Math Is Fun Forum

Stanley_Marsh

Member

Registered: 2006-12-13

Posts: 345

Help arrangements

I was just wondering How many arrangements can be made to form a 9x9 grid Sudoku?

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Numbers are the essence of the Universe

Dross

Member

Registered: 2006-08-24

Posts: 325

Re: Help arrangements

That's an interesting one... I tried it ages ago and it got far, far too complicated! I'm at work at the moment but I'll give it a go when I get home and tell you if I come up with anything again.

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Bad speling makes me [sic]

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Help arrangements

That's a very interesting problem indeed. It's easy enough to say how many ways there are of filling in a line or a box (9!), but then when you start combining lines and boxes together then it gets all nasty.

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Why did the vector cross the road?
It wanted to be normal.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Help arrangements

6,670,903,752,021,072,936,960

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Dross

Member

Registered: 2006-08-24

Posts: 325

Re: Help arrangements

6,670,903,752,021,072,936,960

Could you please post them all, so we can be sure you're right? (there's one I can't find)

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Bad speling makes me [sic]

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Help arrangements

I'm not certain if that number includes isomorphic grids though.  What I mean is say that we changed the label for 1 and 2.  Every place in the original grid there is a 2, we replace it with 1, and we do the same thing with swapping 1 and 2.

The board now has different numbers in different places, so one may call it a different board.  But here we are using the numbers simply as symbols, and so I wouldn't really say that the board has changed.

The number of permutations of 9 numbers is 9!, and thus, just divide the number I posted by 9! to find the number of combinations.

This leaves 18,383,222,420,692,992 combinations.  Since the number is divisible by 9! (which is pretty rare, only 1 out of 9! numbers are divisible by 9!), I would have to say that the above number does not in fact include isomorphic grids.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."