## Sudoku maths - can mortals work it out for the 2x2 square ?

Everything about Sudoku that doesn't fit in one of the other sections

### Sudoku maths - can mortals work it out for the 2x2 square ?

Mathmatic challange for mortals [see the on going 3x3]

Given a 2x2 square

there are 4 boxes [ignore the centrall box !]

1234
2341
4123
3412

What are the total number of possibilities ?

I calculated up to 48 verified grids ! I may well be wrong on this !!!!
coloin

Posts: 1737
Joined: 05 May 2005

Not to be difficult, but your example is not a valid 2x2
e.g
12
23
is not a valid upper left box.

Number of possibilities:
Upper left (UL) box: 4! = 24
UR box given each UL: 2x2 = 4
Lower Left (LL) box given each UL = 4
LR box given LL,UR = Either 0 or 1

Therefore there are >24*4*4 candidates, or >374.

Sample where LR is unfillable:

12 43
34 12

23 ??
41 ??

where two boxes in LL cannot be filled by any number.

If you look at the 16 possibilities for UL and LR, given each arrrangement of UL (of which there are 24), it turns out that 4 of them result in an unfillable LR. Therefore there are 12 that are good, for a total number of valid answers of 12*24=288.

{That's a MUCH easier puzzle to do than the 3x3.}
Guest

### Directory Enquires

Maybe mere mortals can !!!!

I suggested the number 192 some time ago on the programmers forum

I think I "number crunched" it on a sheet of A4 and counted the solutions - I was sure it was right at the time.
coloin

Posts: 1737
Joined: 05 May 2005

### Re: Directory Enquires

coloin wrote:Maybe mere mortals can !!!!

I suggested the number 192 some time ago on the programmers forum

I think I "number crunched" it on a sheet of A4 and counted the solutions - I was sure it was right at the time.

I thought it was 192 - see my post of 5 June within this thread on the programmer's forum.
tannedblondbloke

Posts: 16
Joined: 09 April 2005

The correct number is 288...

You can easily confirm this number by brute forcing it...
Animator

Posts: 469
Joined: 08 April 2005

Are rotations and reflections counted as seperate?
e.g
1234
2341
3412
4123
Is this the same as:-
4321
1432
2143
3214
Guest

Posts: 312
Joined: 25 November 2005

### A counting method

A neat way of counting.

Consider grids of the type

AB | xx
Cx | xx
--------
xx | Dx
xx | x E

where A,B,C are different, D and E are different. There are 288 of these and each gives a unique solution. Therefore 288 solutions.
geoff

Posts: 5
Joined: 07 June 2005

Good work
Ekuuleus

Posts: 1
Joined: 29 November 2005

There are only two essentially different grids. Proof is here.
Nick70

Posts: 156
Joined: 16 June 2005