# Sudoku

Sudoku, called Number Place in some publications, is a type of logic puzzle involving the placement of the digits 1-9 within a 9x9 grid so that each row, column, and 3x3 section of the grid contains each one exactly once. Invented under various names in France in the 1800s, Sudoku puzzles reached a wider audience in the 1980s once Japanese puzzle company Nikoli began publishing them, and an international audience in the late 1990s.

## Background

The first case of a 9x9 grid that could function as a proper Sudoku was published by the French newspaper La France, under the name 'carré magique diabolique'. The goal, as expressed in the paper, was to fill in the 'magic square' based off the numbers already entered so that no number appeared twice in any row, column, or broken diagonal. While the puzzle presented did not mark off 3x3 regions, the final (unique) solution would still have been a valid Sudoku solution as well, as the usual regions did in fact have each number exactly once. Other French newspapers published similar puzzles from 1895 until around 1914, with no clear reason for why they fell out of popularity.

It wouldn't be until 1979 that the Sudoku would be remade, this time in the US by a man named Howard Garns, who published them anonymously in Dell Magazines under the title 'Number Place'. Despite his version being pretty much identical to what is known today as a Sudoku puzzle, it wasn't until the Japanese publisher Nikoli picked it up that it started to gain popularity.

The Japanese Sudoku was originally called 'dokushin ni kagiru', but was shortened by Maki Kaji to 'Sudoku'. In addition, Nikoli registered the name as a trademark, meaning that while Sudoku is still used for the genre as a whole worldwide, there are still many people and publications that use the generic 'Number Place'.

It wasn't until 1997 that the process to spread Sudoku puzzles outside of Japan began, pioneered by a judge from Hong Kong named Wayne Gould. After seeing a partially-solved puzzle, he dedicated six years to writing a computer program to generate unique puzzles, which he then introduced and sold to The Times in the UK and The Conway Daily Sun in New Hampshire, kicking off the Sudoku craze of the 21st century.

Products of the increased interest in Sudoku puzzles worldwide include the World Sudoku Championship (which began in 2006), multiple game shows, and the popular YouTube channel Cracking The Cryptic.

## Puzzle Application

The most basic rules of Sudoku puzzles are that the 9 digits/letters/symbols involved can only occur once in each row/column/3x3square, although some variations exist that introduce more constraints on placement (such as

According to Nikoli, there are only a few rules to creating a Sudoku puzzle, although they aren't strict, and certain publishers may have individual rules that either add to or contradict the ones set by Nikoli.

1. There may be no more than 32 given digits (less than 40% of the grid)
2. The arrangement of numbers must be symmetrical

Both of these rules are reportedly intended to create more enjoyable puzzle, rather than a more difficult/"proper" puzzle. As mentioned, certain publications may put further constraints on the puzzles based on the deductions that need to be made in order to solve it, often to place the puzzle in a particular difficulty range (so that advanced techniques aren't needed to solve a purportedly 'Easy' puzzles).

### Variants

Sudoku variation primarily affect one of two factors: size and constraint. Size variants mostly involve making a puzzle bigger or smaller, within the constraints of making the grid work. For example, while 4x4 and 6x6 sudokus are possible (using 2x2 and 2x3 regions respectively), a 5x5 is not (unless all regions are 1x5, which makes the regional constraint equivalent with the columns/rows constraint). This applies to all prime numbers. It is also possible to make the regions non-rectangular as well, which is usually known as a Jigsaw Sudoku. The content of bigger-than-9x9 puzzles also tends to vary with the size, as some constructors will just choose to stick with numbers even if they extend to double-digits, and others will choose to integrate both numbers and letters (allowing for single-digit/number inputs up to 36x36 if the digits 0-9 and the latin alphabet are both used).

Constraint variations are much more involved, and often involve extra rules unique to those variations. The following are some of the more common variants in puzzle hunts.

