The Scaled Maps Problem

Below are two maps of the United Sates of America. The smaller map is exactly half the original, that is, a 50% scaled copy of the original. The edges of the two maps are parallel to each other (so the smaller is not rotated compared to the first). Imagine that the maps are printed out, with one resting on top of the other. The smaller map must be completely contained within the larger one, but otherwise can lie anywhere.
Believe it or not, you can stick a pin straight through both maps so that the pin simultaneously pierces the identical geographical location on each one! This is obviously true if both maps have been centred. Then the pin must be stuck through this (common) centre.  How can you find this location if the smaller map is placed randomly on the larger?
Various solutions exist and are beautifully discussed in the web-site link given below. Here we mention some of them. In fact, before you read further, try finding the solution yourself! It can be fun.

Method One
Take a scale and mark off the coordinates along the x and y axes of both maps. Since the second map is half the scale of the other, mark off half the distances on this map so that the "size" of both maps is the same: for instance, 13 X 8, as shown in Fig 2. Wrap two strings, one vertically, and the other horizontally across the map, as shown. (In the interactive web model, these lines can be easily dragged.)
First move the horizontal string so that the y-axis readings on both maps is the same (in the Fig 2, they are very close to the right value, somewhere between 3 and 4). Do the same for the vertical string so that the x axis values match. This is the common point where the two maps show the same town. This is where the pin must be pierced. Why do you think this method works?

Method Two
To understand this method, first look at some special cases shown in Fig. 3. On the left, the maps match at the top-left, and on the right, at the top-middle. It is easy to see that A is the common point of the two maps on the left and B for that on the right.  To proceed further, think about the relationship between point A and M_1 and M_2.  And what about the relationship between point B and M_1 and M_2?  Hint: they are equally spaced.
Let us return to the original problem and mark the centres M_1 and M_2 (see Fig 4). Join the centres and extend it to the right by a distance equal to that between the centres. This is the point where a pin will go through the same location on both maps.  Question: Why to the right, why not to the left?  Hint: Go back and understand why point A is to the top left of the large map and not to the bottom right.

Method Three
Another interesting solution is to understand the scaling relation between the two maps. In Fig 5, the points P and Q have been marked so that the distance of P from the bottom left corner of the larger map is the same as the scaled distance of point Q from the same corner but of the smaller map. To do this we first find the coordinates of C.
What we are really trying to find is the common point such that the scaled distances of the point from the bottom left corners of each of the maps is the same.  Hence, we have to keep searching for a point P so that P finally is the same as Q. Clearly we have to keep moving P to the right in Fig 5. The procedure then is to plot P somewhere inside the smaller map region, and find the corresponding Q. Repeat until P is very close to Q.
More generally, if point P‘s coordinates are (x, y), what is the location of point Q? Can you use this information to solve algebraically for the location where points P and Q meet? Remember that the coordinates of points on the smaller map are offset by an amount (x_C,y_C) which are the coordinates of the point C.
There are other methods given in the web-site. If you are fascinated, read on at http://www.sineofthetimes.org/the-scaled-maps-problem/