You are given 6 identical balls . The maximum number of square voids and triangular voids (in separate arrangements ) that can be created respectively are

Options

A. `2, 4`
B. `4, 2`
C. `4, 3`
D. `3, 4`

Answer

To solve the problem of determining the maximum number of square voids and triangular voids that can be created with 6 identical balls, we will analyze the arrangements step by step.

### Step 1: Understanding the Concepts
- **Square voids** are formed when 4 balls are arranged in a square configuration, allowing for empty spaces that are square-shaped.
- **Triangular voids** are formed when 3 balls are arranged in a triangular configuration, allowing for empty spaces that are triangular-shaped.

### Step 2: Arranging for Square Voids
1. **Arrangement of Balls**: To maximize square voids, we can arrange the balls in layers. For example, we can place 3 balls in the upper layer and 3 balls in the lower layer.
2. **Identifying Square Voids**: In this arrangement, the empty spaces between the balls create square voids.
- **Calculation**: In the arrangement of 3 balls on top of 3 balls, we can see that there are 2 square voids formed.
3. **Conclusion for Square Voids**: The maximum number of square voids that can be created with 6 balls is **2**.

### Step 3: Arranging for Triangular Voids
1. **Arrangement of Balls**: For triangular voids, we can arrange the balls in a triangular formation. For example, we can place 3 balls in the first layer, 2 balls in the second layer, and 1 ball in the third layer.
2. **Identifying Triangular Voids**: In this arrangement, we can identify triangular voids formed between the layers.
- **Calculation**: In this triangular arrangement, we can find 4 triangular voids.
3. **Conclusion for Triangular Voids**: The maximum number of triangular voids that can be created with 6 balls is **4**.

### Final Answer
Combining the results from both arrangements, we find that the maximum number of square voids is **2** and the maximum number of triangular voids is **4**.

Thus, the answer is:
**Option A: 2 square voids and 4 triangular voids.**

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