Questions: Pictured on the right are stacks of solid cubes. Determine the number of cubes in the stack and the number of faces that are glued together. There are cubes. There are faces glued together.

Pictured on the right are stacks of solid cubes. Determine the number of cubes in the stack and the number of faces that are glued together.

There are cubes.

There are faces glued together.

Transcript text: Pictured on the right are stacks of solid cubes. Determine the number of cubes in the stack and the number of faces that are glued together. There are $\square$ cubes. There are $\square$ faces glued together.

Solution

Solution Steps

To determine the number of cubes in the stack, we need to count each individual cube. For the number of faces glued together, we need to consider how cubes are arranged and count the shared faces between adjacent cubes.

Step 1: Count the Total Number of Cubes

To find the total number of cubes in the stack, we sum the number of cubes in each layer. Given the layers as $[3, 2, 1]$, the total number of cubes is: \[ 3 + 2 + 1 = 6 \]

Step 2: Calculate the Number of Faces Glued Together

To determine the number of faces that are glued together, we consider both the faces glued within each layer and between layers.

Within each layer: For a single layer with $n$ cubes, there are $n-1$ glued faces. Thus, for the layers:
- First layer: $3 - 1 = 2$ glued faces
- Second layer: $2 - 1 = 1$ glued face
- Third layer: $1 - 1 = 0$ glued faces
Between layers: For two adjacent layers, the number of glued faces is the minimum of the number of cubes in the two layers:
- Between first and second layers: $\min(3, 2) = 2$ glued faces
- Between second and third layers: $\min(2, 1) = 1$ glued face