How to Solve Paint by Pairs Puzzles
Paint by Pairs is another puzzle originally created in Japan. However, this time, It first appeared in a publication other than Nikoli. Publisher Byakuya Shobo included hand-created “Number Net” in many of their puzzle magazines before 2002. This is when Conceptis, a video game company, developed an algorithm allowing them to generate the puzzle more quickly. They called their puzzles “Link-a-Pix,” and soon magazines around the world started carrying these puzzles under different names.
Rules
This is a cell shading puzzle solved in two stages. First, connect pairs of numbers with a path through orthogonally adjacent cells. Next, shade these paths in to reveal a hidden image.
- Number pairs indicate the total count of cells in the path, including those containing the numbers.
- A 1 cell has no pairing, because that path is simply one cell long.
- Paths travel left, right, up, or down without crossing themselves or any other path.
- You must use all of the numbers, but not necessarily every cell.
Numbers must be paired, and show the count of cells along their path. 
A 1 clue has no pair, as that path is only one cell long. 
Paths may not cross or move diagonally, and you must use all numbers.
Basic Techniques to Solve
- Solve the smallest numbers first.
- Search for numbers with fewer options to pair up.
- Count the direct route to eliminate a pairing.
- Avoid blocking other numbers.
- Use partial lines when multiple paths use the same cells.
Start Small
Generally, the smallest pairings are the easiest to solve. As a bonus, marking them early restricts options for longer paths later.
- 1s are simply a shaded single cell.
- Pairs of 2s become shaded dominoes. Be wary of clusters and make sure your shading actually uses all of the 2 pairs.
- 3 pairs form either a straight line or a short L-shape. Look for 3s next to other clues, as they will often wrap around them.
Isolated Clues
Search for numbers with fewer possible pairings near them. Many times, you will find only one possible path between them. Most of the time, a path that hugs either the edge of the grid, or one created by an earlier path is a pretty safe bet. Remember that paths include the number, so for any clue X, you’re looking for a match that is X-1 cells away.
In this case, each of the circled 4s has only one possible pairing that’s 3 cells away. 
Fortunately, short pairings like this often have only one possible path, so we can mark them pretty easily. 
Sometimes, pairings are obvious from the beginning, but you might wait until later to determine the route.
Count Cells Along the Direct Route
This is especially useful for larger numbers. If you have a couple of nearby candidates, try counting the cells along a direct path between them.
If you’re off by exactly 1, then that cannot be the correct pairing. This is because no matter how a potential path changes from the most direct one, it will add at least 2 cells to the count.
Just don’t forget, the cells with the numbers must be included in your count. Use this technique only when you have two or three possible options for a pair, as you don’t want to get bogged down checking too many possibilities.
Here’s a small example. The direct route between these 4s is only three cells. Notice how a reroute makes it five.
Avoid Blocking
Sometimes, a clue will have more than one option for a pair. Try to eliminate options that block other numbers from pairing up.
In addition, paths may not cross each other, so if you find a pair with more than possible route, see if they all pas through the same general area. If so, look at other clues nearby to see if you can restrict the paths by solving those clues first. If you’re stuck, look at another area of the puzzle, as you will likely create restrictions that help you.
Use Partial Lines
Remember that you don’t always need to figure out the entire path at once. Think of it like you’re working on a very large Arukone puzzle. When a number seems to be trapped, look for the escape route it must use to avoid being blocked off. Start drawing a path using only as many cells as necessary to have more than one direction of travel.
Resetting the 10×10 example, let’s look at the 4 and 6 at the bottom. They must both start traveling to the left. 
Of course, the 1 then forces the 4 path to turn upward. Now the 6 has only one direction of travel. 
Yes, you can easily see their pairings and the rest of the route, but this illustrates the principle of partial lines.
Similarly, if a particular pairing has more than one way you can connect them, but all of the options run through some of the same cells, you can mark just those cells used by all options. This technique is most useful for long paths in more difficult puzzles. I don’t have an illustration for this specific situation, but you will see me use this method during the walkthrough puzzle, since it’s more complex.
Solving the Puzzle
In general, start with smaller puzzles first to get a feel for how they work. Our example puzzle for the walkthrough is a larger 20×20, and designed with higher complexity so we can look at more solving techniques. If you’d like to try to complete the example puzzle we’ve been using, or follow along on the walkthrough puzzle, click the links below!
Starting Small
Once again, here’s our 20×20 example puzzle. 
We start by filling in all the single-cell 1 clues. 
After marking all the 2 pairs, we look at the 3s. Here, they’re pretty isolated, and create straight lines.
Now let’s look at some isolated numbers (circled). They each have only one nearby pairing (highlighted). Count the cells – you will see only one possible route for each. 
Looks like we’re starting to see some shapes! Once more, I’m circling numbers with only one possible pairing, and only one way to connect them. 
Much better! And look at this! We only have one pair of 10s left! Let’s examine their possible paths in the next phase of our solution.
Remember Partial Lines?
We can only draw two possible paths between these 10s. Since both pass through these 6 cells, we know at least that part of the line must exist. 
Now we see that this 9 is trapped and must escape downward, because there’s no match above it. Conveniently, that break one of the routes for the 10s. 
After shading in the 10s, we can extend the partial route for the 9, and now we see only one path to a match. Let’s look at a few more examples.
This 9 can’t enter any of the red cells, because they’d be dead ends. Therefore, it must travel upward. 
This 8 has no match below it, and to the left is a dead end. So it, too, must travel upward. 
Now this 8 has only one direction of travel, and it can’t connect to the other line, because that would form a path of only 5 cells. So it must turn upward.
Watch the Count
Sometimes, it helps to count cells in partial lines to see if they connect. Here, we have 4+3 = 7, so no. The actual pairs must be the 8s at the top, color-coded to match. 
At the bottom of the grid, a quick count shows these 6s can’t be paired together, since the direct route is only five cells. 
Each 6 has only one other option within reach, so they must be the correct pairs.
Over on the right side, these 5s are four cells apart, so they can’t pair together. 
Each of them also has only one other option in reach, so they must be the correct pairs. 
Before moving on, let’s knock out a few of the pairings with only one possible route.
A Tricky Path
Let’s look at these three clues. The 8s obviously pair together, since there’s nothing else nearby. They all must exit to the left. 
Now the 9 and the upper 8 each have only one direction of travel, so they must extend one more cell. 
The 9 is still forced to continue to the left, and then it must turn downward, because it’s too early to connect to another 9.
This forces the top 8 to turn downward, which in turn blocks the lower 8, so it also turns down. A quick count, shows 4+3=7, so the 8 lines must connect in this cell. 
From this point in the known 9 path, we have two possible options for a pairing. Let’s see if we can eliminate one of them. 
This route forces the 4s to connect, isolating this 9. Not only does that tell us that the other 9 path is correct, but also that the 4s connect on their right side.
Wrapping Up
Now these 9s must pair up with a crooked path, and the 4s must be straight lines. Any other route would cut off the other pair. 
Connecting the circled 9s isolates the green one. So the first two columns must connect with a simple up and down path. 
The direct route from the 9 in the top row to either of these 9s is only 8 cells. So it must be paired to the last 9 with a simple down and up path.
This 9 can’t travel to the right, so the final path must be left, down, and back up. 
We’ve solved the Paint by Pairs puzzle! But let’s add a little color, just for fun. 
Ah, much better – my terrible minivan pixel art comes to life!
