## Proposed variant puzzle

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

### Proposed variant puzzle

I find the standard rules mathematically inelegant. I propose a variant with an additional unit (containing one of each digit). That unit consists of the cell in the corresponding position in each block. For instance, the top left cell in each 3x3 block has to contain a different digit.
This is mathematically more complete and makes the puzzle more challenging. I have composed several such puzzles by hand. If there's interest, I can post a sample.
cjlt

Posts: 5
Joined: 10 March 2005

Surely this would in fact make it MUCH easier to solve?

I think it's fine as it is...
Guest

The standard Su Doku is a fine puzzle, but I feel the extra dimension I'm proposing makes it even more elegant. Rather like a magic square that can be doubly magic.
The extra rule makes it harder, it seems to me, because you have to look in four directions, not just three. Composing puzzles is also harder. Here, to give you a feel, is one I composed today, based on today's Times Su Doku:
7 - - - 5 - - - 6
- 3 - - - 7 5 - -
- - - - 8 - - 9 -

9 - 7 - - - - - 4
- - - 6 - 8 - - -
6 - - - - - 2 - 8

- 6 - - 7 - - - -
- - 5 4 - - - 2 -
4 - - - 3 - - - 5

It has, of course, only one solution. It requires no trial and error.
cjlt

Posts: 5
Joined: 10 March 2005

I agree with other guest that this SIMPLIFIES the solving, as the solution is MORE restrictive.

Now, if eight of the upper left values are known, I do not automatically know the last one. Your proposal makes this available to me. The number of possibilities is dramatically reduced.
Guest

Well, you may be right, but because there are more restrictions, the puzzle can be posed with fewer given values. As in the example I posted here. Can you solve it? If you can solve it as easily as the ones in The Times, then I'll admit it is no harder, but I'd still claim it is more elegant, mathematically.
cjlt

Posts: 5
Joined: 10 March 2005

There are at least two solutions to this puzzle - see below. I suspect there are many more as I found these very easily...

7 2 9 1 5 4 3 8 6
8 3 6 9 2 7 5 4 1
5 1 4 3 8 6 7 9 2
9 8 7 2 1 3 6 5 4
2 5 1 6 4 8 9 7 3
6 4 3 7 9 5 2 1 8
1 6 8 5 7 2 4 3 9
3 9 5 4 6 1 8 2 7
4 7 2 8 3 9 1 6 5

7 2 9 1 5 4 3 8 6
8 3 6 2 9 7 5 4 1
1 5 4 3 8 6 7 9 2
9 8 7 5 2 3 6 1 4
5 4 2 6 1 8 9 7 3
6 1 3 7 4 9 2 5 8
2 6 1 8 7 5 4 3 9
3 9 5 4 6 1 8 2 7
4 7 8 9 3 2 1 6 5
Guest

Sorry, perhaps I didn't explain my new rule well enough. Those two 'solutions' are both invalid. My extra rule is that, within each 3x3 block in the puzzle, each of the nine positions must contain a digit that does not appear at that position within any of the other 3x3 blocks.
So, for instance in the first proposed solution above, both the top left and the bottom right 3x3 blocks have a 9 in the top-right cell. That is not allowed under my new rule.
Top left 3x3 block:
7 2 9
8 3 6
5 1 4

Bottom right 3x3 block:
4 3 9
8 2 7
1 6 5

These both have '8' and '9' in the same positions within the block.
You must find a solution in which this does not happen. The nine '8's must all be in different positions within their 3x3 blocks. Same for all other digits, of course.

Best wishes
Chris
Guest

Posts: 312
Joined: 25 November 2005

No - you explained it fine. I just didn't read it properly! I got this solution -

7 9 8 1 5 2 4 3 6
1 3 4 9 6 7 5 8 2
5 2 6 3 8 4 1 9 7
9 8 7 5 2 3 6 1 4
2 5 1 6 4 8 3 7 9
6 4 3 7 1 9 2 5 8
3 6 9 2 7 5 8 4 1
8 1 5 4 9 6 7 2 3
4 7 2 8 3 1 9 6 5

I was expecting it to be easier, because there are 4 units (1 row, 1 col, 1 box, and 1 special) placing restrictions on every cell, rather than the normal 3, so you ought to be able to eliminate possibilities much quicker. In practice, it was more interesting than that - I quite enjoyed it. I worked with the normal rules until hitting a dead end, then checked against your cross-box rule, which is quite tricky visually, so a nice challenge. I probably had to use the additional rule 9 or 10 times.

It was certainly harder, and an interesting step up from the Fiendish level. Probably even harder than the "Very Hard" pappocom puzzles. Get's my vote.
Guest

### Re: Proposed variant puzzle

cjlt wrote:I find the standard rules mathematically inelegant. I propose a variant with an additional unit (containing one of each digit). That unit consists of the cell in the corresponding position in each block. For instance, the top left cell in each 3x3 block has to contain a different digit.
This is mathematically more complete and makes the puzzle more challenging. I have composed several such puzzles by hand. If there's interest, I can post a sample.

This is not new -- this is a VERY common variation in most Japanese Number Place magazines. They are not intrinsicly more or less difficult to solve or create, but I think they are a little more fun.

See here:
http://www.setbb.com/phpbb/viewtopic.php?t=57&mforum=sudoku
tso

Posts: 798
Joined: 22 June 2005

Has anyone done more analysis of this variation? It's far more elegant, as it involves square slices through a 3*3*3*3 hypercube in all four dimensions. Because of that additional symmetry, I'd imagine it would be more amenable to nice mathematical results, such as the minimum number of clues required.
Kropotkin

Posts: 2
Joined: 15 October 2005

As many of these you want, plus variarions:

http://www.menneske.no/sudoku/dg/3/eng/
tso

Posts: 798
Joined: 22 June 2005

there is also the variant with 6 constraints:

rows
columns
blocks
positions in a block
3 equidistant minicolumns in a band
3 equidistant minirows in a stack

this one has nice 4 dimensional symmetries as a tesseract.

But maybe it's too much constraint to make nice puzzles here.
dukuso

Posts: 479
Joined: 25 June 2005

Yes, that's the one I was thinking of! I got my maths wrong - there are six directions for getting square slices through a tesseract. But as you say, there may be too many constraints.
Kropotkin

Posts: 2
Joined: 15 October 2005

100 minimal such puzzles with 6 constraints are uploaded to
http://magictour.free.fr/sudoku6

clues-statistics is:
8 582
9 388
10 29
12 1

even the easy ones are not so easy, because it's tedious to search
all the constraints for forced placements !
You probably need some practice with these.

-Guenter
dukuso

Posts: 479
Joined: 25 June 2005