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Sudoku
(From Wikipedia,
the free encyclopedia.)
Sudoku,
sometimes spelled Su Doku, is a logic-based placement puzzle,
also known as Number Place in the United States. The aim of the
canonical puzzle is to enter a numerical digit from 1 through 9 in each
cell of a 9×9 grid made up of 3×3 subgrids (called "regions"), starting
with various digits given in some cells (the "givens"). Each row, column,
and region must contain only one instance of each numeral. Completing
the puzzle requires patience and logical ability. Although first published
in 1979, Sudoku initially caught on in Japan in 1986 and attained
international popularity in 2005.
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Contents
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Carol
Vorderman's Handheld Sudoku
With Carol's
handheld electronic Sudoku game, you can test your mettle against
750 puzzles of varying difficulty. Play as much as you like,
wherever you like. Because you don't need a pen, your grids
won't look like an explosion in an ink factory if you want to
play on the bus or train. From
Boysstuff
Ltd
|
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of page
Introduction
The term Sudoku
implies "numbers singly", or loosely translates to "all the numbers
must remain unmarried", in Japanese; it is a registered trademark of
puzzle publisher Nikoli Co. Ltd in Japan. Other Japanese publishers
generally refer to the puzzle as Nanpure (Number Place),
its original title. Sudoku is pronounced as the English words
"SUE-dough-coo", with the first syllable accented.
The numerals in
Sudoku puzzles are used for convenience; arithmetic relationships
between numerals are absolutely irrelevant. Any set of distinct symbols
will do; letters, shapes, or colours may be used without altering the
rules (Penny Press' Scramblets and Knight Features Syndicate's
Sudoku Word both use letters). Dell Magazines, the puzzle's originator,
has been using numerals for Number Place in its magazines since
they first published it in 1979. Numerals are used throughout this article.
The attraction of
the puzzle is that the completion rules are simple, yet the line of
reasoning required to reach the completion may be difficult. Sudoku
is recommended by some teachers as an exercise in logical reasoning.
The level of difficulty of the puzzles can be selected to suit the audience.
The puzzles are often available free from published sources and also
may be custom-generated using software.
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Rules
and terminology
The puzzle is most
frequently a 9×9 grid, made up of 3×3 subgrids called "regions" (other
terms include "boxes", "blocks", and the like when referring to the
standard variation). Some cells already contain numbers, known as "givens"
(or sometimes as "clues"). The goal is to fill in the empty cells, one
number in each, so that each column, row, and region contains the numbers
1–9 exactly once. Each number in the solution therefore occurs only
once in each of three "directions", hence the "single numbers" implied
by the puzzle's name.
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Solution
methods
The strategy for
solving a puzzle may be regarded as comprising a combination of three
processes: scanning, marking up, and analysing.
The
3×3 region in the top-right corner must contain a 5. By hatching across
and up from 5s located elsewhere in the grid, the solver can eliminate
all of the empty cells in the top-right corner which cannot contain
a 5. This leaves only one possible cell (highlighted in green).
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Scanning
is performed at the outset and periodically throughout the solution. Scans
may have to be performed several times in between analysis periods. Scanning
consists of two basic techniques:
-
Cross-hatching:
the scanning of rows (or columns) to identify which line in a particular
region may contain a certain number by a process of elimination. This
process is then repeated with the columns (or rows). For fastest results,
the numbers are scanned in order of their frequency. It is important
to perform this process systematically, checking all of the digits
1–9.
-
Counting
1–9 in regions, rows, and columns to identify missing numbers.
Counting based upon the last number discovered may speed up the search.
It also can be the case (typically in tougher puzzles) that the value
of an individual cell can be determined by counting in reverse—that
is, scanning its region, row, and column for values it cannot
be to see which is left.
Advanced
solvers look for "contingencies" while scanning—that is, narrowing a number's
location within a row, column, or region to two or three cells. When those
cells all lie within the same row (or column) and region, they
can be used for elimination purposes during cross-hatching and counting
(Contingency example at Puzzle Japan). Particularly challenging puzzles
may require multiple contingencies to be recognized, perhaps in multiple
directions or even intersecting—relegating most solvers to marking up
(as described below). Puzzles which can be solved by scanning alone without
requiring the detection of contingencies are classified as "easy" puzzles;
more difficult puzzles, by definition, cannot be solved by basic scanning
alone.
