Hitori techniques: how to solve every Shadeout board

Technique 1 · basic

Unique numbers stay

Start every board with a scan for numbers that appear only once in both their row and their column. Shading exists to hide a repeat, and such a tile has nothing to repeat with, so it can never be shaded. Circle it straight away. These circles matter more than they look: each one is an anchor the later techniques lean on, and its row and column are where the next duplicates get cleared. Scan row by row, and for each number that is alone in its row, glance up and down its column. A number counts only if it is alone in both directions.

The rule keeps paying off later in the solve. Once every other copy of a number in its row and column has been shaded, the last copy is alone too, and it stays. After each round of shading, look back at the numbers you just shaded and check whether their last surviving twin can now be circled.

Where to look: the rarest numbers on the board. A number that appears only a few times in the whole grid is the likeliest to be alone in its lines.

Technique 2 · basic

The sandwich

Three tiles in a line with the same number on both ends, like 1 2 1, form a sandwich, and the middle tile stays showing. Suppose it were shaded: shaded tiles can't touch, so both ends would have to show, and the 1 would then appear twice in the line. That breaks the first rule, so the middle is safe. The ends are not decided by this. One of them may well end up shaded, but only for some other reason. The sandwich works the same down a column, and it does not matter what the middle number is.

Sandwiches are free information on a fresh board, because they need no marks at all, only the printed numbers. Sweep every row and every column once for the pattern X ? X before anything else, and circle each middle. In the figure, the circled 2 then clears its own duplicates, which is the next technique.

Where to look: any number with the same number on both sides, in rows and in columns. Overlapping patterns like 2 1 2 1 hold two sandwiches at once.

Technique 3 · basic

A circled tile clears its duplicates

A circled tile shows its number, so every other copy of that number in the same row and the same column has to be shaded. Shade them all at once. This is the move that turns safe tiles into shaded ones, and it is where most of a Hitori solve happens: every technique that circles something feeds it. Check both lines through the circled tile, not just the one you were working in. A copy at the far end of the row still counts, and a duplicate in the column is as easy to miss as it is decisive.

If a duplicate is already circled, something went wrong earlier: two copies of one number can never both stay in a line. Take it as a signal to find the slip now, before more marks build on it. The usual cause is a tile circled from one line while a copy in the other line was overlooked.

Where to look: right after you circle anything. Make it a habit: circle a tile, then scan its row and column for its number.

Technique 4 · basic

Neighbors of a shaded tile stay

Shaded tiles never touch side by side, so the up to four tiles above, below, left and right of a shaded tile all stay showing. Circle them. Each new circle then clears its own duplicates, and each tile shaded that way circles its neighbors in turn. This back and forth with the previous technique is the engine of every Hitori: on an easy board it can run for a dozen steps on its own. Diagonal tiles don't count. A tile that touches a shaded tile only at a corner is not decided by this rule.

Some neighbors may already be circled, as in the figure; skip those and circle only the new ones. Then go straight back to the previous technique for each tile you just circled. In the figure, the circled 3 at the top shades the other 3 in the top row, and the chain carries on from there.

Where to look: every tile you have just shaded. Auto circles in Settings marks these for you, which leaves your attention for the other techniques.

Technique 5 · intermediate

Triples in a line

Three equal numbers in a row, like 5 5 5, settle all three tiles at once: both ends are shaded and the middle stays. The middle is a sandwich, so it can't be shaded. Once it shows, the copies on either side are its duplicates and must go. You can also see it from the other end: two of the three 5s have to be hidden, and the only two that don't touch are the outer ones. Spot the triple, and three tiles are done without further thought.

Triples are rare on small boards and more common on large ones, where one number can appear four or five times in a line. The reasoning stretches: with 5 5 5 and another 5 further along the line, that far 5 is shaded too, since the circled middle already shows. A run of four equal numbers can't be solved at all, so a fair puzzle never has one.

Where to look: runs of one number, across and down. A run broken by one tile, like 5 5 2 5, is a different case: a pair plus a third copy, which is the next technique.

Technique 6 · intermediate

Pair isolation

Two equal numbers side by side, like 5 5, can't both show and can't both be shaded, so exactly one of them is shaded and the other shows. You don't know yet which one, but you do know that the number already shows once in that line. So any other copy of it in the same row or column must be shaded. Solvers call this pair isolation. It shades a tile without a single circle to point at, using only the printed numbers, which makes it as valuable on an empty board as the sandwich.

The isolated copy then circles its neighbors as usual. The pair itself stays undecided until something else breaks the tie, often a tile shaded from the crossing column. In the figure, the 3 beside the newly shaded 5 was circled earlier by a sandwich: 5 3 5 holds it in place.

Where to look: any two equal numbers that touch, then the rest of that line for another copy. Check the pair's row or column, not the crossing lines: the pair only tells you about the line it lies in.

Technique 7 · intermediate

Corner patterns

Corners are where the connection rule bites hardest, because a corner tile has only two neighbors. Take a corner number with the same number right beside it on both sides, like the top left 5 here, with a 5 to its right and a 5 below. If the corner stayed showing, both of those 5s would have to be shaded, and the corner would be walled in, cut off from the rest of the board. So the corner tile is shaded, and both of its neighbors stay showing.

The idea behind it works anywhere: count a tile's open neighbors. A corner tile has two, an edge tile three. Whenever keeping a tile would force every one of its neighbors to be shaded, that tile can't stay. Check all four corners on a fresh board, right after the sandwiches and pairs.

Where to look: the four corners of every board, then edge tiles whose number repeats on two or three sides.

Technique 8 · intermediate

Keep the showing tiles connected

All the tiles left showing must form one group, linked through their sides. So a tile that is the only link between part of the board and the rest can't be shaded. The plainest case is a circled tile with only one open way out: shade that way out and the tile is stranded. On a real board it shows up as a shaded tile and the board's edge closing in on a circled tile, or as two shaded tiles meeting at a corner. Whenever an area has a single exit, circle the exit.

Larger areas follow the same rule. When shaded tiles form a near wall across the board, the gap in that wall must stay open, however many tiles sit behind it. After every new shaded tile, look at its diagonal neighbors: two shaded tiles that touch at a corner are the start of a wall.

Where to look: circled tiles on the edge next to a shaded tile, and the tiles between two shaded tiles that meet diagonally.

Technique 9 · advanced

Look-ahead: what if this tile were shaded?

When no pattern is left, pick a tile and ask what would happen if it were shaded, or if it stayed. Follow only the basic moves: the neighbors of a shaded tile stay, a showing tile shades its duplicates. If the chain ends in a broken rule, such as two shaded tiles touching, a number showing twice or a cut-off area, the opposite is true. In the figure, shade the 1 in the top left corner, and the 5 and the 4 beside it must stay. That 5 shades the 5 two tiles further along the top row, so the 4 below that one stays too, and row 2 now shows two 4s. The corner 1 therefore stays showing.

Test tiles where the chain stays short: beside a pair, in a crowded corner, or with many copies in its lines. If one case leads nowhere, try the opposite on the same tile before moving on. Hard Shadeout boards need a longer version, where the chain also uses sandwiches, pairs and connection. This is not guessing: nothing goes on the board until the contradiction proves it.

Where to look: the undecided tile with the most copies of its number in its row and column. The more tiles a case touches, the sooner it breaks or settles.

Common mistakes

• Shading a number that is alone. A tile is only shaded to hide a repeat. If no other copy of its number shows in its row or column, it stays, whatever the tiles around it suggest.
• Checking only one line. A circled tile clears its duplicates in its row and its column. A stuck board very often has a twin left in the line nobody scanned.
• Treating a pair like a sandwich. In 5 5, one of the two is shaded, but the pair alone doesn't say which. Only a third 5 in that line is certain; leave both halves of the pair open for now.
• Leaving connection for the end. Check it after every shaded tile, not as a final pass. A diagonal wall of shaded tiles is easy to build and painful to undo.