Let $\Gamma$ be a circle in the plane and $S$ be a point on $\Gamma$. Mario and Luigi drive around the circle $\Gamma$ with their go-karts. They both start at $S$ at the same time. They both drive for exactly $6$ minutes at constant speed counterclockwise around the circle. During these $6$ minutes, Luigi makes exactly one lap around $\Gamma$ while Mario, who is three times as fast, makes three laps. 

While Mario and Luigi drive their go-karts, Princess Daisy positions herself such that she is always exactly in the middle of the chord between them. When she reaches a point she has already visited, she marks it with a banana.

How many points in the plane, apart from $S$, are marked with a banana by the end of the $6$ minutes.

1. **Understanding the problem**:
   - Mario and Luigi start at the same point \( S \) on the circle \( \Gamma \).
   - Luigi completes one lap in 6 minutes, while Mario completes three laps in the same time.
   - Princess Daisy is always positioned at the midpoint of the chord between Mario and Luigi.
   - We need to determine how many distinct points, apart from \( S \), are marked with a banana by Princess Daisy.

2. **Parameterizing the positions**:
   - Let the circle \( \Gamma \) have a radius of 1 for simplicity.
   - Let \( \theta \) be the angular position on the circle.
   - Luigi's position after \( t \) minutes is given by \( \theta_L(t) = \frac{2\pi t}{6} = \frac{\pi t}{3} \).
   - Mario's position after \( t \) minutes is given by \( \theta_M(t) = \frac{2\pi \cdot 3t}{6} = \pi t \).

3. **Finding Princess Daisy's position**:
   - The coordinates of Luigi's position at time \( t \) are \( (\cos(\frac{\pi t}{3}), \sin(\frac{\pi t}{3})) \).
   - The coordinates of Mario's position at time \( t \) are \( (\cos(\pi t), \sin(\pi t)) \).
   - Princess Daisy's position is the midpoint of the chord between Mario and Luigi:
     \[
     \left( \frac{\cos(\frac{\pi t}{3}) + \cos(\pi t)}{2}, \frac{\sin(\frac{\pi t}{3}) + \sin(\pi t)}{2} \right)
     \]

4. **Identifying distinct points**:
   - We need to find when Princess Daisy's position repeats.
   - Consider the angular positions modulo \( 2\pi \):
     - Luigi's position modulo \( 2\pi \) is \( \frac{\pi t}{3} \mod 2\pi \).
     - Mario's position modulo \( 2\pi \) is \( \pi t \mod 2\pi \).
   - We need to find distinct values of \( t \) in the interval \( [0, 6] \) such that the midpoint position is unique.

5. **Analyzing the midpoint**:
   - The midpoint coordinates are periodic with period \( 6 \) minutes.
   - We need to check the distinctness of the midpoint for \( t \) in \( [0, 6] \).

6. **Checking distinctness**:
   - For \( t = 0 \), the midpoint is \( S \).
   - For \( t = 6 \), the midpoint is again \( S \).
   - For \( t = 2 \) and \( t = 4 \), the midpoint is the same because:
     \[
     \left( \frac{\cos(\frac{2\pi}{3}) + \cos(2\pi)}{2}, \frac{\sin(\frac{2\pi}{3}) + \sin(2\pi)}{2} \right) = \left( \frac{\cos(\frac{4\pi}{3}) + \cos(4\pi)}{2}, \frac{\sin(\frac{4\pi}{3}) + \sin(4\pi)}{2} \right)
     \]
   - For \( t = 1, 3, 5 \), the midpoints are distinct.

7. **Conclusion**:
   - The distinct points marked by Princess Daisy, apart from \( S \), are at \( t = 1, 2, 3, 4, 5 \).

The final answer is \( \boxed{ 5 } \) distinct points.