Marking
up
Scanning
comes to a halt when no further numbers can be discovered. From this point,
it is necessary to engage in some logical analysis. Many find it useful
to guide this analysis by marking candidate numbers in the blank cells.
There are two popular notations: subscripts and dots.
-
In
the subscript notation the candidate numbers are written in subscript
in the cells. The drawback to this is that original puzzles printed
in a newspaper usually are too small to accommodate more than a few
digits of normal handwriting. If using the subscript notation, solvers
often create a larger copy of the puzzle or employ a sharp or mechanical
pencil.
-
The
second notation is a pattern of dots with a dot in the top left hand
corner representing a 1 and a dot in the bottom right hand corner
representing a 9. The dot notation has the advantage that it can be
used on the original puzzle. Dexterity is required in placing the
dots, since misplaced dots or inadvertent marks inevitably lead to
confusion and may not be easy to erase without adding to the confusion.
Using a pencil would then be recommended.
An alternative
technique that some find easier is to mark up those numbers that
a cell cannot be. Thus a cell will start empty and as more constraints
become known it will slowly fill. When only one marking is missing, that
has to be the value of the cell.
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Analysing
The two
main approaches to analysis are "candidate elimination" and "what-if".
- In elimination,
progress is made by successively eliminating candidate numbers from
one or more cells to leave just one choice. After each answer has
been achieved, another scan may be performed—usually checking to see
the effect of the latest number. There are a number of elimination
tactics, all of which are based on the simple rules given above, which
have important and useful corollaries, including:
- A given
set of n cells in any particular block, row, or column
can only accommodate n different numbers. This is the
basis for the "unmatched candidate deletion" technique, discussed
below.
- Each set
of candidate numbers, 1–9, must ultimately be in an independently
self-consistent pattern. This is the basis for advanced analysis
techniques that require inspection of the entire set of possibilities
for a given candidate number. Only certain "closed circuit" or
"n×n grid" possibilities exist (which have acquired peculiar names
such as "X-wing" and "Swordfish", among others; see List
of Sudoku terms and jargon for more information). If these
patterns can be identified, elimination of candidate possibilities
external to the grid framework can sometimes be achieved.
- One of the most
common elimination tactics is "unmatched candidate deletion". Cells
with identical sets of candidate numbers are said to be matched if
the quantity of candidate numbers in each is equal to the number of
cells containing them; essentially, these are perfectly coincident
contingencies. For example, cells are said to be matched within a
particular row, column, or region (scope) if two cells contain the
same pair of candidate numbers (p,q) and no others, or if three cells
contain the same triplet of candidate numbers (p,q,r) and no others.
The placement of these numbers anywhere else in the matching scope
would make a solution for the matched cells impossible; thus, the
candidate numbers (p,q,r) appearing in unmatched cells in the row,
column or region scope can be deleted. This principle also works with
candidate number subsets—if three cells have candidates (p,q,r), (p,q),
and (q,r) or even just (p,r), (q,r), and (p,q), all of the set (p,q,r)
elsewhere in the scope can be deleted. The principle is true for all
quantities of candidate numbers.
- A second related
principle is also true — if the number of cells (in a row, column
or region scope) where a set of candidate numbers only appear is equal
to the quantity of candidate numbers, the cells and numbers are matched
and only those numbers can appear in matched cells. Other candidates
in the matched cells can be eliminated. For example, if (p,q) can
only appear in 2 cells (within a specific row, column, region scope),
other candidates in the 2 cells can be eliminated.
- The first principle
is based on cells where only matched numbers appear. The second is
based on numbers that appear only in matched cells. The validity of
either principle is demonstrated by posing the question 'Would entering
the eliminated number prevent completion of the other necessary placements?'
Advanced techniques carry these concepts further to include multiple
rows, columns, and blocks. (See "X-wing" and "Swordfish" above.)
- In the what-if
approach, a cell with only two candidate numbers is selected, and
a guess is made. The steps above are repeated unless a duplication
is found or a cell is left with no possible candidate, in which case
the alternative candidate is the solution. In logical terms, this
is known as reductio ad absurdum. Nishio is a limited form
of this approach: for each candidate for a cell, the question is posed:
will entering a particular number prevent completion of the other
placements of that number? If the answer is yes, then that candidate
can be eliminated. The what-if approach requires a pencil and eraser.
