The New Sudoku Players' Forum

[Condor]

Having tried my hand at sudoku, the thought occurred to me, I wonder if it is possible to make a sudoku with the additional feature that each digit occurs only once on a diagonal.

With the help of a small computer program I found that it is not possible with all 9 digits. It is however possible to make a sudoku in which for 7 of the 9 digits each occurrence of that digit occurs only once on the 2 diagonals on which it occurs.

There is only 1 unique sudoku with this feature.

So, here is a puzzle in which for 7 of the digits, each occurrence of that digit is also the only occurrence of that digit on the 2 diagonals it occurs.

.5. ..1 .2.
9.. ..3 6..
.76 2.. .53

5.. 1.. .67
... 3.7 ...
38. ..5 ..1

71. .2. 83.
..8 7.. ..9
.4. 5.. .1.

Addenum
After posting this it was discovered by udosuk that the additional feature applies to the 7 digits only. The 2 remaining digits have two ways that they can be placed in the grid. see udosuk below. (after you have solved the puzzle)

[simes]

Have you seen this thread? http://forum.enjoysudoku.com/viewtopic.php?t=595

[Animator]

And this one: http://forum.enjoysudoku.com/viewtopic.php?t=332

Third time this question is asked...

But what exactly do you want? A starting grid that you solve with all traditional rules and results in the number 1-9 on the diagonals? or a grid in which you need to use the knowledge of the diagonals to solve it? (As in, if you have only one empty cell on the diagonal you can fill it in)

I think it is possible to create both of them (a starting grid that is)... but you might need more clues...

[ramnath]

Having tried my hand at sudoku, the thought occurred to me, I wonder if it is possible to make a sudoku with the additional feature that each digit occurs only once on a diagonal.

With the help of a small computer program I found that it is not possible with all 9 digits. It is however possible to make a sudoku in which for 7 of the 9 digits each occurrence of that digit occurs only once on the 2 diagonals on which it occurs.

There is only 1 unique sudoku with this feature.

So, here is a puzzle in which for 7 of the digits, each occurrence of that digit is also the only occurrence of that digit on the 2 diagonals it occurs.

.5. ..1 .2.
9.. ..3 6..
.76 2.. .53

5.. 1.. .67
... 3.7 ...
38. ..5 ..1

71. .2. 83.
..8 7.. ..9
.4. 5.. .1.

I THINK U R 'RONG .THAT CAN'T BE THE CASE

[Condor]

Thanks simes & Animator

That is a good question but a different one. It is concerned with having all 9 digits on both the long diagonals. I was thinking of something more in line with the 8 queens problem on a chessboard. (Put 8 queens on a chess board so that no queen is on the same row, column or diagonal as another queen)

Consider the following where all the asterisks (*) in the diagram represents each occurence of that digit.

Code: Select all

*  .  .  .  .  .  .  .  .
.  .  .  .  .  .  *  .  .
.  .  .  *  .  .  .  .  .
.  .  .  .  .  *  .  .  .
.  .  .  .  .  .  .  .  *
.  *  .  .  .  .  .  .  .
.  .  .  .  *  .  .  .  .
.  .  *  .  .  .  .  .  .
.  .  .  .  .  .  .  *  .
-------------------------
.  .  .  .  .  .  .  *  .
*  .  .  .  .  .  .  .  .
.  .  .  *  .  .  .  .  .
.  .  .  .  .  .  *  .  .
.  .  *  .  .  .  .  .  .
.  .  .  .  .  *  .  .  .
.  .  .  .  .  .  .  .  *
.  *  .  .  .  .  .  .  .
.  .  .  .  *  .  .  .  .
-------------------------
.  .  .  .  .  .  *  .  .
*  .  .  .  .  .  .  .  .
.  .  .  .  .  *  .  .  .
.  *  .  .  .  .  .  .  .
.  .  .  .  *  .  .  .  .
.  .  .  .  .  .  .  *  .
.  .  .  *  .  .  .  .  .
.  .  .  .  .  .  .  .  *
.  .  *  .  .  .  .  .  .


In each case the position of the asterick not only shows where all the digits could go according to the rules, but also no asterick is on the same diagonal as any other.

The sudoku puzzle I gave had 7 digits that conformed to this extra feature. That is the maximum number of digits that will and it is the only example of one.

Ps. I was not asking a question. I stated a fact that there is only 1 unique sudoku with 7 digits that occur only once on any diagonal, then I was trying to produce a puzzle based on it.

[Hammerite]

I think it is possible to create both of them (a starting grid that is)... but you might need more clues...

Surely fewer?

[Condor]

The post above is edited. This was supposed to be deleted

[Animator]

I think it is possible to create both of them (a starting grid that is)... but you might need more clues...

