To mathematicians: How many variants are possible for a Sudoku problem?

Question

How many possible grids are there?

Answer 1

27,704,267,971. Apparently.

Answer 2

Here's my explaination, I managedt o work all the figures through and got to the answer 27,704,267,971 which someone else gave below.

We begin with an empty 9x9 grid, which has 81 squares. Place a 1 into the grid, anywhere. Consider where you can place the next one - it can't be in the same 3x3 as the first, or the same column or row as the first, or the same square as the first. This leaves

81 - 8 - 8 - 1 = 56 free cells for the next 1

Having placed the next 1, we have

(7x7) - 8 - 1 = 40 free cells (as there are now 7 free rows and columns, minus the second 1's 3x3 grid, minus the 2nd 1s cell)

Repeating gives
(6x6) - 8 - 1 = 27
(5x5) - 8 - 1 = 16
(4x4) - 8 - 1 = 7

We eventually get to there being 3 3x3 squares without 1s, so we have 3 choices where to put the remaining three 1s. Now we have filled in all the 1s we move onto 2.


There are 56x40x27x16x7x3 = 20,321,280 different arrangements of the 1s in a 9x9 sudoku grid (although we could have started with any number)

Inserting 2, we use a similar forumla, remembering that 9 cells are already filled with 1s.

(9x9) - 9 = 72, the number of free cells after arranging all the 1s

Place first 2
72 - 7 - 7 - 7 - 1 = 50

Next 2
50 - 7 - 7 - 7 - 1 = 28...

and so on.

If you repeat the method of multiplying the number of available cells at every single entry to finally get to 27,704,267,971

Thanks for giving me something mentally stimulating to do at work!

Answer 3

Even though the 9×9 grid with 3×3 regions is by far the most common, 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. Larger grids are also possible, with Daily SuDoku's 16×16-grid Monster SuDoku [1], the Times likewise offers a 12×12-grid Dodeka sudoku with 12 regions each being 4×3, Dell regularly publishing 16×16 Number Place Challenger puzzles (the 16×16 variant often uses 1 through G rather than the 0 through F used in hexadecimal), 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.

Puzzles constructed from multiple Sudoku grids are common. 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. [2] 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 have also emerged; there is no functional difference in the puzzle unless the letters spell something. Some variants, such as in the TV Guide, include a word reading along a main diagonal, row, or column once solved; determining the word in advance can be viewed as a solving aid. The Code Doku [3] devised by Steve Schaefer has an entire sentence embedded into the puzzle; the Super Wordoku [4] 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 claims this as a feature designed to defeat solving programs.

Answer 4

If you mean the most common grid, a 9x9 grid, and remember they can be smaller, bigger and have slightly different rules, then I think there are 6.67×10 to the 21 ways of filling in a blank sudoku. Is this what you mean?

Answer 5

My first thought is that, if you take any solution and then do a permutation of the digits 1 to 9, e.g. replace 1 by 6, 6 by 5, 5 by 2, 2 by 9, 9 by7, etc, the result is still a solution. Is that what Vijay S means by "only 1"? That they're all basically the same structure?

Anyway, if that is so, the permutations would make it look like 9x8x7x6x5x4x3x2x1 different solutions,

Answer 6

Given a sudoku problem, there is only one solution.

If you want to create a sudoku problem, there are a lot, at least
10!=10X9X8X7X...X1, and then you can have a lot of possibilities
by choosing the numbers you want to leave on...

Answer 7

For a particular combination, only one set of answers.

Answer 8

Only one.

Answer 9

It is endless. As Dave Spikey said "Just fill it in as no one ever checks it".

Answer 10

13!!!