Just how many Sudoku combinations are there?

For most of us, we can just glaze over here, but I thought I’d add this fun little sequence for the mathematicians amongst us.

The question is: If you start from a blank Sudoku grid, just how many different combinations can you have?

There is (wait for it…) 11 pages of forum thread on sudoku.com.

Thankfully a very wise person called Bertram has posted a solution.

The final calculation is:
2^15 * 3^8 * 5*7 * 27,704,267,971 = 9! * 72^2 * 2^7 * 27,704,267,971

And the final answer is (drum roll please…)
6,670,903,752,021,072,936,960.

But…. does anyone know how you say it (are we talking about trillions, gazillions, scadillions…)?

 

22 Responses to “Just how many Sudoku combinations are there?”

  1. 1:02 am on May 26 2005, Roger Walker said:

    Take a deep breath…

    6,670,903,752,021,072,936,960 is:-

    Six sextillion, six hundred and seventy quintillion,
    nine hundred and three quadrillion,
    seven hundred and fifty two trillion,
    twenty one billion, seventy two million,
    nine hundred and thirty six thousand,
    nine hundred and sixty.

    My page here helps; http://www.simetric.co.uk/siprefix.htm

    There were a few variations in the past between UK and US definitions of some *illions but seem to now accept the US version.

    I’m not too sure about this number though. Just compounding the pure possibles may not be the answer.

    When setting a puzzle it is all too easy to make it illogical by choosing the wrong numbers to give to the puzzler.

    So my next challange to the math boffins is:-

    How many Sudoku puzzles are possible when given 24 numbers initially? (only pure puzzles that can be solved without guesswork)

  2. 1:02 am on May 26 2005, Jay Jacob Wind said:

    I do not agree with the number
    6,670,903,752,021,072,936,960

    I think the correct number of
    possible 9×9 Sudoku grids is
    1,834,933,472,251,080,000,000

    Pick any cell, e.g., (1,1)
    It can have 9 values.
    Pick another cell in its row,
    e.g., (1,2).
    It can have 8 values.
    Following that logic,
    (1,3) can have 7 values,
    (1,4) can have 6 values,
    (1,5) can have 5 values,
    (1,6) can have 4 values,
    (1,7) can have 3 values,
    (1,8) can have 2 values, and
    we have solved for (1,9) = 1 value.
    Therefore, a 1×9 Sudoku row
    can have 9*8*7*6*5*4*3*2 patterns =
    9! = 362880

    If we know the first row,
    we can have 8 values in (2,1),
    then 7, 6, 5, 4, 3, 2, 1, and 1 =
    8! = 40320

    Then 7! = 5040
    Then 6! = 720
    Then 5! = 120
    Then 4! = 24
    Then 3! = 6
    Then 2! = 2 and the last cell is solved.

    9!*8!*7!*6!*5!*4!*3!*2! =
    1,834,933,472,251,080,000,000

  3. 1:02 am on May 26 2005, Jay Jacob Wind said:

    I do not agree with the number
    6,670,903,752,021,072,936,960

    I think the correct number of
    possible 9×9 Sudoku grids is
    1,834,933,472,251,080,000,000

    Pick any cell, e.g., (1,1)
    It can have 9 values.
    Pick another cell in its row,
    e.g., (1,2).
    It can have 8 values.
    Following that logic,
    (1,3) can have 7 values,
    (1,4) can have 6 values,
    (1,5) can have 5 values,
    (1,6) can have 4 values,
    (1,7) can have 3 values,
    (1,8) can have 2 values, and
    we have solved for (1,9) = 1 value.
    Therefore, a 1×9 Sudoku row
    can have 9*8*7*6*5*4*3*2 patterns =
    9! = 362880

    If we know the first row,
    we can have 8 values in (2,1),
    then 7, 6, 5, 4, 3, 2, 1, and 1 =
    8! = 40320

    Then 7! = 5040
    Then 6! = 720
    Then 5! = 120
    Then 4! = 24
    Then 3! = 6
    Then 2! = 2 and the last cell is solved.

