Numberlink 1

Here is one Numberlink puzzle.


This construction is easily generalized to show that n clues suffice to force a unique solution on a 2^n - 1 by N grid whenever N \ge 2^n - 1. That the shorter side of a grid which can be forced with n clues grows at least exponentially with n is quite surprising – when I first started wondering about this, I expected linear growth with a slope of about 2 or 4. Is a shorter side larger than 2^n - 1 possible?