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To determine the sum of all possible unit digits that can result from squaring numbers, we start by examining each integer from 0 to 9. We square each one and record the unit digit of the result:

1. \(0^2 = 0\)  → Unit digit: **0**
2. \(1^2 = 1\)  → Unit digit: **1**
3. \(2^2 = 4\)  → Unit digit: **4**
4. \(3^2 = 9\)  → Unit digit: **9**
5. \(4^2 = 16\) → Unit digit: **6**
6. \(5^2 = 25\) → Unit digit: **5**
7. \(6^2 = 36\) → Unit digit: **6**
8. \(7^2 = 49\) → Unit digit: **9**
9. \(8^2 = 64\) → Unit digit: **4**
10. \(9^2 = 81\) → Unit digit: **1**

The unit digits that appear in these squares are **0, 1, 4, 5, 6,** and **9**. Other digits (2, 3, 7, 8) do not appear as unit digits of any square number.

Now, we find the sum of the digits that can be unit digits of square numbers:
\[
0 + 1 + 4 + 5 + 6 + 9 = 25
\]

### Answer:
The sum of all digits that can be the unit digit of a square number is **25**.

Would you like a detailed explanation on any part of this solution, or have further questions?

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Here are some related questions for further exploration:
1. What are all possible last two digits of a square number?
2. Can any number be represented as a sum of two square numbers?
3. What are the properties of numbers that have certain unit digits in their squares?
4. How can we determine if a large number is a perfect square just by looking at its last digit?
5. What other types of numbers have unique patterns in their unit digits?

**Tip:** Noticing patterns in unit digits can help quickly identify potential square numbers, especially in larger computations.

4 can be the unit digits of a square number. For example, 12^2 = 144. However, not all digits from 0 to 9 can be the unit digits of a square number. What is the sum of all these digits?

Explore which digits can be the unit digits of square numbers and learn how to find their sum. This problem covers unit digits, modular arithmetic, and the properties of square numbers, making it suitable for middle school students.

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