This approach may be frowned on by logical purists as trial and error
(and most published puzzles are built to ensure that it will never
be necessary to resort to this tactic,) but it can arrive at solutions
fairly rapidly.
Ideally
one needs to find a combination of techniques which avoids some of the
drawbacks of the above elements. The counting of regions, rows, and columns
can feel boring. Writing candidate numbers into empty cells can be time-consuming.
The what-if approach can be confusing unless you are well organised. The
proverbial Holy Grail is to find a technique which minimises counting,
marking up, and rubbing out.
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Computer
solutions
For computer
programmers, coding the search for cell values based elimination, contingencies
and multiple contingencies (required for harder Sudoku) is relatively
straightforward. These programs emulate the human logic to solve a puzzle
without resorting to guesses. Given the self-imposed constraints of most
Sudoku publishers, this method generally succeeds.
It is
also fairly simple to build a backtracking search. Typically this involves
assigning a value (say, 1, or the nearest available number to 1) to the
first available cell (say, the top left hand corner) and then moves on
to assign the next available value (say, 2) to the next available cell.
This continues until a conflict occurs, in which case the next alternative
value is used for the last cell changed. If a cell cannot be filled, the
program backs up one level (from that cell) and tries the next value at
the higher level (hence the name backtracking). Although far from computationally
efficient, this "brute force" method will find a solution, given sufficient
computation time. A standard 9×9 puzzle can typically be "solved" in under
two seconds using almost any programming language. An extremely difficult
puzzle can take as much as 1 minute. A more efficient program could keep
track of potential values for cells, eliminating impossible values until
only one value remains for a cell, then filling that cell in and using
that information for more eliminations, and so on until the puzzle is
solved.
Another
alternative uses finite domain constraint programming. A constraint program
specifies the constraints of the puzzle (the fact that every number in
each row, each column, and each 3×3 region must be unique, and the provided
"givens"); a finite domain solver applies the constraints successively
to narrow down the solution space until a solution is found. Backtracking
may be applied when alternate values cannot otherwise be excluded.
A highly
efficient way of solving such constraint problems is the Dancing Links
Algorithm, by Donald Knuth. This method can be directly applied to solving
Sudoku problems, counting all possible solutions for most puzzles
in milliseconds. This is the method now preferred by many Sudoku
programmers, mainly by virtue of its speed. A very fast solver is usually
required for most trial-and-error puzzle-creation algorithms.
Difficulty
ratings
Published
puzzles often are ranked in terms of difficulty. Perhaps surprisingly,
the number of givens has little or no bearing on a puzzle's difficulty.
A puzzle with a minimum number of givens may be very easy to solve, and
a puzzle with more than the average number of givens can still be extremely
difficult to solve. It is based on the relevance and the positioning of
the numbers rather than the quantity of the numbers.
Computer
solvers can estimate the difficulty for a human to find the solution,
based on the complexity of the solving techniques required. This estimation
allows publishers to tailor their Sudoku puzzles to audiences of
varied solving experience. Some online versions offer several difficulty
levels.
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Construction
It is
possible to set starting grids with more than one solution and to set
grids with no solution, but such are not considered proper Sudoku
puzzles; like most other pure-logic puzzles, a unique solution is expected.
Building
a Sudoku puzzle by hand can be performed efficiently by pre-determining
the locations of the givens and assigning them values only as needed to
make deductive progress. Such an undefined given can be assumed to not
hold any particular value as long as it is given a different value before
construction is completed; the solver will be able to make the same deductions
stemming from such assumptions, as at that point the given is very much
defined as something else. This technique gives the constructor greater
control over the flow of puzzle solving, leading the solver along the
same path the compiler used in building the puzzle. (This technique is
adaptable to composing puzzles other than Sudoku as well.) Great
caution is required, however, as failing to recognize where a number can
be logically deduced at any point in construction—regardless of how tortuous
that logic may be—can result in an unsolvable puzzle when defining a future
given contradicts what has already been built. Building a Sudoku
with symmetrical givens is a simple matter of placing the undefined givens
in a symmetrical pattern to begin with.