    9!*8!*7!*6!*5!*4!*3!*2! =
    1,834,933,472,251,080,000,000

    — Jay Jacob Wind

  4. 1:02 am on May 26 2005, joe blows anything long said:

    Sorry Jay,

    Let’s say your first row selection is
    1 2 3 4 5 6 7 8 9
    Then you choose a 2 for the first element of the second row.
    2 a b c d e f g h
    The second elemnt of the second row could now be one of 8 (not 7) choices.

    You have undercounted the number of possible permutations.

  5. 1:02 am on May 26 2005, stuart said:

    If the first row is 123456789 and the first element in the second row is a 2 then 2 appears twice in the top box of nine! So should the answer be alot lower

  6. 1:02 am on May 26 2005, Hidenrage said:

    you are all wrong.

    the total number of sudoku combinations is just over 9.75×10^15

    or

    9,751,984,865,280,000

    that’s nine quadrillion seven hundred fifty one trillion nine hundred eighty four billion eight hundred sixtry five million two hundred eighty thousand.

    here is actually why….

    the standard sudoku grid is 9×9

    your first row would be

    9x8x7x6x5x4x3x2x1

    your first box would be

    9x8x7x
    6x5x4x
    3x2x1

    now, your second box, due to the probability of the first row, has to start with 6x5x4. If you use 3 possible numbers for the first row of the second box, and 3 numbers of the second row of box 1, you are left with only 3 possible numbers left.

    see the image here to see what the whole probability schematic looks like

    http://www.kurtism.com/sudoku.jpg

    -me

  7. 1:02 am on May 26 2005, Anna said:

    do these answers take into account the symetry of the 9×9 sudoku board? It seems that one solution may be tranformed into a “different” solution by simply rotating the board by 90, 180, or 270 degrees. taking this into account, does the apparent right answer: 9,751,984,865,280,000 need to be divided by 4 (or something)? clearly I’m not a statistician. I’m just curious.

  8. 1:02 am on May 26 2005, Sylow_P said:

    Still wrong. Hidenrage is close, but your mistake begins with the fourth cell of the second row. You incorrectly assume that everthing to the left [(2,1)-(2,3)] is unique from [(1,4)-(1,6)].

    They could be exactly the same, which means the choices for (2,4) could be 6, 5, or 3 depending on how many of the above are the same.

    Try reading the real math here http://www.afjarvis.staff.shef.ac.uk/sudoku/sudoku.pdf

    The true result is:
    6,670,903,752,021,072,936,960

  9. 1:02 am on May 26 2005, Ellie said:

    This is to Anna
    Sudoku puzzles are not symmetrical at all, if you look at a solved puzzle. The only symmetry is in the placement of the numbers that are placed before you begin. Complete puzzles usually have no symmetry, although in the 6,670,903,752,021,072,936,960 combinations, you might find one that is symmetrical.

    Ellie

  10. 1:02 am on May 26 2005, The Wibberlys said:

    This comment has been removed by the author.

  11. 1:02 am on May 26 2005, Marriage Sucks. Then You Die! said:

    The first post is just slightly wrong. The factors are 2^20 * 3^8 * 5 * 7 * 27704267971.

  12. 1:02 am on May 26 2005, Jesper Uhrenholt said:

    and if we take how many sudoku puzzles there are it is
    6670903752021072936960*80!+1

    the +1 is a blank sudoku

    and the number of puzzles are
    1,9893*10^139

    and just for a comparison, the number of atoms in the universe is 10^80

  13. 1:02 am on May 26 2005, Abrakadeadguy said:

    You are all wrong. the equation is (9x8x7x6x5x4x3x2x1)x 3
    in the first upper left hand box there is 362880. then the middle box can have the same number of combinations because they dont affect the upper left. and the same with the lower right. so the final answer is either 362880 or 1088640. If the first box is filled im not sure if it sets all the other ones for sure or not.

