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The problem asks to determine the center of mass of a square plate, originally uniform, with certain squares removed. Let's analyze each case step by step:

### 1. **Originally (no squares removed):**
The square is symmetric, and since no masses have been removed, the center of mass lies at the **center of the square**, i.e., at the **origin (0, 0)** in this coordinate system, where the x- and y-axes bisect the square.

### 2. **When square 1 is removed:**
- Removing square 1 (top-left corner) reduces the mass in quadrant 2.
- The center of mass will shift slightly to the **right** and **down**, because more mass remains in quadrants 3 and 4.
- Therefore, the new center of mass will be in **quadrant 4**, close to the origin but slightly shifted towards the bottom-right.

### 3. **When squares 1 and 2 are removed:**
- Both squares 1 (top-left) and 2 (top-right) are removed.
- The remaining mass is concentrated in quadrants 3 and 4, which are symmetric about the y-axis but entirely below the x-axis.
- The center of mass will shift **directly down** along the y-axis, and remain on the **negative y-axis** at some point below the origin, likely still near the x-axis.

### 4. **When squares 1 and 3 are removed:**
- Squares 1 (top-left) and 3 (bottom-right) are removed.
- The remaining mass is in quadrants 2 (top-right) and 4 (bottom-left), symmetrically distributed across both axes.
- Because the remaining mass is symmetrically located, the center of mass remains at the **origin (0,0)**, as the symmetry is preserved.

### 5. **When squares 1, 2, and 3 are removed:**
- With squares 1, 2, and 3 removed, only square 4 (bottom-left) remains.
- The center of mass now shifts towards the location of square 4, which is in **quadrant 3**.
- The center of mass will be in **quadrant 3**, at some point near the remaining square's center.

### 6. **When all squares are removed:**
- If all squares are removed, there is no mass left.
- Thus, the concept of the center of mass becomes meaningless, as the plate is entirely removed. We would say the center of mass is **undefined** in this case.

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Would you like further clarification or a more detailed explanation of any part of this problem?

Here are 5 related questions to expand on this topic:
1. How is the center of mass calculated for irregularly shaped objects?
2. What are the effects of symmetry on the center of mass of a system?
3. How does removing mass affect the moment of inertia in a similar system?
4. Can the center of mass lie outside the physical boundaries of an object?
5. How would the center of mass change if only a portion of a square were removed, rather than an entire corner?

**Tip**: When dealing with symmetry in center of mass problems, always consider the axes of symmetry as these provide insight into where the center of mass will likely shift.

The figure shows a uniform square plate from which four identical squares at the corners will be removed. In terms of quadrants, axes, or points (with no calculation), where is the center of mass of the square originally, when square 1 is removed, when squares 1 and 2 are removed, when squares 1 and 3 are removed, when squares 1, 2, and 3 are removed, and when all squares are removed?

Analyze how the center of mass of a uniform square plate changes when different corner sections are removed. This problem involves symmetry and shifts in the center of mass based on mass distribution, suitable for college-level physics students.

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