Alternate Corners is part of the Loop Puzzle genre. Alternate Corners has an unclear origin. It goes under many names : Every Second Turn, Roundabouts, Intermediate Bend, and others that are similar. My best guess is that it was an entry in a puzzle contest at some point and it caught on.
With most loop puzzles, it’s usually a good idea to evaluate the border first. Since the loop must pass through every cell, all corners must have a bend, whether or not they contain a circle. In addition, all circles along the edge must have a line pointing toward the center of the grid, because every one of them is a corner.
Lines Between Circles
There are two basic things to watch for. First, look around your grid for bends outside of circles. Initially, this will probably be in a corner that doesn’t contain a circle. because a line only bends one time between circles, the ends must travel straight until the next circle they touch.
Next, look for a line approaching a cluster of circles that are close together. Has it turned yet? If not, then you know that it must do so before reaching a circle.
Forced Exits
Every line you place acts as a wall for adjacent cells. Eventually, those walls will close off two sides of a cell. Because our loop has to travel through every cell, that means the remaining two sides must be exit points for that cell. Sometimes that means the cell contains a straight line, and sometimes it means a new corner. So when a cell only has two exit points remaining, you can safely draw in the line to fit.
Double-Bends, Loops, and Branches
As you solve, pay attention to potential illegal placements. Remember, a line may only turn one time between circles, and the line of the loop may not create any separate loop or branch. That is, a cell will never have more than 2 exit points. If it helps, place a small X on cell borders where you can’t place a valid line.
Solving the Puzzle
As you can see, Alternate Corners is one of the more simple loop puzzles. Basically, as long as you avoid creating loops, branches, or double-bends, it’s pretty straightforward. After you get the hang of it, you’ll be ready to try a more difficult loop puzzle, such as Masyu. Now, let’s solve the 10×10 example puzzle.
Connect all the circles by darkening lines on the grid. Numbers indicate how many lines are connected. Lines may make no more than one right-angle turn between circles.