A teacher on attempting to arrange the students for mass drill in the form of a solid square found that 24 students were left. When he increased the size of the square by one student, he found that he was short of 25 students. Find the number of students.

Explanation

To solve the problem, we will set up equations based on the information given in the question.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- Let the number of students be \( N \).
- When arranged in a square formation, there are 24 students left over. This means:
\[ N = x^2 + 24 \]
where \( x \) is the number of students along one side of the square.

2. **Increasing the Square Size**:
- When the teacher increases the size of the square by one student (i.e., \( x + 1 \)), he finds that he is short of 25 students. This gives us the equation:
\[ N = (x + 1)^2 - 25 \]

3. **Setting Up the Equations**:
- From the two equations we have:
\[ x^2 + 24 = (x + 1)^2 - 25 \]

4. **Expanding the Second Equation**:
- Expand \( (x + 1)^2 \):
\[ (x + 1)^2 = x^2 + 2x + 1 \]
- Substitute this back into the equation:
\[ x^2 + 24 = x^2 + 2x + 1 - 25 \]

5. **Simplifying the Equation**:
- Combine like terms:
\[ x^2 + 24 = x^2 + 2x - 24 \]
- Subtract \( x^2 \) from both sides:
\[ 24 = 2x - 24 \]

6. **Solving for \( x \)**:
- Add 24 to both sides:
\[ 48 = 2x \]
- Divide by 2:
\[ x = 24 \]

7. **Finding the Total Number of Students**:
- Substitute \( x \) back into the equation for \( N \):
\[ N = x^2 + 24 = 24^2 + 24 \]
- Calculate \( 24^2 \):
\[ 24^2 = 576 \]
- Therefore:
\[ N = 576 + 24 = 600 \]

### Final Answer:
The total number of students is \( \boxed{600} \).