10\times10\times10$, so the big cube is cut into $10$ layers in each of the three directions. Each edge of the big cube consists of $10$ small cubes.
Step 2: Which small cubes have exactly two coloured faces?
A small cube gets a coloured face from every outer face of the big cube it touches.
- At a corner it touches 3 outer faces → 3 coloured faces.
- On an edge but not at a corner it touches 2 outer faces → 2 coloured faces.
- Inside an outer face (not on an edge) it touches 1 outer face → 1 coloured face.
- Completely inside it touches none → 0 coloured faces.
- corner cubes: 3 coloured faces
- edge cubes (not corners): 2 coloured faces
- face-centre cubes: 1 coloured face
Step 3: Count the favourable cubes
A cube has $12$ edges. Each edge has $10$ small cubes, of which the $2$ end ones are corners, so $10-2=8$ cubes have exactly two coloured faces:
\[ m = 12\cdot 8 = 96. \]
Step 4: The probability
All $1000$ small cubes are equally likely to be drawn (they are well mixed), so
\[ \boxed{P = \frac{96}{1000} = \frac{12}{125} = 0.096.} \]
Step 5: Check — count all types
| Coloured faces | Where | Count | Probability |
|---|---|---|---|
| 3 | corners | $8$ | $0.008$ |
| 2 | edges without corners | $12\cdot 8 = 96$ | $0.096$ |
| 1 | faces without edges | $6\cdot 8^2 = 384$ | $0.384$ |
| 0 | interior | $8^3 = 512$ | $0.512$ |
| total | $1000$ ✓ | $1$ ✓ |
The interior is a $8\times8\times8$ cube (remove one layer from each side of the $10\times10\times10$ cube), and each face has an inner $8\times8$ square.
Result
| Answer | |
|---|---|
| Favourable cubes | $12$ edges $\times$ $8$ = $96$ |
| Probability | $P=\dfrac{96}{1000}=0.096$ |