There is a painting brush tool that can paint a circle area of $1$ unit radius. What is the shortest paint track for painting at least a $100*100$ units rectangle area on wall? How much did the brush moved in total?
- Brush must touch wall only once. (It says that painting should be continues and you have only one shot)
- Painting can start and finish anywhere.
- Painting outside the rectangle does not matter. (Actually it is inevitable to paint some of the outer area of rectangle)
Here's a naive try for a $10*10$ rectangle and the same brush with radius of $1$ unit result is $58$ units for $10*10$ rectangle and $(n*m)+2m-2$ for $n*2m$ rectangle