The road that goes from the town to the mountain cottage is $76$ km long. A group of hikers finished it in $10$ days, never travelling more than $16$ km in two consecutive days, but travelling at least $23$ km in three consecutive days. Find the maximum ammount of kilometers that the hikers may have traveled in one day.
1. **Define Variables and Constraints:**
   - Let \( a_1, a_2, \ldots, a_{10} \) be the distances traveled by the hikers on each of the 10 days.
   - The total distance is \( a_1 + a_2 + \cdots + a_{10} = 76 \) km.
   - The constraints are:
     - \( a_i + a_{i+1} \leq 16 \) for \( i = 1, 2, \ldots, 9 \) (never traveling more than 16 km in two consecutive days).
     - \( a_i + a_{i+1} + a_{i+2} \geq 23 \) for \( i = 1, 2, \ldots, 8 \) (traveling at least 23 km in three consecutive days).

2. **Analyze the Constraints:**
   - From \( a_i + a_{i+1} \leq 16 \) and \( a_{i+1} + a_{i+2} \leq 16 \), we get:
     \[
     a_i + 2a_{i+1} + a_{i+2} \leq 32
     \]
   - From \( a_i + a_{i+1} + a_{i+2} \geq 23 \), we get:
     \[
     a_i + a_{i+1} + a_{i+2} \leq 32 - a_{i+1}
     \]
     \[
     23 \leq 32 - a_{i+1}
     \]
     \[
     a_{i+1} \leq 9
     \]

3. **Implications of \( a_{i+1} \leq 9 \):**
   - This implies that for days 2 through 9, the distance traveled each day is at most 9 km.
   - We need to check the boundary conditions for the first and last days.

4. **Calculate the Maximum Distance for One Day:**
   - Let’s assume \( a_1 \) is the maximum distance traveled in one day.
   - From the constraints, \( a_1 + a_2 \leq 16 \) and \( a_2 + a_3 \leq 16 \), and so on.
   - Since \( a_2 \leq 9 \), we have \( a_1 \leq 16 - a_2 \leq 16 - 9 = 7 \).

5. **Verify the Total Distance:**
   - If \( a_1 = 7 \), then \( a_2 \leq 9 \), and similarly for other days.
   - We need to ensure the total distance sums to 76 km:
     \[
     a_1 + a_2 + \cdots + a_{10} = 76
     \]
   - Given the constraints, the maximum distance for one day is 9 km.

The final answer is \( \boxed{ 9 } \) km.