Count cubes by layers

Split a supported stack into horizontal layers and count each cube once.

Start with the whole base

This stack has a complete base: two rows of three cubes. Two more cubes sit on the first two cubes of the back row.

Every upper cube rests on a cube below it. Count horizontal layers, including the base cubes covered by the upper layer. A hidden space alone does not tell you whether a cube is there.

A solid base with two cubes above its back row.

Count one layer at a time

Look down at each layer separately. The bottom layer has six cubes. The upper layer has two.

Add the layers: 6 + 2 = 8. The dashed empty positions in the upper layer contain no cubes. Do not count the visible faces as separate cubes.

Bottom layer: 6 cubes.
Upper layer: 2 cubes.

Try a smaller stack

This new stack has a complete two-by-two base and one cube on its back-left corner. There are no gaps under the upper cube.

Count the bottom layer, then add the upper layer.

A complete square base with one cube above a corner.

Try it

How many cubes are in this smaller stack?

  1. 5
  2. 6
Show the answer

5

There are five cubes: four in the complete base and one above it. The covered base cube still belongs to the bottom layer.

Enable JavaScript to answer interactively and save your place.

Practice this method