How to Solve
CORALINE

       Coraline is the new CORners And LINEs logic puzzle from the Contest Center. The object is to draw a single continuous path through every square in the grid, once each. Your path may contain only horizontal and vertical line segments and box corners. It may turn only at the center of a square, and it must end in the same square as it started.

       Some lines and corners will already be placed in the grid. They are needed to make sure the puzzle has only one solution. You must use all of these given lines and corners in your path.

       Typically, your solution will begin as a set of small isolated sections of the path. The solution will proceed by connecting these sections into larger pieces, until there is a single path going through all of the squares.

       Be patient. Simply because two short sections are close and can be immediately connected, that does not mean they should be connected that way. Don't connect two sections of the path until you are certain they must be connected. Connecting the wrong sections will prevent you from finishing the path, and it may be difficult to figure out which connection was in error.

       You can figure out the correct connections using just a few basic rules of deduction.

RULE 1:   Extend the lines

       The first rule is that any time a cell contains a line or a corner, that may be extended to the center of the next square. In this example, the line in Grid 1 can be extended to the centers of the two adjacent squares.

Grid 1

Grid 2

RULE 2:   Two neighbors

       The second rule is that any time a square has only 2 available neighbors, that is, only 2 squares that it can connect to, then it must connect to both of them.

       In any Coraline puzzle, the 4 corner squares have only two neighbors, so you can always begin solving in the corners. In this example, Grid 1 shows the corner squares empty. Grid 2 shows connections from the centers of the corner squares to the neighboring squares. Then using Rule 1, these are extended to the centers of the neighboring squares, as shown in Grid 3.

Grid 1

Grid 2

Grid 3

       In the next example, in Grid 1 the middle square on row 3 (shown in yellow) has only 2 neighbors. The middle square cannot be connected to the squares on its left or on its right, so it must connect to the squares above and below, marked in pink. Grid 2 shows connections from the center of the middle square to its neighboring squares. Then using Rule 1, these are extended to the centers of the neighboring squares, as shown in Grid 3.

Grid 1

Grid 2

Grid 3

RULE 3:   Don't block the path

       The next rule is that you must not block your path. All of the parts of your path must eventually be connected. If you block any of the endpoints, you can't complete your path. In the following example, the section of the path starting in the upper left corner is blocked. You need to keep all of the endpoints open.
       A special case of Rule 3 is that you cannot form a closed loop unless it is the entire path. In this example, completing the loop blocks both endpoints of the path section.

Grid 1

Grid 2

RULE 4:   Dead ends

       Don't create dead ends. Any square or group of squares that can be reached from only one other square creates a dead end. You can get in, but you can't get back out. In these examples the yellow squares show types of dead ends.

Example 1

Example 2

Example 3

RULE 5:   Separate regions

       Any time the pieces of your path divide the puzzle into separate regions, each of these regions must contain at least one endpoint, and the number of endpoints must be even. A closed region cannot contain an odd number of endpoints. There is no way to connect them together. There will always be one endpoint left unconnected.

       In this example the white region at the top of the grid contains 3 endpoints of the path. Any two of them can be connected, but the third one will always remain unpaired. Similarly, the white region at the bottom of the puzzle has only one endpoint. There is nothing to which it can connect.

RULE 6:   Unique solutions

       Every Coraline puzzle has a unique solution. You can use this knowledge to help you eliminate some possibilities. Consider Grid 1 below. The lines in the upper left cannot connect horizontally, as shown in Grid 2, because this leaves two squares, marked in yellow, which cannot be reached.

Grid 1

Grid 2

       There are two ways to change these horizontal paths to fill in the two yellow squares. The upper line could dip down, as shown in Grid 3, or the lower line could bend up, as shown in Grid 4. These two methods are exactly equivalent in the way that they connect up the separate pieces of the path. So neither one of them could be part of the solution, since the solution would not be unique.
       Instead, the parts of the path must connect vertically, as shown in Grid 5.

Grid 3

Grid 4

Grid 5

       Here is another example where knowing that the solution must be unique can help you decide how to connect the pieces. Look at the 4 pieces of the path shown in Grid 1. You cannot connect the pieces vertically because that leaves two squares that cannot be reached, shown in yellow in Grid 2.

Grid 1

Grid 2

       As before, there are two ways to change these vertical connections to fill in the two yellow squares. The left connector could be extended to the right, as shown in Grid 3, or the right connection could be extended to the left, as shown in Grid 4. These two methods are exactly equivalent in the way that they connect up the separate pieces of the path. So neither one of them could be part of the solution, since the solution would not be unique.
       Instead, the parts of the path must connect horizontally, as shown in Grid 5.

Grid 3

Grid 4

Grid 5

SAMPLE PUZZLE
       To illustrate the use of these rules, let's solve a sample puzzle. To make the explanation easier, let's call the rows of the puzzle A, B, C, through H. The squares in each row will be numbered 1 to 8, so the upper left corner is A1, the upper right corner is A8, the lower left corner is H1, and the lower right corner is H8. Here is the sample puzzle.

	1	2	3	4	5	6	7	8
A
B
C
D
E
F
G
H

       Let's begin with Rule 2. The four corners always have two neighbors. In addition, the three squares A5, B8 and E8 (in yellow) also have only 2 neighbors.


       Let's put in the correct lines and corners in these squares.


       Next let's use Rule 1, and extend all of these lines and box corners to the centers of the adjacent squares.


       Now square A6 has only 2 neighbors, so we can connect A6 to square B6.


       Rule 3 prevents connecting square C8 to C7. That means that C8 has to be connected to D8.


       Square D7 now has only 2 neighbors, so D7 must be connected to C7 and D6.


       Now square C6 now has only 2 neighbors, so C6 must be connected to C5 and D6.


       If square D4 connected to either C4 or E4, that would create a dead end at D5. By Rule 4, D4 must connect to D5.


       Square C4 now has only 2 open neighbors, so those can be connected.


       Now it's time for a little thought experiment. Suppose that F8 connected to F7. That would lead to the following situation.


       This is an impossible situation. You cannot avoid forming a closed loop in the lower right corner, which is not allowed by Rule 3. This means that F8 must connect to G8, like this


       In the same manner, square H7 cannot connect to G7, because that would leave H6 disconnected. Therefore, H7 has to connect to H6, leading to this configuration.


       Using Rule 2 at G7 leads to this.


       E6 cannot connect to E5, since that would leave a dead end at F5 and F6. Thus E6 must connect to F6, which results in this.


       Let's try another thought experiment. Suppose that F1 connects to G1. Applying Rule 2 repeatedly leads to this layout.


       It would be impossible to complete this path. The region in the upper left has 3 endpoints, and the region in the lower left has 3 endpoints. By Rule 5 these cannot be connected.

       It follows that F1 must connect to E1, so that G1 must connect to G2, like this



       Using Rule 5, E2 cannot connect to E3, otherwise that would split the remaining squares into 2 regions, each one having 5 endpoints. So E2 must connect to D2.



       From this point on, applying Rule 2 and Rule 3 repeatedly leads directly to the solution.






Solved!!

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