If at least how many soldiers are removed from 56728 soldiers or at least how many more soldiers are added to them, the army can be arranged in a square?

To determine how many soldiers need to be removed or added so that the army can be arranged in a square, we need to find the nearest perfect square number to \( 56728 \).

### Step 1: Calculate the square root of 56728.
\[
\sqrt{56728} \approx 238.17
\]
This means that the perfect squares closest to 56728 are \( 238^2 \) and \( 239^2 \).

### Step 2: Calculate the closest perfect squares.
\[
238^2 = 56644
\]
\[
239^2 = 57121
\]

### Step 3: Determine how many soldiers to add or remove.

- **To remove soldiers**:  
  Soldiers to be removed to reach 56644:
  \[
  56728 - 56644 = 84
  \]
  So, 84 soldiers need to be removed.

- **To add soldiers**:  
  Soldiers to be added to reach 57121:
  \[
  57121 - 56728 = 393
  \]
  So, 393 soldiers need to be added.

### Conclusion:
- **At least 84 soldiers must be removed** to arrange the army in a square.
- **At least 393 soldiers must be added** to arrange the army in a square.

Would you like more details or have any questions?

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**Relative Questions:**
1. What is the significance of perfect squares in arranging objects?
2. How can we quickly estimate the square root of a large number?
3. What other shapes besides squares can be formed by arranging a certain number of objects?
4. How would the problem change if arranging soldiers in a rectangular formation?
5. What if you have a number that is already a perfect square—how does that affect the solution?

**Tip:** Understanding how to quickly estimate square roots can be very helpful in solving problems like this one without needing a calculator.

Learn how to arrange 56728 soldiers into a perfect square by adding or removing soldiers. The solution involves calculating square roots and finding the nearest perfect square numbers. This problem introduces the concept of square numbers and is suitable for Grades 6-8.

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