How to Solve Alternate Corners Puzzles

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.

Rules
Basic Techniques to Solve
Solving the Puzzle

Rules

Draw a single loop which passes through every cell of the grid. Naturally, you have a few restrictions:

You must make a 90-degree turn when passing through a circle. The loop never exits the opposite side of a circle it enters.
While traveling between circles, you must make exactly one 90-degree turn.
All lines must be placed in orthogonal directions, connecting the centers of adjacent cells.
The solution has only one loop which may not cross itself.

Start at the Border

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.

All corners have a turn, whether or not there is a circle.
A circle on the edge holds a turn from either direction.
So a line must extend toward the middle of the grid.

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.

Here, we already marked the easy border lines. Note the bent lines in these corners.
They must extend to reach the circles in their paths.
These lines can’t go straight, because they must bend before reaching a circle.
So we turn the lines and connect to different circles.

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.

These lines walled off this cell. A line can’t exit up or down without creating a branch.
Therefore, the only possible line is straight through.
These cells are walled off by other lines or the puzzle border.
Since the only two exits form a corner, we can draw those in.

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.

Alternate Corners Forced Loop — For example, if we place this yellow line, the line in the red circle is forced into a loop which also creates a branch..

If this line turns to go down, it creates a loop. Up creates a double bend. So it must travel straight.

Before we move on, let’s finish this small puzzle. First, these lines already turned, so they must go straight.

The line in each of these circles must turn. Each has only one direction that doesn’t create a loop or branch.

This line can’t turn upward to create a loop, so it must go straight. These two cells only have one exit point left.

That completes the small puzzle. Now let’s tackle the larger one.

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.

First Steps

First, we place lines in all the corners. We can also draw lines from all border circles toward the middle of the grid.
These lines have already bent between circles, so they must go straight until they hit a circle.
These cells each have only two exit points. They are all corners, so we can extend the lines to circles.

Turn these lines before hitting the circles directly ahead. They each have only one option.
Next, notice the circles with only one possible corner. We can draw them in.
This segment on the right must turn before the next circle. Down would create a double-bend, so it must turn up.

This line at the bottom can’t turn without creating a branch or a loop. Therefore, we go straight one more cell.
We have a few forced corners, so we can draw lines to connect circles.
Now this line can only turn one direction, to avoid a branch or loop.

Forcing Corners

These circles all have only one possible corner.
This cell is interesting. Its line can’t exit down without creating a branch.
It also can’t exit right, because the circle must turn. So we found another corner!

By looking at where we can’t place lines, we find several less-obvious corners.
Now we can simply complete these corners.
Neither of these segments can turn and reach a circle. Therefore, they connect in the highlighted cell.

Finishing Touches

Now this line has a straight path through these cells. However, we don’t know yet where it turns.
This line has to turn before reaching a circle.
This creates two forced corners.

To avoid a loop or branch, this line turns downward.
Now the final two connections are obvious.
We’ve solved the Alternate Corners puzzle!

How to Solve Alternate Corners Puzzles

Rules