Nonogram

Example

Solution techniques

To solve a puzzle, one needs to determine which cells will be boxes and which will be empty. Solvers often use a dot or a cross to mark cells they are certain are spaces. Cells that can be determined by logic should be filled. If guessing is used, a single error can spread over the entire field and completely ruin the solution. An error sometimes comes to the surface only after a while, when it is very difficult to correct the puzzle. The hidden picture may help locate and eliminate an error, but otherwise it plays little part in the solving process, as it may mislead.

Many nonograms can be solved through line-by-line reasoning, analyzing one row or column at a time before moving to the next and repeating until the grid is complete. More difficult puzzles often require additional inference techniques, such as probing: tentatively assuming a value for an uncertain cell and checking whether it leads to a contradiction across more than one row or column. If a contradiction results, the opposite value can be deduced with certainty.

Simple boxes

At the beginning of the solution, a simple method can be used to determine as many boxes as possible. This method uses conjunctions of possible places for each block of boxes. For example, in a row of ten cells with only one clue of 8, the bound block consisting of 8 boxes could spread from

• the right border, leaving two spaces to the left;
• the left border, leaving two spaces to the right;
• or somewhere in between.

As a result, the block must spread through the six centermost cells in the row.

The same applies when there are more clues in the row. For example, in a row of ten cells with clues of 4 and 3, the bound blocks of boxes could be

• crowded to the left, one next to the other, leaving two spaces to the right;
• crowded to the right, one just next to the other, leaving two spaces to the left;
• or somewhere between.

Consequently, the first block of four boxes definitely includes the third and fourth cells, while the second block of three boxes definitely includes the eighth cell. Boxes can therefore be placed in the third, fourth and eighth cells. When determining boxes in this way, boxes can be placed in cells only when the same block overlaps; in this example, there is overlap in the sixth cell, but it is from different blocks, and so it cannot yet be said whether or not the sixth cell will contain a box.

Simple spaces

This method consists of determining spaces by searching for cells that are out of range of any possible blocks of boxes. For example, considering a row of ten cells with boxes in the fourth and ninth cell and with clues of 3 and 1, the block bound to the clue 3 will spread through the fourth cell and clue 1 will be at the ninth cell.

First, the clue 1 is complete and there will be a space at each side of the bound block.

Second, the clue 3 can only spread somewhere between the second cell and the sixth cell, because it always has to include the fourth cell; however, this may leave cells that may not be boxes in any case, i.e. the first and the seventh.

Note: In this example, all blocks are accounted for; this is not always the case. The player must be careful for there may be clues or blocks that are not bound to each other yet.

Forcing

In this method, the significance of the spaces will be shown. A space placed somewhere in the middle of an uncompleted row may force a large block to one side or the other. Also, a gap that is too small for any possible block may be filled with spaces.

For example, considering a row of ten cells with spaces in the fifth and seventh cells and with clues of 3 and 2:

• the clue of 3 would be forced to the left, because it could not fit anywhere else.
• the empty gap on the sixth cell is too small to accommodate clues like 2 or 3 and may be filled with a space.
• finally, the clue of 2 will spread through the ninth cell according to method Simple Boxes above.

Glue

Sometimes, there is a box near the border that is not farther from the border than the length of the first clue. In this case, the first clue will spread through that box and will be forced outward from the border. In the simplest case, whenever a box is present in the first or last cells of a row or column, the first or last clue must be aligned to the edge of that row or column.

Considering a row of ten cells with a box in the third cell and with a clue of 5, the clue of 5 will always span from the third to the fifth cell (but not necessarily to the second or the sixth). It is therefore possible to mark the third, fourth and fifth cell as belonging to the 5.

Note: This method may also work in the middle of a row, farther away from the borders.

• A space may act as a border, if the first clue is forced to the right of that space.
• The first clue may also be preceded by some other clues, if all the clues are already bound to the left of the forcing space.

Joining and splitting

Boxes closer to each other may be sometimes joined together into one block or split by a space into several blocks. When there are two blocks with an empty cell between, this cell will be:

• A space if joining the two blocks by a box would produce a too large block
• A box if splitting the two blocks by a space would produce a too small block that does not have enough free cells remaining

For example, considering a row of fifteen cells with boxes in the third, fourth, sixth, seventh, eleventh and thirteenth cell and with clues of 5, 2 and 2:

• The clue of 5 will join the first two blocks by a box into one large block, because a space would produce a block of only 4 boxes that is not enough there.
• The clues of 2 will split the last two blocks by a space, because a box would produce a block of 3 continuous boxes, which is not allowed there.

Note: The illustration picture also shows how the clues of 2 are further completed. This is, however, not part of the Joining and splitting technique, but the Glue technique described above.

Punctuating

To solve the puzzle, it is usually also very important to enclose each bound or completed block of boxes immediately by separating spaces as described in Simple spaces method. Precise punctuating usually leads to more Forcing and may be vital for finishing the puzzle. Note: The examples above did not do that only to remain simple.

Mercury

Mercury is a special case of Simple spaces technique. Its name comes from the way mercury pulls back from the sides of a container.

