A map may be rotated, shifted, and placed almost any way you like. Even so, one coincidence cannot be avoided.
The Map on the City
Riddle statement
You have an exact, scaled-down map of a city.
You place the map in any position and orientation, always entirely inside the real city.
Must there be some point on the map lying exactly on top of the real place it represents?
Show solution
Solution
Yes.
Placing the map defines a transformation: each real point of the city is assigned to the physical spot where that same point appears on the placed map.
Because the map is a reduced copy, this transformation shortens all distances by the same factor, which is less than 1. Therefore it has a unique fixed point: a point that the transformation leaves exactly where it was.
Here is an intuitive way to see it. Choose any point in the city and find where it is represented on the placed map. Then repeat the same operation with the new point. Each jump is shorter than the previous one, because the map reduces distances. The sequence converges to a point that coincides with its own representation.
That fixed point belongs to the city because all intermediate positions remain inside it and the city is being treated as a closed region.
Answer: yes. There is always at least one point on the placed map lying exactly on top of the real place it represents.