Wednesday, October 28, 2015

Four Paths Problem



For the solution, click "Read More" below.


It seems counterintuitive, but the first three paths are all the same size.  If the squares are all unit squares, then the lengths of each of the first 3 paths in 2 units.   One way to understand this is that if we have a stair-step pattern made of right-angled turns, then the sum of the lengths of all of the horizontal lines must 1 and the sum of the lengths of all of the vertical lines must 1 -- no matter how many "steps" there are in the path.  


The diagonal, on the other hand, is shorter.  Its length is the square root of 2 units.

 

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