Place numbers 1 to 64 in the cells of this 8 x 8 board in such a way that consecutive numbers occupy neighboring cells (either vertically or horizontally). Shaded cells must be occupied by prime numbers.
The start has to be:
There's only a handful of primes that are separated by 2. And, apart from 3-5-7, there's no 3 odd numbers in a row that are prime (well-known result, but easily verified up to 64). So that shows us this:
Note the corners have to be connected to the two squares on either side. This means the top-left and bottom-right corners can't be looping back onto the shaded areas (that would be two primes separated by 1). And the bottom left is one of the 5 composites in a row. The lack of 2 primes in a row add a few more restrictions to the path, which I've shown as a red line (i.e. the path can't go through the red lines).
Next I took an intuition that the bottom left must surround the 47-53-59 sequence (with 53 on the isolated shaded spot). I tried a few ways to get the end sequence (which needs 59-61-and the final tail of 3 composites), and eventually hit on using up the space on the right (I almost discarded it because it's not unique). But doing this and the rest fell in place: