Overview

A hitori puzzle consists of a grid of numbers, generally square. The goal is to black out certain squares so that the resulting grid satisfies these rules:

1. No numeral appears twice in any row or column.
2. No black square is horizontally or vertically adjacent to another black square.
3. It is possible to move from any white square to any other white square by moving only through white squares.

Here is a sample unsolved hitori grid and the solution:

Fig. 1: Unsolved grid		Fig. 2: Solved grid

Strategy Summary

Basic Strategy

Simpler hitori grids can be solved by first identifying some white squares, then repeatedly using a basic two-step strategy.

1. If a numeral is between two identical numerals, the square in the middle is white.
2. Apply rule 1.
3. Apply rule 2.
4. Repeat steps 2-3 as much as possible.

Advanced Strategies

Harder puzzles generally also require the advanced strategies, and some trial-and-error.

1. If there is some other square that is a specific color in both cases, then you can color that square.
2. Remember the third rule: If coloring a square black will split white squares into two groups, color that cell white.

Basic Strategy

If a numeral is between two identical numerals, the square in the middle is white.

For instance, assume that three side-by-side squares are 1 2 1. If the leftmost 1 is black, then the 2 must be white (rule 2). If the leftmost 1 is white, on the other hand, the right most 1 must be black (rule 1), which also means that the 2 must be white (rule 2). Either way, the 2 must be white.

Applying this strategy to the sample puzzle gives us the following grid:

Fig. 3: Cells between identical numbers

Each of the white 5s is between identical digits in the same row; the white 2 is between identical digits in the same column.

Apply rule 1.

Once the grid has some white squares, matching numerals can be blacked if they are in the same row or column. For instance, the 5 in the first row should be black, since there's a white 5 in that column:

Fig. 4: Cells in rows/columns with matching numbers

Apply rule 2.

All neighbors of black squares must be white. The black 5 has three neighbors are all white:

Fig. 5: Cells around black cells

Repeat the previous two steps as much as possible.

Repeating these steps multiple times to the sample yields the following:

Fig. 6		Fig. 7

Fig. 8		Fig. 9

At this point, consider the third rule. If the 4 next to the 3 in the lower left is black, then the 3 would be cut off from the rest of the grid. Therefore, that 4 is white.

Fig. 10: Avoiding islands

Likewise, we can determine that the 2 and 5 in the bottom row are both white, because if they were black, some white squares would be cut off from the rest.

Fig. 11		Fig. 12

We can then complete the grid, using these strategies iteratively:

Fig. 13		Fig. 14

Fig. 15		Fig. 16

Fig. 17

Some hitori grids (like this example) can be solved using just this technique alone. However, there are some additional strategies that sometimes need to be used.

Advanced Strategies

Overview

Once you've exhausted the possibilities with the basic strategy, you can try these strategies. These are specific examples of how to use a trial-and-error strategy on remaining squares. Select a square and ask, "What if this were white, what color would other squares have to be? What if this were black?"

1. If there is some other square that is a specific color in both cases, then you can color that square.

For instance, if two identical numbers appear against a wall near a corner, a neighboring square is white.

Let's go back to Fig. 12 and work on solving the grid another way. If the square in the corner is next to an identical number, then the square below (or above) it is white. For instance, in the grid, the 4 in the upper right corner is next to another 4. Let's say the leftmost 4 is black; in that case, the 1 is white.

Fig. 18

Let's say the leftmost 4 is white; in that case, the rightmost 4 is black. If the 1 were also black, then the 4 in the corner would be stranded. Therefore, the 1 must be white.

Fig. 19

Either way, then, the 1 is white.

Fig. 20

2. Remember the third rule: If coloring a square black will split white squares into two groups, color that cell white.

Now consider the group of 3s. Imagine the uppermost 3 is black:

Fig. 21: Assume 3 is black

This would lead to the following situation:

Fig. 22		Fig. 23

The orange cells are now blocked from the rest of the white cells:

Fig. 24: Invalid solution

Therefore, the uppermost 3 must be white, leading to this situation:

Fig. 25: Hence, 3 is white		Fig. 26

Likewise, imagine the still uncolored 2 were black:

Fig. 27: Assume 2 is black

The orange cells are separated from the rest of the white cells:

Fig. 28: Invalid solution

Hence, the 2 is white:

Fig. 29: Hence, 2 is white

Conclusion

These strategies should be sufficient to solve most any hitori puzzles you choose to tackle. As you solve more puzzles, you will begin to see other patterns that will simplify solving.

Avoid the pitfalls:

• While the presence of a white square allows you to black all the squares in the same row or column with the same number, the presence of a black square doesn't do the opposite. In face, once a square is black, its numeric value is meaningless. If there are three 4's in a row, there can be at most one white, but there could be none.
• If all the numbers in a row or column are all different, don't assume they're all white. Some could be black because of white squares in the other direction.

All hitori puzzles will have a unique solution which can be entirely determined by logic. Have fun!

All content copyright Paul and/or Valerie Hartzer, all rights reserved, except where indicated. Contact Paul Hartzer with comments. This file was last modified on Wednesday, January 6, 2010.