Yajilin ヤジリン

Yajilin (ヤジリン) is a loop puzzle from the Japanese publisher Nikoli that mixes shading with line drawing. Some cells of the grid are gray clue cells holding a number and an arrow, and each number tells you how many shaded cells lie in the direction of its arrow, all the way to the edge. You shade cells so that every arrow is right and no two shaded cells share a side, then draw a single closed loop through the centers of all the cells that are left.

The loop may turn, but it never crosses itself, branches or visits a cell twice, and it never enters a shaded cell or a clue cell. That pairing is what makes Yajilin special: every shading decision changes the space the loop can travel through, and every dead end in the loop tells you where a shaded cell must go. Nikoli first published the puzzle in 1999, and in English it has also appeared as Arrow Ring.

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Start with the arrows that leave no room. A 0 arrow means nothing in that direction is shaded, so every cell along it that is not a clue carries the loop. A count that equals the most shaded cells its line can hold is just as strong. Because shaded cells cannot touch, a run of free cells holds at most half of them, rounded up, so a 2 with three cells to the edge shades the first and the third. A clue cell in the line is never shaded and splits the run in two.

Every shaded cell forces its neighbors to be white, so dot them at once. Each white cell has exactly two segments, so a white cell with only two open neighbors, such as a corner or a cell boxed in by clues, uses both of them. The reverse finds shading: a cell left with fewer than two open neighbors cannot be on the loop, so it must be shaded. And if shading a cell would leave one of its neighbors with a single exit, that cell stays white.

In harder spots, use two global ideas. The loop is one piece, so a segment that would close a small loop while other white cells are still unvisited can be crossed off. And on a checkerboard coloring, a closed loop alternates colors, so it visits exactly as many dark cells as light ones; late in a puzzle, the last shaded cells must fall on the color that restores that balance.

Common mistakes are stopping an arrow’s count at another clue instead of the edge of the grid, assuming every shaded cell must be seen by an arrow, and keeping shaded cells apart diagonally, which is not required. Arrows turn red when they see too many shaded cells, and the Check button flags touching shaded cells, loop branches and anything that is not in the solution.

Yajilin is an original Nikoli puzzle. It first appeared in 1999 in issue 86 of Puzzle Communication Nikoli, taking the cells that hold both an arrow and a number from Yajisan-Kazusan, a Nikoli shading puzzle introduced shortly before, and adding a loop to be drawn through the remaining cells. From issue 107 it ran as a page in every issue of the magazine. The name is a contraction of yajirushi, the Japanese word for arrow, and rinku, the Japanese rendering of the English word link. English-language puzzle tests have also printed it as Arrow Ring, for example the 2005 United States qualifier for the World Puzzle Championship. Researchers have since shown that solving Yajilin on grids of arbitrary size is NP-complete.