Question

A cube is created by stacking 27 smaller cubes as shown in the figure. A plane, going through vertices A, B and C, cuts the cube as shown. How many smaller cubes will get cut?

Hint:

For planar cuts through a \(3\times3\times3\) cube, intersect each \(3\times3\) layer with the plane and count grid cells the line crosses; then multiply by the number of layers.

Answer 1

Step 1: Slice the big cube into 3 layers.
Think of the \(3\times3\times3\) cube as three parallel \(3\times3\) layers. The plane is oblique (not parallel to any face), so it passes across all three layers.
Step 2: Count per layer.
In each \(3\times3\) layer, the plane appears as a straight line that crosses a \(3\times3\) grid. Such a line necessarily passes through exactly \(\,3\) unit squares (it enters one side and exits the opposite side without running along a grid line).
Step 3: Total across layers.
There are 3 layers, so the number of small cubes cut is \(3 \times 3 = \boxed{9}\).