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...)?
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...)?
posted by The Navigator
13 Comments:
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)
I do not agree with the number
6,670,903,752,021,072,936,960
I think the correct number of
possible 9x9 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 1x9 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
I do not agree with the number
6,670,903,752,021,072,936,960
I think the correct number of
possible 9x9 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 1x9 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
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.
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
you are all wrong.
the total number of sudoku combinations is just over 9.75x10^15
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 9x9
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
do these answers take into account the symetry of the 9x9 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.
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
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
The first post is just slightly wrong. The factors are 2^20 * 3^8 * 5 * 7 * 27704267971.
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
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.
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