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To solve this problem, let’s analyze the configuration and properties of the moments of inertia \( I_1 \), \( I_2 \), and \( I_3 \) with respect to the masses and axis locations:

### Problem Breakdown
1. **Masses**:
   - There are five spherical masses arranged at the corners of a hexagon.
   - White spheres have mass \( 2M \), and gray spheres have mass \( M \).

2. **Axes of Rotation**:
   - \( I_1 \), \( I_2 \), and \( I_3 \) represent the moments of inertia about three different axes passing through the center of the hexagon.
   - Each axis orientation and corresponding masses affects the moment of inertia.

3. **Moment of Inertia Calculation**:
   - The moment of inertia \( I \) for a system of masses about an axis is calculated as:
     \[
     I = \sum m_i r_i^2
     \]
     where \( m_i \) is the mass of each sphere, and \( r_i \) is the perpendicular distance of each mass from the axis.

### Comparison of Moments of Inertia
Let’s evaluate the contributions of masses for each axis:
- **Axis \( I_1 \)**:
  - This axis passes through two white spheres (mass \( 2M \) each), which are closer to the center, and three gray spheres (mass \( M \) each) positioned further away.
  - As a result, \( I_1 \) will include significant contributions from both white and gray spheres.

- **Axis \( I_2 \)**:
  - This axis also passes through two white spheres (mass \( 2M \)), but at a different orientation than \( I_1 \).
  - Similar to \( I_1 \), it includes contributions from both sets of masses, potentially with a different arrangement affecting the moment.

- **Axis \( I_3 \)**:
  - This axis passes through a white sphere and two gray spheres.
  - With fewer contributions from the higher-mass white spheres, \( I_3 \) is likely to be lower compared to \( I_1 \) and \( I_2 \).

### Conclusion
After evaluating the arrangements, the correct ranking is likely:
\[
I_3 < I_1 < I_2
\]
since \( I_3 \) has fewer large masses and lower distances compared to \( I_1 \) and \( I_2 \), and \( I_2 \) has a higher contribution due to its alignment with both white spheres.

### Answer
The correct answer is:
- **\( I_3 < I_1 < I_2 \)**

Would you like further details on how each axis contributes to the moment of inertia? Here are some questions for further exploration:

1. How does the distance of each mass from the center affect the moment of inertia?
2. Why do moments of inertia differ when axes pass through different sets of masses?
3. What role does mass distribution play in determining the rotational inertia?
4. How does symmetry in mass arrangement simplify moment of inertia calculations?
5. What would change if all spheres had the same mass?

**Tip**: Always consider both mass and distance when calculating moments of inertia for complex systems.

Five spherical masses are arranged on the corners of a regular hexagon. The white spheres have mass 2M while the gray spheres have mass M. Let the I1, I2, and I3 be the moment of inertia of the system about the corresponding axes of rotation as shown. Which of the following choices correctly ranks I1, I2, and I3 from lowest to highest?

Explore how to compare moments of inertia for a hexagonal arrangement of masses with varying values. This problem covers concepts in rotational dynamics, including moment of inertia and mass distribution effects. Suitable for undergraduate physics students.

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