#### Killer Sudoku

Despite the name, Killer Sudoku is not just 'a very difficult Sudoku'. In reality, it's a kind of combination of a Sudoku puzzle and a Kakuro puzzle. A Killer Sudoku, on top of the usual 3x3 divisions in the grid, also contains 'cages' that surround up to 9 cells and have a number in the corner indicating the sum of the numbers contained within. Within these cages, numbers still may not be duplicated, and all other regular Sudoku rules still apply (no duplications within 3x3s, rows, or columns).

#### Samurai Sudoku

A mix of a constraint and size variations, Samurai Sudoku presents multiple normal Sudoku puzzles (or other variations) fused together at the corners, extending them out diagonally. Each grid needs to be solved independently, but since each puzzle shares a 3x3 box with at least one other puzzle, any numbers placed in those boxes must work for both puzzles' rows/columns. Otherwise, there are no extra constraints, and even though rows and columns may appear to go on for more than 9 cells in certain spots, solvers must make sure to only take into account each 9x9 grid's individual constraints.

#### Thermo Sudoku

Thermo Sudoku puzzles contain a series of lines and bulbs, or 'thermometers', that put additional constrains on what numbers can be placed where in the grid. When presented with a thermometer, solvers must place numbers along the thermometer's path (from bulb to tip) so that they only increase in size. For example, if a thermometer had a length of three, and the bulb had the number 7 on it, solvers could deduce that the other numbers going outwards had to be an 8 and a 9. Because of the 'strictly-increasing' rule, thermometers can usually only contain one of each number at most.

There are some exceptions to the one-of-each assumption when it comes to thermometers, particularly that thermometers can have both multiple bulbs and multiple paths. In these cases, the only rule that applies is that regardless of which bulb is started at or which path is taken, the sequence of numbers must always increase.

#### Wordoku

Primarily a visual change, Wordoku swaps out the digits 1-9 for a particular selection of 9 letters. Functionally, these puzzles work the same as a normal sudoku, but the inclusion of letters makes combination with word puzzle genres much more possible. In fact, many Wordoku puzzles end up having a particular word reading down one of the diagonals.

Wordoku also opens the door to other non-number variants, like ColorDoku and Symbolic Sudoku, which use colored squares and non-alphanumeric symbols instead of numbers. As long as there are only 9 different things present in a puzzle, anything can be used in place of numbers.

TO DO

## Notable Examples

### Played Straight

• Color Sudoku (MITMH 2013) (web) - A relatively straightforward set of Sudoku puzzles, with the only real gimmick being that the colored regions within each puzzle must also have a unique set of numbers.
• Blocks (MITMH 2015) (web) - A basic ColorDoku puzzle. No gimmicks in how things are placed, aside from possibly getting help from Click to revealthe letters of a particular color spelling out the name of that color.
• Pentoku (MITMH 2017) (web) - While not a normal variant, the puzzle provides all of the instructions needed to solve 'Pentoku'. Extraction is a different story, but ignoring that it's a fairly straight puzzle.

### Notable Twists

• Build Your Own Sudoku (MITMH 2014) (web) - Solvers are presented with a blank grid and a BUNCH of extra constraints on each individual region, row, and column, allowing solvers to uniquely place the puzzle's givens, before actually solving the puzzle they've created. The title certainly lives up to its name as a Build-Your-Own-Puzzle.
• Replication (MITMH 2016) (web) - A Samurai Sudoku taken to extreme levels (far too large to solve by hand). This puzzle is 45-46 puzzles wide at any given point, and an unknown (but presumably very large) height. Ultimately, solvers need to Click to revealtreat it not as a Sudoku, but as an entire computer program in Sudoku form.
• Fun With Sudoku (MITMH 2021) (web) - What appears to be a series of variant Sudoku puzzles is...well, it's still a bunch of variant Sudoku puzzles. The twist, however, is that Click to revealmost of them can't be solved uniquely. Rather than being tasked with solving all of the puzzles, solvers are instead tasked with determining exactly how many solutions each puzzle has. This starts out relatively simply, with a puzzle with only 4 solutions, but quickly devolves into massive numbers when presented with puzzles with no givens at all.