# The Binary Box

Consider an integer $$n$$ in base 10, whose binary form can be expressed as a string $$S$$. Let the minimum value of $$n$$ be $$N$$, such that the sub-strings of $$S$$ include binary numbers from $$1$$ to $$10000_2$$. What is the sum of digits of $$N$$?

Details and Assumptions

• For example, the integer $$9$$, whose binary form can be expressed as a string $$1001$$ includes sub-strings of $$0,1,10,1001$$, but not $$101$$.
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