How many possible 2x2 soduku's?

fractal · Posted: Mon Apr 24, 2006 6:09 pm Post subject: How many possible 2x2 soduku's?

How many possible 2x2 soduku's (4x4 grid)?

Hydro · Posted: Mon Apr 24, 2006 9:50 pm Post subject:

I would probably say 2^(4!), or basically- 16,777,216 - I'm probably wrong, but this seems quite reasonable.

nero · Posted: Tue Apr 25, 2006 12:11 am Post subject:

Hydro · Posted: Tue Apr 25, 2006 4:23 am Post subject:

frazer · Posted: Tue Apr 25, 2006 7:33 am Post subject:

Actually, the answer was worked out elsewhere on this forum - there are 288 2x2 sudokus.

fractal · Posted: Tue Apr 25, 2006 8:06 am Post subject:

Let me try
Row1 : 4x3x2x1
Row2: 3x2x1x1
Row3: 2x1x1x1
Row4: 1x1x1x1
So 24x6x2 = 288. Correct?

nero · Posted: Tue Apr 25, 2006 9:57 pm Post subject:

frazer · Posted: Wed Apr 26, 2006 2:30 pm Post subject:

nero's most recent answer misses 4 possibilities, making 24 Latin squares with top row 1234. There is a ninth possibility for the second row, namely, 3142, which then has two more completions to a full grid. Further, the second row 3412 should have four completions, not two. In total, we get an extra four Latin squares, giving 4!*24=576, and exactly half of these are 2x2 sudoku, giving 288.

The number of essentially different ones, where we treat as the same those coming from rotations, reflections, relabelling and row/column swaps is just 2.

As for 3x3 sudokus, these were indeed enumerated by computer. Indeed, sufficiently many people have done this independently (and agreed with the answer that we first got!) that this number seems certain. The approach is quite mathematical. We listed all possibilities for the top three rows (nearly 10^12), and split them into equivalence classes, two possibilities being equivalent if we could prove that they had the same number of ways to complete to a full grid. Computer work showed that there were just 44 equivalence classes (all of different size); for each, we counted (by computer) the possible ways to complete the grid. Multiplying everything together and adding gives the total. Subsequent methods improved our original calculation by counting the number of ways using further simplifications to count the number of ways to complete these to a full grid, along the lines of our method for the top three rows. These methods have been pushed much further, to count sudoku grids of size 2x3, 2x4, 2x5 and 3x4, although the latter two have not been independently counted. The number of essentially different grids in the 2x3 and 2x4 cases are also known.

I doubt that you could get the 3x3 number without a computer, but that's something you should try to convince yourself!

See http://en.wikipedia.org/wiki/Mathematics_of_Sudoku for more.

nero · Posted: Wed Apr 26, 2006 3:24 pm Post subject:

serge asselman · Posted: Fri Jul 14, 2006 9:06 am Post subject: How many 2x2 sudoku's?

1234 2134 3124 4123
1243 2143 3142 4132

1324 2314 3234 4213
1342 2341 3243 4231

1423 2413 3423 4312
1432 2431 3432 4321

# = 4! = 24

I have made this more visually @ :

http://groups.msn.com/overallesenniks/sudoku.msnw?action=ShowPhoto&PhotoID=275

with these combinations i try to make some 2x2 sudoku's @ :

http://groups.msn.com/overallesenniks/sudoku.msnw?action=ShowPhoto&PhotoID=276

If anyone shares the same interest, you are very welcome to
post me some 2 x 2 solutions to my msn group.

Serge Asselman
student elektronica

tpgames · Posted: Wed Mar 07, 2007 5:54 pm Post subject:

I found 40 possible 2x2 sudoku solutions, but I can not come up with the other 248 possibilities. googling it did not come up with a list of all the possibilities either. Yes, I'm doing this by hand, as math at this level is a little beyond me. I flunked P^... in college.

Any one got a link to a list of all 288 possibilites? Thanks!
(no one jumped or commented when I asked for someone to write a JavaScript 2x2 sudoku program using images for my 4 yr. old nephew, so I'm trying to have 288 pages, all with a different sudoku game on it - at least until I figure out Javascript.) Laughing

Thanks!

Pat · Posted: Wed Mar 14, 2007 11:26 am Post subject: re: 2x2 (24x12=288)

tpgames · Posted: Thu Mar 15, 2007 3:07 pm Post subject:

I read that post earlier, and tried to come up with all 288.

However, I came up with a problem with my idea of making 288 different pages...I don't understand JavaScript well enough to write the code to allow people to choose an image and put it in the box. No matter how, I thought it out, I'd still have that little issue. I'm using code from both a 3x3 and a 2x3 sudoku game to try and learn the code. So far, I've not been able to convert either into a 2x2 sudoku game. I'm going to look at this further after April. I've got too much stuff to do on my site before I can tackle this issue. Rolling Eyes

Thanks though! (I'll reread that entire thread too after April)

mathrec

The number of filled 2x2 solution grids is 288, and there are only two meaningfully distinct filled grids (distinct under operations of swapping rows or columns of boxes, swapping individual rows or columns, or permuting the number values).

But I'm trying to figure out the number of distinct 2x2 sudoku puzzles. For example, I found the following valid puzzles with the minimum four clues doing a search by hand (and no others that are meaningfully distinct, although this was only a one-time search by hand):

JPF · Posted: Sat Mar 29, 2008 1:19 pm Post subject:

Properties of 2x2 puzzles have probably already been posted in the players forum, but I can't remember where exactly.

Edit 1: see here
Edit 2 --------------
There are 288 grids ; only 2 are essentially different.

192 are isomorphic to :