The New Sudoku Players' Forum

[susan]

We wanted to reply in the other topic, but it was impossible (every time we clicked on 'post reply' we got the home page ), that's why we started this one.

We are 2 dutch students, and we have a math assignment. We have to find out how many Su Doku grids are possible
in different messurements.

On a search for the best way to figure this out, we were send to this site. We tried to read the topic about Su Doku maths, but 31 pages of math in English is allot for us. That is why we hoped that someone can tell us what the right/best way is to see how many Su Doku's (9x9) there are. It would help us allot!

Thanks!
Simone&Susan

[simes]

have you seen the calculation by Bertram Felgenhauer and Frazer Jarvis?

It should be just what you want.

S

[susan]

Yes, Frazer sended us to this site. He said we could find methods here wich are more easy and better to understand..his article already helped us allot though!

[Pi]

There is a post called "Sudoku's maths" which you would probably be interested in

http://forum.enjoysudoku.com/viewtopic.php?t=44&highlight=maths

This should be what you want

[Pi]

I would recomend this as your approach

Remember that the numbers in sudoku are not numbers but mereley characters and so their value means nothing

Try applying one sudoku rule at a time

firstly the colums

How many ways are there are arranging number is a 9X9 grid with of each number in each collumn
Then find out the proportion of those that obey the row rule and then the proportion of those that obey the box rule

This should work for grids of all sizes

I'll start

Firstly for 2X2X2 grids (4X4)

There are 98 ways of making a 4X4 grid which obey the colum rule

Edit: sorry there are 331776 ways

I will not be much more help to you as i am only 15, but i will try

[susan]

We tried to read that topic, but we couldn't make up what was relevant and, what was not right. Our mathematical english isn't that good
So if anyone thinks to know what helpfull is from that topic, plz let us know!

Susan & Simone

[susan]

I would recomend this as your approach

Remember that the numbers in sudoku are not numbers but mereley characters and so their value means nothing

Try applying one sudoku rule at a time

firstly the colums

How many ways are there are arranging number is a 9X9 grid with of each number in each collumn
Then find out the proportion of those that obey the row rule and then the proportion of those that obey the box rule

This should work for grids of all sizes

I'll start

Firstly for 2X2X2 grids (4X4)

There are 98 ways of making a 4X4 grid which obey the colum rule

I will not be much more help to you as i am only 15, but i will try

You're age doesnt matter, we are also 'just' 17. The smaller Su Doku of 4x4 isn't the problem, that was quite easy. We found that it was harder to figure it out when you have a 'inner loop' (as mentioned in the article of Bertram Felgenhauer and Frazer Jarvis)...

[Pi]

Again with the 4X4's

if you keep the top two rows the same there are 4 ways of completing the sudoku

If you keep the top row the same there are 4 ways of completing the second row

There are 24 ways of making the top row

therefore the number of pussible 4X4 sudoku's is 24X4X4

which is 384

There are 384 ways of completing a 4X4 sudoku

[Pi]

Another Thought

You only need to be able to calculate the number of ways of making a grid which is missing either

One Box
One Row
One Collum
All of One Number
One of Each number

I intend to use the first of these[/list]

[susan]

You forgot that with relabelling entries,reflection, rotation and permutation there a lot of the same Su Doku's.

[Pi]

I am aware of that but i don't now the specification of your project

I will remove those at the end of my calculation i think


I have prgressed.
I am trying to find out the possible ways of making a sudoku with a box missing.

If 7 boxes are kept the same then there are 8 ways of making the 8th box as there are 2 ways of making each mini colum within the box and 2 cubes is 8

so

If 7 boxes are the kept constant there are 8 ways of making the 8th box and one way of making the 9th

Code: Select all

CCC|CCC|CCC|
CCC|CCC|CCC|
CCC|CCC|CCC|
-----------|
CCC|CCC|CCC|
CCC|CCC|CCC|
CCC|CCC|CCC|
-----------|
CCC|111|111|
CCC|222|111|
CCC|111|111|



I hope you know what i mean from this diagram

C = constant
number = number of choices for that box

[Pi]

I think that there are 216 ways of making the sixth box

Code: Select all

|CCC|CCC|CCC|
|CCC|CCC|CCC|
|CCC|CCC|CCC|
------------|
|CCC|CCC|CCC|
|CCC|CCC|CCC|
|CCC|CCC|CCC|
------------|
|333|111|111|
|222|222|111|
|111|111|111|


The key is the same as before

If this is true then there are 1728 ways of making the bottom three rows in a 9X9 grid

[Pi]

Bos six

Code: Select all

---
--6
---


is similar top box 8

Code: Select all

---
---
-8-


when looking for the amount of ways of completing it and the amount is the same, 8

again for each mini row each cell has two candidatesa and selecing one defined the others and so you have the same 2X2X2 which =8

I believe you are now left with this

Code: Select all

|CCC|CCC|CCC|
|CCC|CCC|CCC|
|CCC|CCC|CCC|
|CCC|CCC|121|
|CCC|CCC|121|
|CCC|CCC|121|
|333|111|111|
|222|222|111|
|111|111|111|


C=constant
number = number of possibilities

Now if 5 boxes are constant then i believe that there are 13824 ways of completing the puzzle from there

As i said before i am only 15 and it is likeley that someone here is an experienced mathematiciam. If i am mistaken i would like to know

[Pi]

In Box 3

Code: Select all

XX3
XXX
XXX


There are the same ways of completing this box as there are of completing box 7

Code: Select all

XXX
XXX
7XX


There are 216 ways of completing box 3
If you see each row whithin these collums as independent there are 6 ways of completing each row and so 6 cubed ways of completing the box.

My Overall Grid now looks like this

Code: Select all


|CCC|CCC|321|
|CCC|CCC|321|
|CCC|CCC|321|
|CCC|CCC|121|
|CCC|CCC|121|
|CCC|CCC|121|
|333|111|111|
|222|222|111|
|111|111|111|




If the four boxesa containing C's are constant then there are 2985984
ways of completing the grid from this stage[/quote]

[Pi]

I Will now work on the middle box

If the top two rows in the middle box are constant

Code: Select all

CCC
CCC
---


Then the numbers which go into the bottom row are pre-decided
The only variable is the order in which they are placed
Each number can go in two boxes as one box will be blocked by that number in the square above and so that is the first variable
2

After that two numbers must be placed in two boxes.
Before that first number was placed each number had two possible locations and so one of the remaining numbers had the filled box as a possible location. This is now ruled out and so there are now no more variables

This means that if the top two rows are filled there are two ways of filling the bottom row.

Code: Select all

|CCC|
|CCC|
|211|


I now need to find out the number of ways of competing the second row if the top row is constant

I believe that there are 9 ways of calculating the middle row but i can't remember why, i used both logic and trial and error to get this

I believe that there are 47 ways of making the top row of the middle box, agfain i used partially trial and error here.

I am not left with this permutations grid

Code: Select all

|CCC|CCC|321|
|CCC|CCC|321|
|CCC|-47|121|
|CCC|-9-|121|
|CCC|211|121|
|333|111|111|
|222|222|111|
|111|111|111|


I now believe that if squares 1,2 an 4 are constant

Code: Select all

|CC-
|C--|
|---|


then there are 2526142464 ways of finishing the grid