# 74: Su Doku

Su Doku |

Title text: This one is from the Red Belt collection, of 'medium' difficulty |

## Explanation

Su Doku (Japanese for "Single number", and now usually written as "Sudoku") is a type of number puzzle, in which the player must place digits in a matrix playfield in such a way that no digit appears twice in a horizontal and vertical row, and in a region of nine digits in the said matrix. The most common arrangement is a 9x9 grid subdivided into nine 3x3 grids, each of which must contain the digits 1-9. The number and combination of pre-filled squares determines the difficulty of the puzzle. The image text refers to the "Red Belt"-collection, which is a series of extremely difficult puzzles.

In this comic, Randall presents us with a binary Sudoku puzzle. A normal Sudoku is "decimal" like our normal counting system (ten digits) counting from one to nine. Some Sudoku puzzles use the hexidecimal system with 16 digits (0-9 and A-F) and a 16x16 grid for more difficulty. The joke is that binary system has only two digits (0 and 1), and therefore binary Sudoku puzzles would be infinitely easy and thus pointless. There really are only two possible puzzles in a 4x4 grid. The puzzle in the comic would be completed by filling 0 in the top-left and 1 in the bottom-left empty box. The only other possible grid would have the 0s and 1s swapped. This fulfills the criteria of having no repeated digits in any row or column, although digits would repeat in the 4x4 grid. Presumably binary Sudoku has no sub-regions.

The image text appears to reference a series of published Sudoku puzzle books called the "Martial Arts Sudoku" series. The difficulty of each book is denoted by a belt colour which itself references the fact that in many Martial Arts, participants are awarded coloured belts when they reach certain skill levels, with each colour representing a certain skill level. It appears Judo was the first to use this system. A black belt is the stereotypical "highest" belt, although this is not always in fact the case, depending on the Martial Arts discipline. In the Sudoku series, red belt was level 8 (of 9) just below black belt. Thus the red belt reference therefore implies that this puzzle was found in the second-highest level of binary Sudoku books.

The puzzle is labeled as "medium" difficulty in the title text because it has two of the squares filled. In a 2x2 binary Sudoku puzzle, only one square is required to solve the puzzle. By labeling this puzzle as "medium" difficulty, it implies that a puzzle with three squares filled would be labeled "easy," and a puzzle with only one square filled would be labeled "hard," with the irony being that none of the puzzles could possibly be considered difficult.

## Transcript

- [A square divided into 2x2 squares, the top-right one has an 1 in it, the bottom-right one has a 0, the two left ones are empty]

- Binary Su Doku

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# Discussion

If that puzzle is 4 (i.e. 2x2) domains of 1x1 cells or 1 domain of 4x4 cells then it's actually an impossible puzzle. Sudoku grids for 'n' symbols (ignoring some very interesting variants) need to be of n² cells in total with n cells in each direction, composed of n 'domains', each of n cells so as to contain *one and only one* of each symbol in use. That's 81 cells in a traditional 1-9 digit 9x9 format, being 3x3 array of 3x3 individual cells in typical ~~Sudokus~~ ~~Sudokii~~ ~~Sudoka~~ puzzles, but can be irregularly domained instead as long as the domains still have nine cells. In a 12-digit that's often 3x4 cells in each domain, arrayed 4x3 (or 4x3 arrayed 3x4) to make a 12x12 full grid but can be 2x6 6x2s (or <=>) or of irregular, but still equally-sized, subdivisions. ("Killer" variations typically augment the row, column and domain parities with a 'fourth dimension' of *unequally*-sized irregular domains (no larger than any other domain, containing a *maximum* of one of each digit, but possibly zero of some) labelled as having a stated sum total within (more than or equal to m*(m-1)/2 for m cells in that given sum-zone, assuming the lowest digit is 1, and less than or equal to (n*(n-1)/2)-((n-m)*(n-m-1)/2), if n digits are being used as unique symbols throughout the whole grid), but that's generally in leiu of *all* pre-existing clue digits, using Kakuro-like calculations to break ground on the puzzle's answer.)

Realistically, therefore, the comic must be 1x2 domains of 2x1 cells. Or the other way round. Although it's not obvious from the line-weighting which it might be. As each subdivision is the same as the row-grouping or column-grouping it could effectively be just a 'simpler' puzzle that abandons or considers redundant domains *other* than the basic rows and columns, given that each possible domain-type would be congruent with one or other of the two implicit groupings. However, it definitely could *not* be an "X" variant of the puzzle type (repetition dissallowed across the major diagonals, as well as across rows and columns), otherwise it reverts to being impossible again...

However, none of what I've just said is particularly entertaining, so please feel free to ignore it and instead try the following Unary Sudoku.... (Hint: its major diagonals are also valid domains to solve!)

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178.98.31.27 15:07, 24 June 2013 (UTC)

## Major update

I think we even can discus PIxPI grids here at discussion page, but the explain should be simple as possible. Please help on that bad remaining language. AND: Since Randall is from the US we have AE (American English) here.--Dgbrt (talk) 21:19, 1 July 2013 (UTC)

I think this description of the alt text is a little inadequate. The reason the puzzle is 'medium difficulty' is because any given puzzle in a binary sudoku is going to be essentially the same... you'll either have a one and a zero, two ones, or two zeroes since those are the only ways to ensure a unique solution. So all puzzles have the same difficulty which is why it is 'medium'.--108.162.238.162 08:28, 28 July 2015 (UTC)

- I don't follow your sentence "you'll either have a one and a zero, two ones, or two zeroes", please can you try and explain more clearly. Essentially though, there are two possible solutions to the puzzle:

0 1 or alternatively 1 0 1 0 0 1

- The only thing which can make the difficultly change is how many cells are pre-filled. If 0 are filled then either both solutions are correct, or it is impossible to know which solution is correct. If 1 cell is filled, then it is easy to complete the rest of the grid. If 2 are filled, then it is even easier to complete, and easier again with 3 filled. --Pudder (talk) 11:01, 28 July 2015 (UTC)