You are given 4 identical balls. What is the maximum number of square voids and triangular voids (in separate arrangements) that can be created ?

Options

A. 1,2
B. 2,1
C. 3,1
D. 1,3

Answer

To solve the problem of determining the maximum number of square voids and triangular voids that can be created using four identical balls, we will analyze the arrangements of these balls step by step.

### Step 1: Understanding Square Voids
A square void is formed when four balls are arranged in such a way that they create a square-shaped empty space in the center. This arrangement requires exactly four balls.

- **Arrangement**: Place the four balls at the corners of a square.
- **Result**: This arrangement creates one square void in the center.

### Step 2: Counting Square Voids
Since we only have four balls, and they can only form one square void when arranged at the corners, the maximum number of square voids is:
- **Maximum Square Voids = 1**

### Step 3: Understanding Triangular Voids
A triangular void is formed when three balls are arranged in such a way that they create a triangular-shaped empty space in the center. This arrangement requires exactly three balls.

- **Arrangement**: Place three balls in a triangular formation (e.g., one ball at the top and two balls at the bottom).
- **Result**: This arrangement creates one triangular void in the center.

### Step 4: Counting Triangular Voids
With the remaining ball, we can create additional triangular voids by rearranging the balls. Here’s how we can do it:

1. **First Triangular Void**: Arrange three balls in a triangular formation.
2. **Second Triangular Void**: The fourth ball can be placed in such a way that it does not interfere with the triangular void created by the first three balls. For example, it can be placed on top of one of the balls or outside the triangular formation.

- **Result**: We can create two distinct triangular voids with the four balls.

### Conclusion
After analyzing both arrangements, we conclude:
- **Maximum Square Voids = 1**
- **Maximum Triangular Voids = 2**

Thus, the final answer is:
**1 square void and 2 triangular voids.**

### Final Answer:
**1, 2**

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