It is
commonly believed that Dell Number Place puzzles are computer-generated;
they typically have over 30 givens placed in an apparently random scatter,
some of which can possibly be deduced from other givens. They also have
no authoring credits — that is, the name of the constructor is not printed
with any puzzle. Wei-Hwa Huang claims that he was commissioned by Dell
to write a Number Place puzzle generator in the winter of 2000; prior
to that, he was told, the puzzles were hand-made. The puzzle generator
was written with Visual C++, and although it had options to generate a
more Japanese-style puzzle, with symmetry constraints and fewer numbers,
Dell opted not to use those features, at least not until their recent
publication of Sudoku-only magazines.
Nikoli
Sudoku are hand-constructed, with the author being credited beside
each puzzle; the givens are always found in a symmetrical pattern. Dell
Number Place Challenger (see Variants below) puzzles also list
author credits. The Sudoku puzzles printed in most UK newspapers
are apparently computer-generated but employ symmetrical givens; The
Guardian licenses and publishes Nikoli-constructed Sudoku puzzles,
though it does not include authoring credits. The Guardian famously claimed
that because they were hand-constructed, their puzzles would contain "imperceptible
witticisms" that would be very unlikely in computer-generated Sudoku.
The challenge to Sudoku programmers is teaching a program how to
build clever puzzles, such that they may be indistinguishable from
those constructed by humans; Wayne Gould required six years of tweaking
his popular program before he believed he achieved that level.
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Variants
A nonomino Sudoku
puzzle
Although
the 9×9 grid with 3×3 regions is by far the most common, numerous variations
abound: sample puzzles can be 4×4 grids with 2×2 regions; 5×5 grids with
pentomino regions have been published under the name Logi-5; the
World Puzzle Championship has previously featured a 6×6 grid with 2×3
regions and a 7×7 grid with six heptomino regions and a disjoint region;
Daily SuDoku features new 4×4, 6×6, and simpler 9×9 grids every day as
Daily SuDoku for Kids. Even the 9×9 grid is not always standard,
with Ebb regularly publishing some of those with nonomino regions; the
2005 U.S. Puzzle Championship had a Sudoku with parallelogram regions
that wrapped around the outer border of the puzzle, as if the grid were
toroidal. Larger grids are also possible, with Daily SuDoku's 12×12-grid
Monster SuDoku , Dell regularly publishing 16×16 Number Place
Challenger puzzles, and Nikoli proffering 25×25 Sudoku the Giant
behemoths.
Another
common variant is for additional restrictions to be enforced on the placement
of numbers beyond the usual row, column, and region requirements. Often
the restriction takes the form of an extra "dimension"; the most common
is for the numbers in the main diagonals of the grid to also be required
to be unique. The aforementioned Number Place Challenger puzzles
are all of this variant, as are the Sudoku X puzzles in the Daily
Mail, which use 6×6 grids. The Daily Mail also features Super
Sudoku X in its Weekend magazine: an 8×8 grid in which rows,
columns, main diagonals, 2×4 blocks and 4×2 blocks contain each number
once. Another dimension in use is digits with the same relative location
within their respective regions; such puzzles are usually printed in colour,
with each disjoint group sharing one colour for clarity.
Other
kinds of extra restrictions can be mathematical in nature, such as requiring
the numbers in delineated segments of the grid to have specific sums or
products (an example of the former being Killer Su Doku in The
Times), demarcating all places arithmetically adjacent digits appear
orthogonally adjacent in the grid, providing the parity of all cells,
requiring the Lo Shu Square to appear in the solution, and so on. Some
such variants forsake standard givens entirely.
Puzzles
constructed from multiple Sudoku grids are commonly seen. Five
9×9 grids which overlap at the corner regions in the shape of a quincunx
is known in Japan as Gattai 5 (five merged) Sudoku. In The
Times and The Sydney Morning Herald this form of puzzle is
known as Samurai SuDoku. Puzzles with twenty or more overlapping
grids are not uncommon in some Japanese publications. Often, no givens
are to be found in overlapping regions. Sequential grids, as opposed to
overlapping, are also published, with values in specific locations in
grids needing to be transferred to others.
Alphabetical
variations, which use letters rather than numbers, have also emerged;
of course, there is no functional difference in the puzzle unless the
letters spell something. Recent variants have just that, often in the
form of a word reading along a main diagonal once solved; determining
the word in advance can be viewed as a solving aid. The Code Doku
devised by Steve Schaefer has an entire sentence embedded into the puzzle;
the Super Wordoku from Top Notch embeds two 9-letter words, one
on each diagonal. It is debatable whether these are true Sudoku
puzzles: although they purportedly have a single linguistically
valid solution, they cannot necessarily be solved entirely by logic, requiring
the solver to determine the embedded words. Top Notch claim this as a
feature designed to defeat solving programs.
