Yajilin

Yajilin, sometimes called Arrow Ring or occasionally Yalooniq, is a path-drawing and grid-shading logic puzzle where solvers must fill in cells based on numerical clues and then draw a continuous loop through the resulting grid.

Background

The name 'Yajilin' is a portmanteau of the Japanese word 'yajirushi', which means 'directing arrow' and the start of the word 'link'.

Yajilin as a genre was first published in June of 1999, in issue 86 of Puzzle Communication Nikoli and was invented by 天歩 ("Tenpo"). The genre was inspired by Yajisan-Kazusan, which appeared about a year earlier.[1]

Puzzle Application

Yajilin puzzles are divided into two steps, although those steps may overlap in order to make certain logical deductions.

First, solvers must use the numbered arrows to place shaded 'obstacle' squares. As a rule, the arrows indicate exactly the number of shaded cells are between it and the edge of the puzzle (in the given direction), and those shaded cells cannot touch each other orthogonally.

The other step is creating a continuous loop that goes through every unshaded/non-clue cell in the puzzle. While not necessarily a heavily-constrained task, this must be possible after shading in all of the correct cells, and as a result care must be taken when doing the previous task to not create a grid where this is impossible.

TO DO

Strategy

• All squares adjacent to an unfilled square that has only two neighbors must be unshaded. This is because if a square were shaded, then the two-neighbor cell would be unshaded, but now this cell only has one neighbor. This applies to both "corners" as well as "corridors" where the neighbors are opposite of each other.
• A clue with a number N pointing towards a region with 2N-1 squares in a row gives a lot of information, as you can shade these squares alternatingly in only one way. This also gives a large amount of loop information!
• Two clues in the same row/column pointing horizontally/vertically allows you to do some math:
• If the clues are pointing in the same direction, then you can subtract the clues to get the number of shaded cells in between the two clues.
• If the clues are pointing towards each other, then you can get the difference between the clues to get the difference of the number of shaded cells past each respective clue.
• While useful in any loop-drawing genre, in Yajilin it's particularly useful to consider region parity, since the clues often pinch the grid, constraining the shape of the loop heavily. That is, given any particular region, the loop must enter and exit an even number of times. As an example where this is invoked, a 1x4 region with 2 known shaded cells and 1 entry must have the shaded cells at the ends of this region.
• Any empty 2x3 region can only support two shaded squares. This is sometimes used with two adjacent clues with the same number N both pointing at a 2x3N region. There are only two ways to complete this pattern.

Examples

• The Neverending Story (MITMH 2022) (web) - This puzzle consists of 8 sub-puzzles spanning 4 different version of Yajilin: normal, One Off (where clues are one off the real number), Full Lane (where arrows indicate entire columns or rows rather than a direction), and Clued Loop (where the arrow clues count the number of line segments in that direction rather than shaded squares).