TITLE: Length of wire needed to graph a function
QUESTION [5 upvotes]: Question:

$$S = {(x,y): \left|||x|-2|-1\right| + \left|||y|-2|-1\right| = 1}$$
  If S is made out of wire, find the length of the wire required.

I have no clue as to where to begin this question. Please note that this was a question on an exam and we weren't allowed any graphing equipment or calculator.

REPLY [5 votes]: To draw the graph, start with the graph of $$|x|+|y|=1$$ which is a square with corners at $(1,0),(0,1),(-1,0),(0,-1)$ 
The total length of wire is $4\sqrt{2}$.
Then consider the transformation to $$||x|-1|+||y|-1|=1$$ which has the effect of translating the square so it has its centre at $(1,1)$ but is repeated in all four quadrants. Now the length is $16\sqrt{2}$.
For the final graph, translate this whole shape so that its centre is at $(2,2)$ and again replicate this lattice of four squares in all four quadrants. You now have $16$ of the original squares, and the total length of wire is $64\sqrt{2}$
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