Here
are some of the more notable unique variations:
- A three-dimensional
Sudoku puzzle was invented by Dion Church and published in
the Daily Telegraph in May 2005.
- The 2005 U.S.
Puzzle Championship includes a variant called Digital Number Place:
rather than givens, most cells contain a partial given—a segment of
a number, with the numbers drawn as if part of a seven-segment display.
- Wei-Hwa Huang
created a meta-Sudoku, where the object is to finish drawing
the 5×5 grid's pentomino-region borders so as to leave a uniquely
solvable puzzle with no identically-shaped regions.
Mathematics
of Sudoku
The general
problem of solving Sudoku puzzles on n2 x n2
boards of n x n blocks is known to be NP-complete. This
gives some indication of why Sudoku is difficult to solve, although
on boards of finite size the problem is finite and can be solved by a
deterministic finite automaton that knows the entire game tree.
For full
mathematical results and conjectures, see the Mathematics
of Sudoku page.
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History
Maki
Kaji, president of Nikoli (right) and colleague Dave Green, president
of Conceptis, Tokyo, April 16, 2004
The puzzle
was designed by Howard Garns, a retired architect and freelance puzzle
constructor, and first published in 1979. Although likely inspired by
the Latin square invention of Leonhard Euler, Garns added a third dimension
(the regional restriction) to the mathematical construct and (unlike Euler)
presented the creation as a puzzle, providing a partially-completed grid
and requiring the solver to fill in the rest. The puzzle was first published
in New York by the specialist puzzle publisher Dell Magazines in its magazine
Dell Pencil Puzzles and Word Games, under the title Number Place
(which we can only assume Garns named it).
The puzzle
was introduced in Japan by Nikoli in the paper Monthly Nikolist
in April 1984 as Suuji wa dokushin ni kagiru (????????), which
can be translated as "the numbers must be single" or "the numbers must
occur only once" (?? literally means "single; celibate; unmarried"). The
puzzle was named by Kaji Maki (?? ??), the president of Nikoli. At a later
date, the name was abbreviated to Sudoku (??, pronounced SUE-dough-coo;
su = number, doku = single); it is a common practice in Japanese to take
only the first kanji of compound words to form a shorter version. In 1986,
Nikoli introduced two innovations which guaranteed the popularity of the
puzzle: the number of givens was restricted to no more than 32 and puzzles
became "symmetrical" (meaning the givens were distributed in rotationally
symmetric cells). It is now published in mainstream Japanese periodicals,
such as the Asahi Shimbun. Within Japan, Nikoli still holds the
trademark for the name Sudoku; other publications in Japan use
alternative names.
In 1989,
Loadstar/Softdisk Publishing published DigitHunt on the Commodore
64, which was apparently the first home computer version of Sudoku.
At least one publisher still uses that title.
Yoshimitsu
Kanai published his computerized puzzle generator under the name Single
Number (the English translation of 'sudoku') for the Apple Macintosh
in 1995 in Japanese and English, and in 1996 for the Palm (PDA).
Bringing
the process full-circle, Dell Magazines, which publishes the original
Number Place puzzle, now also publishes two Sudoku magazines:
Original Sudoku and Extreme Sudoku. Additionally, Kappa
reprints Nikoli Sudoku in GAMES Magazine under the name
Squared Away; the New York Post, USA Today, The
Boston Globe, Washington Post, and San Francisco Chronicle
now also publish the puzzle. It is also often included in puzzle anthologies,
such as The Giant 1001 Puzzle Book (under the title Nine Numbers).
Within
the context of puzzle history, parallels are often cited to Rubik's Cube,
another logic puzzle popular in the 1980s. Sudoku has been called
the "Rubik's cube of the 21st century".
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Popularity
in the media
In 1997,
retired Hong Kong judge Wayne Gould, 59, a New Zealander, was enticed
by seeing a partly completed puzzle in a Japanese bookshop. He went on
to develop a computer program to produce puzzles quickly; this took over
six years. Knowing that British newspapers have a long history of publishing
crosswords and other puzzles, he promoted Sudoku to The Times
in Britain, which launched it on 12 November 2004 (calling it Su Doku).
The puzzles by Pappocom, Gould's software house, have been printed daily
in the Times ever since.
