Numberlink and Nurikabe

This puzzle solves both as a Numberlink and a Nurikabe.

Numberlink 4 and Nurikabe 6

This puzzle solves both as a Numberlink and a Nurikabe.

Numberlink 3 and Nurikabe 5

This puzzle solves both as a Numberlink and a Nurikabe.

Numberlink 2: Wiggly

Here is one Wiggly Numberlink puzzle.

Numberlink 1

Here is one Numberlink puzzle.

This construction is easily generalized to show that clues suffice to force a unique solution on a by grid whenever . That the shorter side of a grid which can be forced with clues grows at least exponentially with 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 possible?