Question

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.

Explanation:

Step1: Count number of cubes

Assume the stack is a rectangular - prism - like structure. If we count the number of cubes layer - by - layer or use the volume formula for a rectangular prism of unit - cube construction. Let's say the dimensions of the stack are length $l$, width $w$, and height $h$. If we assume the stack has length $3$, width $2$, and height $2$. The number of cubes $N=l\times w\times h$. So $N = 3\times2\times2= 12$.

Step2: Count glued - together faces

When two cubes are adjacent, one face of each cube is glued together. Consider the internal structure of the stack. In a $3\times2\times2$ stack of cubes:

• For the length - width layers, there are $2$ layers. In each layer, the number of adjacent - cube connections in the length - width plane: In a $3\times2$ layer, the number of horizontal connections (along the length) is $2\times(3 - 1)=4$ and the number of vertical connections (along the width) is $3\times(2 - 1)=3$. So in one layer, there are $4 + 3=7$ connections. Since there are $2$ layers, the number of connections between layers is $3\times2 = 6$. The total number of glued - together faces is $12$.

Answer:

There are 12 cubes.
There are 12 faces glued together.