How to Solve Arukone Puzzles
Arukone has a long and storied history. It appears to have been invented by Sam Loyd, an American who wrote columns for the Brooklyn Daily Eagle. On February 28, 1897, he published Puzzleland Park, which required solvers to connect 8 houses with 4 convoluted paths.
Later, in 1917’s Amusements in Mathematics, by Henry Ernest Dudeney, we see a more familiar grid in A Puzzle for Motorists, though the endpoints were letters instead of numbers. Eventually, Nikoli puzzle magazine in Japan popularized this puzzle as Arukone, again using letters for endpoints. Nanbarinku is the same puzzle, but with numbers, and turned out to be the more popular name when the puzzle returned to English-speaking countries, as Number Link.
Rules
Simply connect each pair of numbers with a single continuous line that passes through the center points of adjacent cells.
- Lines run horizontally or vertically, and may travel straight through a square, or turn 90 degrees.
- Lines may not pass through number cells, or cross another line.
- No more than one line may pass through any cell.
- The solution might not use all cells.
If you find Arukone or Number Link puzzles elsewhere, you should be aware of some common variations to the rules. Always be careful to know all the rules a puzzle designer is using, so that you don’t apply incorrect logic. If I use either of these variations, they will be clearly stated with the puzzle.
In its original rules, Arukone does not require that all cells be used. However, Nikoli magazine has a custom of using all cells in their different puzzles, and it became rare to see an Arukone puzzle with unused cells. Later, they introduced a new form that borrows a rule from Nurikabe puzzles – that a line may not cover a 2×2 area. That is, it won’t make a u-turn or pass itself in adjacent cells.
Basic Techniques to Solve
Arukone puzzles are pretty straightforward to solve, and only have a couple of very basic things to look for. Rather than creating individual illustrations, we’ll see them in use as we solve the example puzzle.
- Look for numbers and line segments with only one path.
- Pay attention to choke points, and avoid isolating other numbers with your path.
- Don’t assume the shortest path.
- Work both ends of the path when you can.
If you happen to be solving a Number Link puzzle that has a rule requiring the use of all cells, there is one additional technique. Remember that every time you create a corner or otherwise use two cells around another one, you’ve cut off two exit points of that cell. As a result, in those puzzles, you also know the path through the cell – either straight through, or another corner. You will frequently see this technique in loop puzzles.
Solving the Puzzle
The First Paths
In our example puzzle, this 11 has a single exit point, so we’ll start there. 
Don’t assume the shortest path. If this line turns downward, it will eventually have to pas through the lower two cells. This will isolate the 10 and 2 pairs. 
Therefore, the 11 must continue to the left and turn. Now the 10 and 4 each have a single exit point.
To get past the 4, the 11 must continue down two cells. This forces both the 2 and 7 to exit to the right. 
Here we have a fun cascade. Because of the 7, the 2 must continue to the right and turn down. The 2 blocks the 10, which must go right, completing it. 
Because paths can’t cross, the 7 and 2 lines must both move downward. The 2 completes its path in the process.
Choke Points and Parallel Paths
Let’s look at the other end of the 7 path. It it exits to the left, it will have to pass through these two cells. This will isolates the 9., so the 7 must exit upward and turn. 
Now the 9 and 6 each have a single exit point, both downward. The 9 will then turn to the left, and then up. 
Because of the 6 segment, the 7 line must continue down and complete. The 11 has one exit point, so it goes up and turns left immediately.
This set of lines must all travel left, which completes the 6 path. 11 and 9 stay parallel through a choke point.. 
Whenever you see that an endpoint has a single exit, start drawing that path. These lines will create restrictions for your other paths. We’ll start the 3 and 8. 
That 3-segment forces the 11 and the 1 to move left. Once they start, they can’t turn, so we again have a set of parallel lines.
The Final Paths
At this point, the 11 can’t continue left, because any direction it takes afterward would cut off the 3 path. So it must turn up and 3 moves left while 1 turns down. 
We have 2 sets of forced parallel lines. We start the 13 and 12 paths. This also completes the 4, 1, and 8 paths. 
13 and 12 continue upward for one more cell. 9 turns up and completes. Finally, 5’s only path completes the pair.
Paths 13 and 11 follow each other left, completing the 13. Meanwhile, the other half of path 11 must zig-zag upward. 
At last, the final segments of paths 11 and 3 become apparent. 
We’ve solved the Arukone puzzle!
