Shortest Sierpinski Walk

Geometry Level 5

This is a Sierpinski carpet, bounded by a unit square.

The largest white square has \(\frac{1}{3}\) the side length of the unit square, the next largest white squares have \( \frac{1}{3}\) the side length of the largest, etc. The construction of this fractal starts with dividing a black unit square into 9 equal squares, of which the center is made white and all the rest are left black; then each of the remaining 8 squares is likewise divided into 9 equal squares, of which the center is made white and all the rest are left black, etc., which continues infinitely many times.

What is the length of the shortest path between 2 opposite vertices of the unit square, which does not travel across any of the white squares? The distance can be expressed as \[\dfrac{a\sqrt{b}}{c}, \] where \(a, b, c\) are positive integers, \(a\) and \(c\) are coprime, and \(b\) is square-free.

What is the sum \(a+b+c\)?

Note: Only the interior of the white squares are removed, i.e. the boundaries are left available for any pathway. For example, pathways may include vertices of "white squares."

Note of interest: This Sierpinski carpet has an area of \(0\), which may not be relevant to this problem.

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