Three
days later The Daily Mail began to publish the puzzle under the
name "Codenumber". The Daily Telegraph introduced its first Sudoku
by its puzzle compiler Michael Mepham on 19 January 2005 and other Telegraph
Group newspapers took it up very quickly. Nationwide News Pty Ltd began
publishing the puzzle in The Daily Telegraph of Sydney on 20 May
2005; five puzzles with solutions were printed that day. The immense surge
in popularity of Sudoku in British newspapers and internationally
has led to it being dubbed in the world media in 2005 variously as "the
Rubik's cube of the 21st century" or the "fastest growing puzzle in the
world".
There
is no doubt that it was not until The Daily Telegraph introduced
the puzzle on a daily basis on 23 February 2005 with the full front-page
treatment advertising the fact, that the other UK national newspapers
began to take real interest. The Telegraph continued to splash
the puzzle on its front page, realizing that it was gaining sales simply
by its presence. Until then the Times had kept very quiet about
the huge daily interest that its daily Sudoku competition had aroused.
That newspaper already had plans for taking advantage of their market
lead, and a first Sudoku book was already on the stocks before
any of the other national papers had realised just how popular Sudoku
might be.
By April
and May 2005 the puzzle had become popular in these publications and it
was rapidly introduced to several other national British newspapers including
The Independent, The Guardian, The Sun (where it
was labelled Sun Doku), and The Daily Mirror. As the name
Sudoku became well-known in Britain, the Daily Mail adopted
it in place of its earlier name "Codenumber". Newspapers competed to promote
their Sudoku puzzles, with The Times and the Daily Mail
each claiming to have been the first to feature Sudoku, and The
Guardian claiming (though perhaps ironically) that its hand-made puzzles,
licensed from Nikoli, offered a superior experience (complete with "almost
imperceptible witticisms") to the computer-generated grids found in other
papers.
The rapid
rise of Sudoku from relative obscurity in Britain to a front-page
feature in national newspapers attracted commentary in the media (see
References below) and parody (such as when The Guardian's
G2 section advertised itself as the first newspaper supplement
with a Sudoku grid on every page). Sudoku became particularly
prominent in newspapers soon after the 2005 general election leading some
commentators to suggest that it was filling the gaps previously occupied
by election coverage. A simpler explanation is that the puzzle attracts
and retains readers—Sudoku players report an increasing sense of
satisfaction as a puzzle approaches completion. Recognizing the different
psychological appeals of easy and difficult puzzles The Times introduced
both side by side on 20 June 2005. From July 2005 Channel 4 included a
daily Sudoku game in their Teletext service (at page 142). On 2nd
August 2005 the BBC's programme guide Radio Times started to feature a
weekly Super Sudoku.
The world's first
live TV Sudoku show, 1 July 2005, Sky One.
As a
one-off, the world's first live TV Sudoku show, Sudoku Live,
was broadcast on 1 July 2005 on Sky One. It was presented by Carol Vorderman.
Nine teams of nine players (with one celebrity in each team) representing
geographical regions competed to solve a puzzle. Each player had a hand-held
device for entering numbers corresponding to answers for four cells. Conferring
was permitted although the lack of acquaintance of the players with each
other inhibited an analytical discussion. The audience at home was in
a separate interactive competition. A Sky One publicity stunt to promote
the programme with the world's largest Sudoku puzzle went awry
when the 275 foot (84 m) square puzzle was found to have 1,905 correct
solutions. The puzzle was carved into a hillside in Chipping Sodbury,
near Bristol, England, in view of the M4 motorway.
CBS has
run several stories concerning Sudoku, including on the Early Show
in Summer 2005, and on the CBS Evening News that autumn, on October 26th.
Most
recently, Dr. House was clearly seen working on a Sudoku puzzle
on his office computer in one scene of the December 13, 2005 episode of
House, M. D..
|
Carol
Vorderman's Touchscreen Sudoku
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The
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The
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The
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The
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Total
Su Doku is exactly what the title suggests... a whole magazine
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Total Su Doku comes with comprehensive instructions and solutions and
prides itself with the careful thought that has gone into each puzzle.
This really is a magazine for all levels. It involves no maths, no knowledge
and no amount of genius. It's all about logical thinking and lots and
lots of practice.
 You
can now visit our sister site - The
Shopping Center, especially for visitors from the US.
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