How many different Sudoku puzzles can there be mathematically?

Question

how did you figure 1 trillion or 3.5 trillion?

Answer 1

One.

Take the letters A, B & C, and apply them three times to a 3x3 grid so as to not repeat in a row or column. Now do it differently. Sure, you can, but you will find only a variation of the original. It might be mirrored sideways, or mirrored vertically, or rotated, or both, but it essentially the same singular solution.

While some may claim that the variations in Sudoku are on the level of an exponential factorial number of solutions, one would not get very far into that factorial before redundancy sets in. When one considers mirrors and rotations of a single solution as being a copy of the original, then serious limitations are placed on how many are unique.

Mirror a solved puzzle. Is it a different answer or a variation of the first? Move the top three rows to the bottom. Is that not just a variation of the first answer?

There are only so many ways that the numbers 1-9 can be applied in a 3x3 square such that it can mesh with its "neighboring squares" in accordance with Sudoku rules. This is not a factorial number; not even an exponential number.

Let us look at this another way.

How many numbers contain the digit "3"?

Answer: almost all of them.

Sure, the numbers from 1 to 10 witness the digit 3 only 10% of the time, but from 1-100, the incidence is 19%. from 1-1000 we approach 27%, and before we reach a billion we are pushing past 90%.

Some answers here are suggesting that there is not even one redundant solution in several billion? That is not only unlikely, but absurd to believe. Are we to accept that all such entries in a factorial are actually valid solutions to a distinctly ruled 4D matrix?

How many numbers from one to one billion actually have all nine digits? 512. Apply those to the rules of Sudoku in unique ways.

OKAY...maybe there is more than one valid solution even when given liberal consideration to mirrors and rotations. However, I contend quite readily that my answer of ONE is far closer to the truth than billions.

Almost all numbers have the digit three, and almost all Sudoku solutions resemble but a few genuinely unique solutions.

Actually just one.

Answer 2

This has been answered: http://answers.yahoo.com/question/index;_ylt=Ah6wyriBej34IWuEZSPIimvsy6IX?qid=20061014171429AAQzveY

"This is an interesting topic, which I looked at a while ago but didn't get far.

The other number of interest is the number boards where each digit 1-9 occurs 9 times (without the Sudoku conditions). How many of these are sudoku boards? There are 81!(9!^9) of them. That is a big number.

Followup: look at http://www.afjarvis.staff.shef.ac.uk/sud... It says (with links to explain):

There are 6670903752021072936960 Sudoku grids (Bertram Felgenhauer and Frazer Jarvis)

See
Details of the enumeration
A summary page of results from Ed Russell's program
Original write-up
Recent rewrite
There are 5472730538 essentially different Sudoku grids (Ed Russell and Frazer Jarvis)

Source(s):
http://www.afjarvis.staff.shef.ac.uk/sud...

Answer 3

i'd say for a 9x9 there are 9!*9! = 1,361,681,894,400 possibilities for putting 1-9 in each small square and in then all those possibilities could go into every big square. However, not all would be solutions, because every line and every column would also have to be 1-9. I'm too tired to think about it so I guess I'll go to bed instead.

PS somebody worked this out at the linked website below. I dont' know if this is accurate, but maybe it will help.

Answer 4

In my hometown newspaper they give you a daily Sudoku puzzle and it starts off easy and gets harder through the week. I can only do Monday and Tuesday's puzzle the rest are too hard!!!

Answer 5

mathmatically there can be over 1 trillion sudoku puzzles taht are 9x9 squares

Answer 6

6561
or 9 power of 9

Answer 7

3,546,146,300,288

♥