Some cubes are missing from a 4 x 8 x 8 tightly packed structure made of alternating white and blue solid cubes as shown below. No cubes have been removed from the portion that is not visible. What is the total number of blue cubes in this structure?
Show Hint
In alternating cube structures, if the cubes alternate perfectly, half of them will be one color and the other half will be the other color.
Step 1: Understanding the structure.
We are given a 4 x 8 x 8 cube structure with alternating white and blue cubes. The visible portion of the structure shows the alternating pattern of the cubes, but no cubes have been removed from the hidden portion.
Step 2: Finding the total number of cubes.
The total number of cubes in the structure is given by the volume of the cube:
\[
\text{Total number of cubes} = 4 \times 8 \times 8 = 256 \, \text{cubes}
\]
Step 3: Determining the number of blue cubes.
Since the cubes alternate in color, half of the cubes will be blue. Therefore, the number of blue cubes is:
\[
\text{Number of blue cubes} = \frac{256}{2} = 128 \, \text{blue cubes}
\]
Step 4: Conclusion.
Thus, the total number of blue cubes in the structure is 128.
\[
\boxed{128}
\]