If there is a box in a row that is in the same distance from the border as the length of the first clue, the first cell will be a space. This is because the first clue would not fit to the left of the box. It will have to spread through that box, leaving the first cell behind. Furthermore, when the box is actually a block of more boxes to the right, there will be more spaces at the beginning of the row, determined by using this method several times.

Contradictions

Some more difficult puzzles may also require advanced reasoning. When all simple methods above are exhausted, searching for contradictions may help. It is wise to use a pencil (or other color) for that to facilitate corrections. The procedure includes:

1. Trying an empty cell to be a box (or then a space).
2. Using all available methods to solve as much as possible.
3. If an error is found, the tried cell will not be a box for sure. It will be a space (or a box, if space was tried).

In this example a box is tried in the first row, which leads to a space at the beginning of that row. The space then forces a box in the first column, which glues to a block of three boxes in the fourth row. However, that is wrong because the third column does not allow any boxes there, which leads to a conclusion that the tried cell must not be a box, so it must be a space.

The problem of this method is that there is no quick way to tell which empty cell to try first. Usually only a few cells lead to any progress, and the other cells lead to dead ends. Most worthy cells to start with may be:

• cells that have many non-empty neighbors;
• cells that are close to the borders or close to the blocks of spaces;
• cells that are within rows that consist of more non-empty cells.

Mathematical approach

It is possible to get a start to a puzzle using a mathematical technique to fill in blocks for rows/columns independent of other rows/columns. This is a good "first step" and is a mathematical shortcut to techniques described above. The process is as follows:

1. Add the clues together, plus 1 for each "space" in between. For example, if the clue is 6 2 3, this step produces the sum 6 + 1 + 2 + 1 + 3 = 13.
2. Subtract this number from the total available in the row (usually the width or height of the puzzle). For example, if the clue in step 1 is in a row 15 cells wide, the difference is 15 - 13 = 2. Note: If spaces can be used on the left or right (top or bottom) borders, this "shrinks" the available area. If it is known that the rightmost cell is a space, the difference is 14 - 13 = 1.
3. Any clues that are greater than the number in step 2 will have some blocks filled in. In the example, this applies to the clues 6 and 3, but not 2.
4. For each clue in step 3, subtract the number in step 2 to determine the number of blocks that can be filled in. For example, the 6 clue will have (6 - 2 =) 4 blocks filled in and the 3 clue will have (3 - 2 =) 1. Note: Applying the same procedure to a clue that "failed" step 3 will produce a non-positive number, indicating that no blocks will be filled in for this clue. The clue 2 produces the number (2 - 2 =) 0; if there were a 1 clue, it would produce the number (1 - 2 =) -1.
5. To fill in the blocks, assume the blocks are all pushed to one side, count from that side "through" the blocks, and backfill the appropriate number of blocks. This can be done from either direction. For example, the 6 clue can be done either of two ways as follows:
   1. From the left: Since the 6 is the first number, count 6 blocks from the left edge, ending in the 6th block. Now "backfill" 4 blocks (the number obtained in step 4), so that cells 3 through 6 are filled.
   2. From the right: Starting from the right, the clues that are to the right of the 6 clue must be accounted for. Starting from cell 15, count 3 cells for the 3 clue (to cell 13), then a space (12), then the 2 clue (10), then a space (9), then the 6 clue (3). From the 3rd cell, "backfill" 4 blocks, filling cells 3 through 6. The results are the same as doing it from the left in the step above.
6. Repeat step 5 for all clues identified in step 3.

In the illustration, row 1 shows the cells that are filled under this procedure, rows 2 and 4 show how the blocks are pushed to one side in step 5, and rows 3 and 5 show the cells backfilled in step 5.

Using this technique for all rows and columns at the start of the puzzle produces a good head start into completing it. Note: Some rows/columns won't yield any results initially. For example, a row of 20 cells with a clue of 1 4 2 5 will yield 1 + 1 + 4 + 1 + 2 + 1 + 5 = 15. 20 - 15 = 5. None of the clues are greater than 5. Also, this technique can be used on a smaller scale. If there are available spaces in the center or either side, even if certain clues are already discovered, this method can be used with the remaining clues and available spaces.

Deeper recursion

Some puzzles may require to go deeper with searching for the contradictions. This is, however, not possible simply by a pen and pencil, because of the many possibilities that must be searched. This method is practical for a computer to use.

Multiple rows

In some cases, reasoning over a set of rows may also lead to the next step of the solution even without contradictions and deeper recursion. However, finding such sets is usually as difficult as finding contradictions.

Nonograms in computing

Deciding whether a nonogram puzzle has a solution is an NP-complete problem. This places the related problem of constructing a solution (or demonstrating that no solution exists) for a given puzzle into the category of NP-hard problems, meaning that there is no polynomial time algorithm that solves all nonogram puzzles unless P = NP.

However, certain classes of puzzles, such as those in which each row or column has only one block of cells and all cells are connected, may be solved in polynomial time by transforming the problem into an instance of 2-satisfiability.

