A circle of radius $1$ rolls around a circle of radius $\sqrt{2}$. Initially, the tangent point is colored red. Afterwards, the red points map from one circle to another by contact. How many red points will be on the bigger circle when the center of the smaller one has made $n$ circuits around the bigger one?
1. **Understanding the problem**: We have a smaller circle of radius \(1\) rolling around a larger circle of radius \(\sqrt{2}\). We need to determine how many distinct red points will be on the larger circle after the smaller circle has made \(n\) circuits around the larger circle.

2. **Initial setup**: When the smaller circle rolls around the larger circle, the point of contact between the two circles changes. Initially, the point of contact is marked red.

3. **Distance traveled by the smaller circle**: As the smaller circle rolls around the larger circle, it travels a distance equal to the circumference of the larger circle for each circuit. The circumference of the larger circle is:
   \[
   2\pi \sqrt{2}
   \]

4. **Number of red points**: Each time the smaller circle completes one full rotation around its own center, it paints a new red point on the larger circle. The circumference of the smaller circle is:
   \[
   2\pi \times 1 = 2\pi
   \]

5. **Calculating the number of new red points**: After \(n\) circuits around the larger circle, the smaller circle will have traveled a total distance of:
   \[
   n \times 2\pi \sqrt{2} = 2n\pi \sqrt{2}
   \]
   The number of times the smaller circle completes a full rotation around its own center during this distance is:
   \[
   \frac{2n\pi \sqrt{2}}{2\pi} = n\sqrt{2}
   \]

6. **Counting distinct red points**: The number of distinct red points on the larger circle is the integer part of \(n\sqrt{2}\), which is:
   \[
   \lfloor n\sqrt{2} \rfloor
   \]
   Additionally, we need to consider the initial red point, so we add 1 to this count.

7. **Final calculation**: Therefore, the total number of distinct red points on the larger circle after \(n\) circuits is:
   \[
   \lfloor n\sqrt{2} \rfloor + 1
   \]

The final answer is \(\boxed{\lfloor n\sqrt{2} \rfloor + 1}\).