Determine the last four digits of a perfect square of a natural number, knowing that  the last three of them are the same.
To determine the last four digits of a perfect square of a natural number, knowing that the last three of them are the same, we need to analyze the possible values for the last four digits.

1. **Identify the possible last three digits:**
   - The last three digits of the number must be of the form \( aaa \) where \( a \) is a digit from 0 to 9.
   - Therefore, the possible last three digits are \( 000, 111, 222, 333, 444, 555, 666, 777, 888, 999 \).

2. **Check the congruence conditions for perfect squares:**
   - A number is a perfect square if and only if it satisfies certain congruence conditions modulo 4 and modulo 5.
   - We will check each of the possible last three digits to see if they can be the last three digits of a perfect square.

3. **Check each case:**
   - For \( 000 \):
     \[
     000 \equiv 0 \pmod{4} \quad \text{and} \quad 000 \equiv 0 \pmod{5}
     \]
     Since \( 0 \) is a perfect square modulo both 4 and 5, \( 000 \) is a valid candidate.

   - For \( 111 \):
     \[
     111 \equiv 3 \pmod{4} \quad \text{and} \quad 111 \equiv 1 \pmod{5}
     \]
     Since \( 3 \) is not a perfect square modulo 4, \( 111 \) is not a valid candidate.

   - For \( 222 \):
     \[
     222 \equiv 2 \pmod{4} \quad \text{and} \quad 222 \equiv 2 \pmod{5}
     \]
     Since \( 2 \) is not a perfect square modulo 4 or 5, \( 222 \) is not a valid candidate.

   - For \( 333 \):
     \[
     333 \equiv 1 \pmod{4} \quad \text{and} \quad 333 \equiv 3 \pmod{5}
     \]
     Since \( 3 \) is not a perfect square modulo 5, \( 333 \) is not a valid candidate.

   - For \( 444 \):
     \[
     444 \equiv 0 \pmod{4} \quad \text{and} \quad 444 \equiv 4 \pmod{5}
     \]
     Since \( 4 \) is a perfect square modulo 5, \( 444 \) is a valid candidate.

   - For \( 555 \):
     \[
     555 \equiv 3 \pmod{4} \quad \text{and} \quad 555 \equiv 0 \pmod{5}
     \]
     Since \( 3 \) is not a perfect square modulo 4, \( 555 \) is not a valid candidate.

   - For \( 666 \):
     \[
     666 \equiv 2 \pmod{4} \quad \text{and} \quad 666 \equiv 1 \pmod{5}
     \]
     Since \( 2 \) is not a perfect square modulo 4, \( 666 \) is not a valid candidate.

   - For \( 777 \):
     \[
     777 \equiv 1 \pmod{4} \quad \text{and} \quad 777 \equiv 2 \pmod{5}
     \]
     Since \( 2 \) is not a perfect square modulo 5, \( 777 \) is not a valid candidate.

   - For \( 888 \):
     \[
     888 \equiv 0 \pmod{4} \quad \text{and} \quad 888 \equiv 3 \pmod{5}
     \]
     Since \( 3 \) is not a perfect square modulo 5, \( 888 \) is not a valid candidate.

   - For \( 999 \):
     \[
     999 \equiv 3 \pmod{4} \quad \text{and} \quad 999 \equiv 4 \pmod{5}
     \]
     Since \( 3 \) is not a perfect square modulo 4, \( 999 \) is not a valid candidate.

4. **Conclusion:**
   - The only valid candidates for the last three digits are \( 000 \) and \( 444 \).
   - Therefore, the last four digits of the perfect square could be \( 0000 \) or \( 1444 \).

The final answer is \( \boxed{0000 \text{ or } 1444} \).