Software solvers

An extensive comparison and discussion of nonogram solving algorithms is found at the WebPBN site (Web Paint-By-Number). A nonogram solver written in C++ and published in the journal Pattern Recognition solves lines in quadratic time at most.

Video games

Nintendo has published several nonogram video games using the name "Picross" (ピクロス, Pikurosu). The Game Boy game Mario's Picross was initially released in Japan on March 14, 1995 to decent success. However, the game failed to become a hit in the U.S. market, despite a heavy advertising campaign by Nintendo. The game is of an escalating difficulty, with successive puzzle levels containing larger puzzles. Each puzzle has a limited amount of time to be cleared. Hints (line clears) may be requested at a time penalty, and mistakes made earn time penalties as well (the amount increasing for each mistake). Picross 2 was released later for Game Boy and Mario's Super Picross for the Super Famicom, neither of which were translated for the U.S. market (Mario's Super Picross was, however, later released on the Wii Virtual Console's PAL service on September 14, 2007, as part of its Hanabi Festival, as well as on the Nintendo Switch Online service worldwide on September 23rd, 2020). Both games introduced Wario's Picross as well, featuring Mario's nemesis in the role. These rounds vary by removing the hint function, and mistakes are not penalized — at the price that mistakes are not even revealed. These rounds can only be cleared when all correct boxes are marked, with no mistakes. The time limit was also removed. Nintendo also released eight Picross volumes on the Japanese Nintendo Power peripheral in Japan, called NP Picross, each with a new set of puzzles, including puzzles based around various Nintendo characters, such as Mario, The Legend of Zelda, and Pokémon.

Nintendo has released Picross DS for the Nintendo DS portable system in 2007. It contains several stages of varying difficulty, from 5x5 grids to 25x20 grids. Normal mode tells players if they made an error (with a time penalty) and free mode does not. A hint is available before starting the puzzle in all modes; the game reveals a complete row and column at random. Additional puzzles were available through Nintendo Wi-Fi Connection; some of the original Mario's Picross puzzles were available. However, the service was shut down on 20 May 2014. Nintendo made new releases available bi-weekly. Picross DS was released in Europe and Australia on 11 May 2007 and in the United States on July 30, 2007 and has been received well by critics, including Craig Harris, Jessica Wadleigh and Dave McCarthy labelling the game "Addictive". A 3D version of the game, titled Picross 3D, was also released for the DS in Japan in 2009 and internationally in 2010. A sequel, Picross 3D: Round 2, was released for the Nintendo 3DS in 2015. Another downloadable version of the game was released for Nintendo 3DS's Nintendo eShop, called Picross e, Picross e2, and Picross e3 released in 2013, with Picross e4 released in 2014. Nintendo has also released a Pokémon spinoff on December 7, 2015 in the form of the freemium game of Pokémon Picross for Nintendo 3DS. My Nintendo Picross: The Legend of Zelda: Twilight Princess was released for Nintendo 3DS on March 31, 2016, exclusively as a premium reward for My Nintendo.

Other companies have also released nonogram video games, such as Falcross on iOS, Across-Stitch by Knitwit Studios on Microsoft Windows and Apple macOS, and the Color Cross series of games by Little Worlds Studio on the Nintendo DS, Microsoft Windows, and iOS. In addition, nonogram puzzles have appeared in non-nonogram puzzle games, such as in Deadly Rooms of Death's fifth installment, The Second Sky. In it, nonogram puzzles (referred to as "Picross puzzles") representing in-game objects are optional, unlockable puzzles late into the game that can be played in the level "The Central Station", and solving them unlocks bonus levels in the game. In 2018, Konami released a game titled Pixel Puzzle Collection, or Picroji Puzzle (ピクロジパズル), featuring classic Konami characters and sprites.

Other picture logic puzzles

Pentomino paint-by-numbers is a variant in which the twelve pentomino shapes must be placed in the grid, without touching each other (even diagonally).

Triddlers are an offshoot that uses triangle shapes instead of squares.

Paint by pairs or Link-a-Pix consists of a grid, with numbers filling some squares; pairs of numbers must be located correctly and connected with a line filling a total of squares equal to that number. There is only one unique way to link all the squares in a properly-constructed puzzle. When completed, the squares that have lines are filled; the contrast with the blank squares reveals the picture. (As above, colored versions exist that involving matching numbers of the same color.)

Fill-a-Pix also uses a grid with numbers within. In this format, each number indicates how many of the squares immediately surrounding it, and itself, will be filled. A square marked "9," for example, will have all eight surrounding squares and itself filled. If it is marked "0" those squares are all blank.

Maze-a-Pix uses a maze in a standard grid. When the single correct route from beginning to end is located, each 'square' of the solution is filled in (alternatively, all non-solution squares are filled in) to create the picture.

Tile Paint is another type of picture logic puzzle by Nikoli. It works like regular nonograms except that it only specifies the total number of squares in each row or column that will be filled in and irregular sections within the grid have borders around them that indicate that, if one of the squares within it is filled in, all of them must be filled in.