Pearls, Explained

In the first step, we draw the following lines:

1) Any white pearls on the edge get a line right through them, with no angle on the ends (because we don't know if one end, or both ends will have the required bend).

2) Any pairs of white pearls, or two white pearls with a single space between them (still on the edge) get a line between them with a bend away from the edge on both ends. This is required. as the line going straight through the adjacent pearls on one end means it HAS to bend on the other end.

3) Any black pearls within one space of the edge get a line coming out of them perpendicular to that edge. This is because a line cannot run straight through the pearl parallel to the edge, and it cannot go into the edge (or go the required 1 straight square before turning).

The next step (which can be iterrated as needed) draws the following lines:

1) And adjacent black pearls (see right-of-center, 2nd row from bottom) get a line coming out of them, away from the other pearl. That is because the line cannot go into the other black pearl, as that would require it to bend instead of going straight).

2) Any line terminating in a white pearl continues straight through it.

3) Any black pearl within 1 space of a line it cannot join (see black pearl, 4th row, just left of center), must get a line going the other way. That line is no more "crossable" than the edge.

4) Any line which has two options, one of which makes a small loop, must go the other way. See the Top left cluster of 3 black pearls. My options are right and left, from the pearl closest to the center of the grid. Going left gives me a small loop, which would be cut off from the larger string. Therefore, I must go right.

Now, we have to connect some of those line segments. This is where HARD puzzles get tricky. Easy puzzles force you almost to the final conclusion. Rules are harder, other than the overall "Make a single closed loop" rule. So let's look at some decisions:

1) The line along the lower edge, connecting the white pearls:
Not too hard - the line has nowhere else to go. But since it comes straight out of the white pearl on the bottom, it forces the other edge to make a turn.

2) Top-right corner:
The black pearl has two options, right or left. Both are legal. But if it goes left, the line segment in the top left corner gets "stranded" with nothing to connect to. Therefore, it must go right. The same is true for the top center group. Once the black goes right, the white pearl must connect to the vertical segment from the black pearl, or it would become stranded.

This leads to cascading effects, if you are lucky. Once the vertical path from the black pearl in the top row goes to the left, the black pearl directly beneath it must have it's path turn south. Which means the black pearl next to that path must now go to the right.

Once you get it all figured out, you get something like this: