Map Folding Problem

The lines in the map above show how it was folded.  Dotted lines show a “valley fold” and solid lines show a “mountain fold.”  This map was folded in half horizontally, then vertically, then vertically again.

Question:  Each of the maps below was folded in four steps – in half each time.  Which of the maps is an “impossible map”?

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For the solution to the map problem, click here:

As with many spatial problems, we can solve this one both spatially (by folding the paper mentally or physically) or by analyzing the diagrams logically.

A good way to start analyzing a novel problem is to start with a smaller version of the problem.  In this case we can do that literally by looking at a smaller map:

What can we tell from the little map?  The map above must have been folded twice: once horizontally and once vertically.  Notice that it is not very important that we keep track of whether the folds are folding in or out because the map could be turned to its backside and the folds would be reversed:  mountain folds would become valley folds and vice versa.  The only thing that is important is to keep track of whether two folds go in the same direction or opposite directions.

How can we tell that the little map was folded first horizontally?  The horizontal fold is the one that goes straight through the map without “flipping” directions. There is no single fold that would produce the following pattern:

Also notice that something always happens when a map is folded twice so that fold #2 goes through fold #1.  We always get one of the two following fold patterns at the intersection.  (We are assuming perpendicular folds, and of course they can differ by a rotation.)  Nothing else is possible.

Once we understand a little about how folds work, we can easily pick out the impossible map in the problem at hand because it has two impossible fold intersections.