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The image shows three rectangular cuboid diagrams composed of smaller cubes. Each diagram has a portion of the cubes shaded, while the rest are unshaded. The task asks for a formula that relates the number of shaded cubes (denoted as \( S \)) to the number of white (unshaded) cubes (denoted as \( W \)) in each diagram.

### Let's break it down:

1. **First Diagram**:
   - It appears to be a \( 3 \times 3 \times 3 \) cube.
   - The central \( 2 \times 2 \times 1 \) block is shaded.

2. **Second Diagram**:
   - It appears to be a \( 4 \times 4 \times 3 \) cuboid.
   - The central \( 2 \times 2 \times 1 \) block is shaded.

3. **Third Diagram**:
   - It appears to be a \( 5 \times 5 \times 3 \) cuboid.
   - The central \( 2 \times 2 \times 1 \) block is shaded.

### Formula Explanation:

- The **total number of cubes** in each diagram can be calculated by multiplying the dimensions (length, width, height). For example, in the first diagram, there are \( 3 \times 3 \times 3 = 27 \) total cubes.
- The **number of shaded cubes** is a constant in all diagrams, as the shaded block is always \( 2 \times 2 \times 1 = 4 \) cubes.
- The **number of white cubes** \( W \) can be calculated by subtracting the number of shaded cubes \( S \) from the total number of cubes.

Thus, for each diagram, the number of white cubes \( W \) is given by:

\[
W = \text{Total cubes} - S
\]

where \( S = 4 \) for all diagrams. For each specific diagram:
- First diagram: \( W = 27 - 4 = 23 \)
- Second diagram: \( W = 48 - 4 = 44 \)
- Third diagram: \( W = 75 - 4 = 71 \)

### General Formula:
For a cuboid with dimensions \( L \times W \times H \) and a constant shaded volume of \( S = 4 \), the formula linking shaded cubes \( S \) and white cubes \( W \) is:

\[
W = (L \times W \times H) - S
\]

where \( L \), \( W \), and \( H \) are the dimensions of the cuboid.

Would you like further details or have any questions? Here are five related questions to expand this:

1. How does the formula change if the dimensions of the shaded block change?
2. How would you calculate the total surface area of the cuboids?
3. What is the volume of the shaded region in terms of the overall cuboid volume?
4. How can we generalize the formula for any arbitrary shaded volume?
5. How do the formulas change if the cuboids are not uniform?

**Tip:** Always remember to visualize the cuboid's structure and carefully count the cubes before deriving a formula.

Find the formula that links the shaded cubes (S) and the white cubes (W) in each diagram of cubes.

This math problem demonstrates how to calculate the relationship between shaded cubes (S) and white cubes (W) in 3D diagrams of cubes. Learn how to use simple geometry and volume formulas to break down cuboids into their parts and find the correct formula for any set of cubes.

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