  14. 1:02 am on May 26 2005, Christer Falchen said:

    isn’t the answer: how many different ways you can write the numbers 1-9 wich is 9,999,999,999

  15. 1:02 am on May 26 2005, profDanielCz said:

    the basic sudoco is
    123456789
    456789123
    789123456
    234567891
    567891234
    891234567
    345678921
    678921345
    921345678

  16. 1:02 am on May 26 2005, profDanielCz said:

    the basic sudoco is
    123456789
    456789123
    789123456
    234567891
    567891234
    891234567
    345678921
    678921345
    921345678

  17. 1:02 am on May 26 2005, profDanielCz said:

    the basic sudoco is
    123456789
    456789123
    789123456
    234567891
    567891234
    891234567
    345678921
    678921345
    921345678
    so divide the number by 9 factorial since i can make 9! from this one by renamimg the numbers. so the interesting question is how many different sudokos with the first line 123456789. So I can build 9!=3628800 form this basic one all of them structurally identical, since this is a class of eqivalence, better the given number is divisible by 9! and it is not so I do not like this number, I reject it
    Dr Cazalis

  18. 1:02 am on May 26 2005, profDanielCz said:

    The number must have at least two zeroes at the end since sudokos can be easelly divided into equivalence classes of 9 factorial elements each by the renaming of the numbers and there are 9! renaming so the number must be divisible by 3628800.

  19. 1:02 am on May 26 2005, Eliezer Garza said:

    I made an Excel Spreadsheet that gave all the posible values for each empty Sudoku cell, then I started filling from left->right top->bottom, with the same general secuence [9,8,7,6,5,4,3,2,1], and these factors came out:
    ____________________________
    |9!|8!|7!|6!|5!|4!|3!|2!|1!|
    |–+–+–+–+–+–+–+–+–|
    |6!|5!|4!|6!|5!|4!|3!|2!|1!|
    |–+–+–+–+–+–+–+–+–|
    |3!|2!|1!|3!|2!|1!|3!|2!|1!|
    |–+–+–+–+–+–+–+–+–|
    |6!|6!|5!|4!|4!|3!|2!|2!|1!|
    |–+–+–+–+–+–+–+–+–|
    |4!|4!|3!|4!|4!|3!|2!|2!|1!|
    |–+–+–+–+–+–+–+–+–|
    |2!|2!|1!|2!|2!|1!|2!|2!|1!|
    |–+–+–+–+–+–+–+–+–|
    |3!|3!|3!|2!|2!|2!|1!|1!|1!|
    |–+–+–+–+–+–+–+–+–|
    |2!|2!|2!|2!|2!|2!|1!|1!|1!|
    |–+–+–+–+–+–+–+–+–|
    |1!|1!|1!|1!|1!|1!|1!|1!|1!|
    —————————-
    at first the factors seemed counterintuitive, but after carefull exaination they make sense, only the other elements in your row, column and big cell affect te possible values for a cell.
    this is the solved puzzle:
    +—————–+
    |9|8|7|6|5|4|3|2|1|
    |-+-+-+-+-+-+-+-+-|
    |6|5|4|3|2|1|9|8|7|
    |-+-+-+-+-+-+-+-+-|
    |3|2|1|9|8|7|6|5|4|
    |-+-+-+-+-+-+-+-+-|
    |8|7|6|5|4|3|2|1|9|
    |-+-+-+-+-+-+-+-+-|
    |5|4|3|2|1|9|8|7|6|
    |-+-+-+-+-+-+-+-+-|
    |2|1|9|8|7|6|5|4|3|
    |-+-+-+-+-+-+-+-+-|
    |7|6|5|4|3|2|1|9|8|
    |-+-+-+-+-+-+-+-+-|
    |4|3|2|1|9|8|7|6|5|
    |-+-+-+-+-+-+-+-+-|
    |1|9|8|7|6|5|4|3|2|
    +—————–+

  20. 1:02 am on May 26 2005, Beastick said:

    2,580,481

  21. 1:02 am on May 26 2005, Patrick Bergeron said:

    No, you are all wrong.

    The number of possible combinations is 42.

  22. 1:02 am on May 26 2005, The Sudoku Experience » Blog Archive » Number of Possible Sudoku Grids said:

    […] to my previous post that gave an idea on how many combinations of Sudoku puzzles we can have, Alex Rothenberg has come […]

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