Three students write on the blackboard next to each other three two-digit squares. In the end, they observe that the 6-digit number thus obtained is also a square. Find this number!
1. Let \( a^2, b^2, c^2 \) be the three two-digit squares written by the students, where \( a, b, c \in \{4, 5, 6, 7, 8, 9\} \). The goal is to find a 6-digit number \( n^2 \) such that \( n^2 = \overline{a^2b^2c^2} \).

2. We need to find \( n \) such that \( n^2 \) is a 6-digit number and can be decomposed into three two-digit squares. This means:
   \[
   100000 \leq n^2 < 1000000
   \]
   Taking the square root of both sides, we get:
   \[
   316 \leq n < 1000
   \]

3. We need to ensure that \( n^2 \) can be written as \( \overline{a^2b^2c^2} \), where \( a^2, b^2, c^2 \) are two-digit numbers. This implies:
   \[
   10 \leq a^2, b^2, c^2 < 100
   \]

4. We will test each possible value of \( a \) from 4 to 9 and find the corresponding range for \( n \).

5. If \( a = 4 \):
   \[
   4^2 = 16
   \]
   We need \( n \) such that \( n^2 \) starts with 16. The range for \( n \) is:
   \[
   400 \leq n < 500
   \]
   Testing values within this range:
   \[
   404^2 = 163216 \quad (\text{not valid})
   \]
   \[
   408^2 = 166464 \quad (\text{valid, as } 16, 64, 64 \text{ are all two-digit squares})
   \]
   \[
   410^2 = 168100 \quad (\text{not valid})
   \]

6. If \( a = 5 \):
   \[
   5^2 = 25
   \]
   We need \( n \) such that \( n^2 \) starts with 25. The range for \( n \) is:
   \[
   500 \leq n < 600
   \]
   Testing values within this range:
   \[
   504^2 = 254016 \quad (\text{not valid})
   \]
   \[
   509^2 = 259081 \quad (\text{not valid})
   \]

7. If \( a = 6 \):
   \[
   6^2 = 36
   \]
   We need \( n \) such that \( n^2 \) starts with 36. The range for \( n \) is:
   \[
   600 \leq n < 700
   \]
   Testing values within this range:
   \[
   604^2 = 364816 \quad (\text{not valid})
   \]
   \[
   608^2 = 369664 \quad (\text{not valid})
   \]

8. If \( a = 7 \):
   \[
   7^2 = 49
   \]
   We need \( n \) such that \( n^2 \) starts with 49. The range for \( n \) is:
   \[
   700 \leq n < 800
   \]
   Testing values within this range:
   \[
   704^2 = 495616 \quad (\text{not valid})
   \]
   \[
   707^2 = 499849 \quad (\text{not valid})
   \]

9. If \( a = 8 \):
   \[
   8^2 = 64
   \]
   We need \( n \) such that \( n^2 \) starts with 64. The range for \( n \) is:
   \[
   800 \leq n < 900
   \]
   Testing values within this range:
   \[
   804^2 = 646416 \quad (\text{valid, as } 64, 64, 16 \text{ are all two-digit squares})
   \]

10. If \( a = 9 \):
    \[
    9^2 = 81
    \]
    We need \( n \) such that \( n^2 \) starts with 81. The range for \( n \) is:
    \[
    900 \leq n < 1000
    \]
    Testing values within this range:
    \[
    904^2 = 817216 \quad (\text{not valid})
    \]
    \[
    905^2 = 819025 \quad (\text{not valid})
    \]

Conclusion:
The valid 6-digit numbers that are squares and can be decomposed into three two-digit squares are \( 166464 \) and \( 646416 \).

The final answer is \( \boxed{166464, 646416} \).