To find the total number of squares in a $10 \times 10$ square grid, we use the formula:

$\sum_{k=1}^{n} (n-k+1)^2$

where $n$ is the grid size (in this case, $n = 10$), and $k$ represents the side length of the squares.

Step-by-step Calculation:

1. Count $1 \times 1$ squares:
   $(10 - 1 + 1)^2 = 10^2 = 100$

2. Count $2 \times 2$ squares:
   $(10 - 2 + 1)^2 = 9^2 = 81$

3. Count $3 \times 3$ squares:
   $(10 - 3 + 1)^2 = 8^2 = 64$

4. Count $4 \times 4$ squares:
   $(10 - 4 + 1)^2 = 7^2 = 49$

5. Count $5 \times 5$ squares:
   $(10 - 5 + 1)^2 = 6^2 = 36$

6. Count $6 \times 6$ squares:
   $(10 - 6 + 1)^2 = 5^2 = 25$

7. Count $7 \times 7$ squares:
   $(10 - 7 + 1)^2 = 4^2 = 16$

8. Count $8 \times 8$ squares:
   $(10 - 8 + 1)^2 = 3^2 = 9$

9. Count $9 \times 9$ squares:
   $(10 - 9 + 1)^2 = 2^2 = 4$

10. Count $10 \times 10$ squares:
    $(10 - 10 + 1)^2 = 1^2 = 1$

Summing all these:

$100 + 81 + 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 = 385$

Final Answer:

There are 385 squares in a $10 \times 10$ grid.

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Would you like a visualization or more details? Here are some related questions:

1. How many rectangles are in a $10 \times 10$ grid?
2. What is the formula for counting squares in an $n \times n$ grid?
3. How does this formula generalize to a rectangular $m \times n$ grid?
4. What is the number of squares in a $8 \times 8$ chessboard?
5. How do you count non-overlapping squares in a grid?

Tip: The formula for the total number of squares in an $n \times n$ grid is:

$\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$

Let me know if you need further explanations! 😊