Surely fewer?

I guess that it depends...

If you require the person that is solving the grid to look at the diagnoals then fewer clues could be a possibility.

If you want to have a solution with 1 - 9 on the diagnols without ever looking at them then I guess you need more clues.

[Condor]

I THINK U R 'RONG.

Thanks

How am I supposed to reply to that? You have not given me anything to consider. At least you should have said why you do not think I am correct.

If you think I am wrong then please give a counter-example. All you would have to do is produce 1 example where for all 9 digits, each occurrence is the only 1 on the 2 diagonals it occurs. You could even produce an example with 8 digits. Or an example with 7 digits that cannot be transformed by rotating, reflecting, or renumbering to the one given.

Anyway. I will explain how I came up with 1 unique sudoku.

I started by having my computer make a list of all arrangements that meet the rules of sudoku, plus the extra feature of no 2 occurrence of a digit on the same diagonal. Here is a example:

Code: Select all

  .  .  .  .  .  .  .  *  .
  *  .  .  .  .  .  .  .  .
  .  .  .  *  .  .  .  .  .
  .  .  .  .  .  .  *  .  .
  .  .  *  .  .  .  .  .  .
  .  .  .  .  .  *  .  .  .
  .  .  .  .  .  .  .  .  *
  .  *  .  .  .  .  .  .  .
  .  .  .  .  *  .  .  .  .

There was a total of 144 such arrangements, counting rotations and reflections.

Then I had the computer search for 9 of those arrangements such that every asterisk was on a different square. This was not successful. I did notice that up to 7 was possible. So I changed my program a bit to give me all sets of 7. Then I grouped the sets so that each member of the set was a rotation and/or reflection of one another. There was 4 groups. Taking 1 representative of each group I attempted to insert the last 2 digits. It was possible for only 1 of the sudokus. So that is how I came up with that result.

[Hammerite]

I must admit that I'm not sure I've understood what you posted correctly.

If you meant that you believe that it's not possible to produce a completed Su Doku grid that has the additional property of each of the 9 symbols appearing precisely once in each of the 2 diagonals... well, then you ARE wrong. You can find a counterexample in the second post in the topic simes links to, in the second post in this topic. If that isn't what you meant, I'm sorry - I'm not sure what you mean.

EDIT: I wrote "columns" instead of "diagonals" in the main paragraph. Whoops.

[johnw]

If I may butt in, I think you need to define what is meant by diagonals.

There are two "main diagonals" with 9 cells each, every other diagonal that you would find on the grid is shorter than 9.

You could partition the grid into 9 sets of 9 if you include sets like this (below), but it would not make a puzzle that a human player could get to grips with.

Code: Select all

. . . . . * . . .
. . . . * . . . .
. . . * . . . . .
. . * . . . . . .
. * . . . . . . .
* . . . . . . . .
. . . . . . . . *
. . . . . . . * .
. . . . . . * . .


[Hammerite]

That's a good point, you could come up with extra "diagonals" like that. (But how do you know a human wouldn't be able to get to grips with it?)

[Condor]

I must admit that I'm not sure I've understood what you posted correctly.

If you meant that you believe that it's not possible to produce a completed Su Doku grid that has the additional property of each of the 9 symbols appearing precisely once in each of the 2 diagonals... well, then you ARE wrong. You can find a counterexample in the second post in the topic simes links to, in the second post in this topic. If that isn't what you meant, I'm sorry - I'm not sure what you mean.

EDIT: I wrote "columns" instead of "diagonals" in the main paragraph. Whoops.

You have not understood my post correctly. Simes's link is concerned only with the long diagonals.

To show what I mean, consider the example that I was trying to show. It could represent any of the digits. I will show it again where it represents the 5 digit

Code: Select all

  .  .  .  .  .  .  .  5  .
  5  .  .  .  .  .  .  .  .
  .  .  .  5  .  .  .  .  .
  .  .  .  .  .  .  5  .  .
  .  .  5  .  .  .  .  .  .
  .  .  .  .  .  5  .  .  .
  .  .  .  .  .  .  .  .  5
  .  5  .  .  .  .  .  .  .
  .  .  .  .  5  .  .  .  .

If you check any 5 in the diagram above, you will not find any 5 on the same diagonal as any other 5.

Please note that the diagonals I am refering to can be from 1 square in length (short diagonal) up to the long diagonals (9 squares in length).

[Hammerite]

Aah, ok, I get what you mean now.

[Condor]

Aah, ok, I get what you mean now.

I glad about that. I was starting to rack my brains as to how to explain it better.

Anyway, we can get onto the more important part which is - Has anybody been able to verify my result.