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This problem describes two hollow cubes \(C_1\) and \(C_2\), both of negligible mass, partially filled with liquids of densities \(\rho_1\) and \(\rho_2\), respectively, and floating in water of density \(\rho_w\). We need to determine the relationship between the densities \(\rho_1\), \(\rho_2\), and \(\rho_w\).

### Key insights:
1. **Floating Condition**: Since both cubes are floating, the buoyant force on each cube must equal the weight of the liquid inside. This means:
   \[
   \text{Buoyant force} = \text{Weight of the liquid in the cube}
   \]
   The buoyant force is proportional to the volume of water displaced, and the weight of the liquid is proportional to the volume of liquid in the cube.

2. **Buoyant Force for Each Cube**:
   - For cube \(C_1\), the buoyant force is \(V_1 \rho_w g\), where \(V_1\) is the volume of water displaced by cube \(C_1\), \(\rho_w\) is the density of water, and \(g\) is the acceleration due to gravity.
   - For cube \(C_2\), the buoyant force is \(V_2 \rho_w g\).

3. **Weight of the Liquid**:
   - The weight of the liquid in cube \(C_1\) is \(V_{l_1} \rho_1 g\), where \(V_{l_1}\) is the volume of liquid in cube \(C_1\) and \(\rho_1\) is the density of that liquid.
   - The weight of the liquid in cube \(C_2\) is \(V_{l_2} \rho_2 g\), where \(V_{l_2}\) is the volume of liquid in cube \(C_2\) and \(\rho_2\) is the density of that liquid.

4. **Equilibrium Condition**:
   - For cube \(C_1\), the equilibrium condition is:
     \[
     V_1 \rho_w g = V_{l_1} \rho_1 g
     \]
     \[
     \Rightarrow V_1 \rho_w = V_{l_1} \rho_1
     \]
   - Similarly, for cube \(C_2\):
     \[
     V_2 \rho_w g = V_{l_2} \rho_2 g
     \]
     \[
     \Rightarrow V_2 \rho_w = V_{l_2} \rho_2
     \]

5. **Relative Submersion**: If cube \(C_1\) is submerged more deeply than cube \(C_2\), it means \(V_1 > V_2\), indicating that \(\rho_1\) is less than \(\rho_2\).

### Conclusion:
The cubes are in equilibrium due to the balance of buoyant forces and weights. Thus, the relationships between the densities can be expressed based on the equilibrium conditions:
\[
\frac{\rho_1}{\rho_w} = \frac{V_1}{V_{l_1}}, \quad \frac{\rho_2}{\rho_w} = \frac{V_2}{V_{l_2}}
\]
In general, if one cube is more submerged, the liquid inside that cube has a lower density compared to the other.

Would you like more details or have any questions on this? Here are some related questions to consider:

1. What is Archimedes' principle, and how does it relate to floating objects?
2. How can the densities of different liquids affect the submersion depth of floating objects?
3. What happens if the density of the liquid inside the cube equals the density of water?
4. How can you experimentally determine the density of an unknown liquid using floating objects?
5. What other factors could influence the floating condition of hollow objects?

**Tip**: When working with floating objects, always consider the balance between buoyant forces and the weight of the object to analyze equilibrium conditions.

Understand the relationship between densities p1, p2, and pw using the buoyant force and weight equilibrium condition. The problem explores the floating conditions of hollow cubes partially filled with liquids, based on Archimedes' Principle and density calculations. Suitable for high school students in Grades 11-12.

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