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Overview
This is a selection of some of the more common types of Japanese-style logic problems. There are many variants, as well as other types of puzzles not listed here.
Pictures
Paint-by-Numbers (Paint-a-Pix, Nonograms, Paint-doku)
Start with a blank grid. Along the left side and top are numbers. These numbers indicate the number of contiguous black squares. For instance, if the values to the left of a row are 3 2, then there is a block of 3 black squares, a gap of white space, then a block of 2 black squares on that line. The strategy is in determining where the white space is so that the values on the left and top are satisfied.
The result is typically a picture, although automatic puzzle generators create random images.
Variant: Color puzzles do not require white space between blocks of different colors, but do require white space between blocks of the same color.
Start with a grid, some squares of which have numbers in them. Color squares black so that each number in the grid describes the number of black squares touching it, including itself. For instance, a 7 would have seven black squares in the nine squares around it (including itself).
Start with a grid, some squares of which have numbers in them. Link pairs of numbers by drawing a line between them. For instance, a pair of 5s would be linked by a line crossing three other squares. 1s can be colored black without being paired; otherwise, numbers can only be paired with one other number. The challenges are in pairing the numbers correctly and in drawing lines appropriately (each square can have at most one line).
Once all the lines have been drawn, color in the squares to reveal a picture.
Variant: Color puzzles require the linked pairs to be of the same color.
Number Fill-in
Calc-doku (Kenken®)
Start with a square grid divided into zones. In the upper left of each zone is a number and a mathematical operator (some puzzles use only one operator, in which case that's marked outside the grid); possible operators are +, -, x, and /. Fill in the puzzle so that the numbers in each zone satisfy the clue, and so that each row and each column contain different digits.
For instance, in a 4x4 Calc-doku, each row and each column would contain each digit from 1 to 4. If a zone with 3 squares contains a 5+, then the zone could contain a 1, 1, 3 or a 1, 2, 2. Note that a zone can contain the same digit more than once, as long as they're not in the same row or column. Also note that - and / are only used in zones with 2 squares.
Kakuro (Cross Sums)
Kakuro superficially resembles a crossword grid with numbers to the left and top of each block of white squares. The goal is to enter the digits 1 to 9 so that no digit appears twice in a horizontal or vertical block of squares, and so that the sum of the digits is the number at the left or top of the block. For instance, if there is a 5 next to two squares, then those squares must contain either a 1 and a 4 (in some order) or a 2 and a 3 (in some order). The strategy is based on using a process of elimination to find the unique solution.
Sudoku (Number Place)
Start with a 9x9 grid, with some numbers placed. In standard sudoku, the numbers are placed in a rotationally symmetrical pattern. Place the digits 1 to 9 so that each row, column, and 3x3 subgrid contains one of each digit.
Variant: There are quite a few variants of sudoku. The original version (Number Place, invented in the U.S.) also required that the long diagonals contain one of each digit, something which is now called Sudoku-X (Number Place also did not have a rotationally symmetrical pattern at the outset). Popular variants include different grid sizes (such as 6x6 and 16x16), overlapping grids, and irregularly shaped subgrids. Odd/even marks certain cells which can only contain odd numbers (or even numbers). Sum sudoku provides marked-off areas that must add up to an indicated value (kakuro-style). There are other variants as well.
Connections
Start with a grid partially filled with numbers. Connect the numbers with horizontal and vertical lines, or bridges. The number represents the total number of bridges connecting it to other numbers. At most two lines can connect two numbers.
Start with a grid of numbers. Color the numbers black or white so that no neighboring cells are both black, and so that no line contains two white cells with the same value.
Nurikabe (Islands)
Start with a grid with some squares containing numbers. An island is a group of white squares surrounded by black ones, i.e., the sea. Fill in the grid such that each number is on its own island, each island has a number, and the sea is contiguous.
Start with a grid with some squares containing numbers. Draw a single closed loop through the grid such that the numbers represent the number of adjoining grid segments.
Placement
Start with a grid with numbers on both axes, and possibly with some cells filled in. The goal is to place "ships" of various specific sizes in the grid such that any existing grid clues are satisfied, the numbers on the axes match the number of ship pieces in each row and column, and no two ships touch, even diagonally.
Light up (Akari)
Start with a grid of black and white cells. Some of the black cells contain numbers. The goal is to place light bulbs in some of the white cells such that the numbers are directly in line (horizontally or vertically) with that number of light bulbs, all white cells are directly in line with at least one light bulb, and no light bulb is in line with any other light bulb (without an intervening black square).
Tents
Start with a grid with some squares filled in with trees. Along both axes are numbers. The goal is to place tents in empty grid cells so that each tree neighbors a tent (excluding diagonally), no two tents are in neighboring cells (including diagonally), and the number of tents in each row and column is indicated on the axes.