The New Sudoku Players' Forum

[Condor]

Previously in another post, (Extra Feature) I looked at sudokus with the extra feature of a digit occuring once on a diagonal.

Just after that, I noticed that some diagonals passed through 5 boxes. They are 7 or 8 cells in length. Here are 2 example templates, 1 of 7 cells in length, the other 8 cells that show this.

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

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

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

It occured to me that it might be possible to make a sudoku where all 8 of those diagonals has 5 occurances of the same digit.

With the help of a small computer program I found 3 different sudokus that have this feature. 2 of the sudokus are very similar, in fact I had to look closely to see the difference.

So, based on that I have produced some puzzles for people to try.

Code: Select all

Puzzle 1
.  .  .    7  .  .    .  .  .
7  .  5    .  .  .    6  .  .
.  .  .    .  3  .    .  .  4

.  1  .    .  .  .    8  .  .
.  .  2    .  9  .    3  .  .
.  .  6    .  .  .    .  2  .

2  .  .    .  4  .    .  .  .
.  .  9    .  .  .    4  .  5
.  .  .    .  .  5    .  .  .


Puzzle 2
.  .  6    .  3  .    .  4  .
.  .  .    6  .  .    3  .  .
.  3  .    .  .  .    .  .  .

.  .  3    .  .  .    .  7  .
.  .  .    .  5  .    .  .  .
.  6  .    .  .  .    8  .  .


.  .  .    .  .  .    .  5  .
.  .  1    .  .  2    .  .  .
.  9  .    .  8  .    1  .  .

Puzzle 3
.  .  1    .  .  .    .  .  .
.  .  .    4  .  9    1  .  .
7  .  .    .  .  .    .  3  .

.  .  .    .  5  .    .  .  .
.  2  .    8  .  3    .  9  .
.  .  .    .  4  .    .  .  .

.  8  .    .  .  .    .  .  1
.  .  7    2  .  5    .  .  .
.  .  .    .  .  .    7  .  .

Hope you enjoy these Sudokus.

[tso]

Excellent -- the puzzles are only solvable using the additional stipulation. Lots of room for experimenting with other stipulations.

[Spoiler removed upoon request]

[Condor]

Glad you liked it.

And thanks tso for removing the spoiler.

[dukuso]

when there are only 3 sudokus with this feature, shouldn't
it be possible to make a puzzle with only 8 clues, but
which has a unique solution ?

[udosuk]

I think I've verified that Condor's results are definitely correct this time using the "simple sudoku" solver. Another very remarkable discovery, it's like a triplet with 2 identical twins and an odd one.

The first 2 grids, after norming, differ by 15 cells only. And you can build them using the following 2 "skeletons":

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1 - 3 - - - 7 - 9
- - - 1 9 7 - 3 -
9 7 - - 3 - - - 1
- 9 - 7 - 3 - 1 -
- 1 7 - - - 3 9 -
- 3 - 9 - 1 - 7 -
7 - 9 - 1 - - - 3
- - - 3 7 9 1 - -
3 - 1 - - - 9 - 7

- 2 - 4 5 - - 8 -
4 - 8 - - - 2 - 5
- - 5 8 - 2 4 - -
- - 4 - 8 - 5 - 2
8 - - 2 - 5 - - 4
5 - 2 - 4 - 8 - -
- 8 - 5 - 4 - 2 -
2 4 - - - - - 5 8
- 5 - - 2 8 - 4 -

Superimposing them will create the 1st grid, and if you mirror the 1st skeleton horizontally and then superimpose you will get the 2nd grid.

The 3rd one is very different to the other 2 in structure, and is 180 degree symmetrical. One of the diagonals is comprised of 3 digits only, and those 3 digits form a very symmetrical skeleton as follows:

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- - 3 - - 6 - - 9
- - - 3 9 - - 6 -
9 - 6 - - - 3 - -
- 9 - - - 3 - - 6
- 3 - - 6 - - 9 -
6 - - 9 - - - 3 -
- - 9 - - - 6 - 3
- 6 - - 3 9 - - -
3 - - 6 - - 9 - -

And the 8-clue puzzle requested by dukuso:

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

Believe it or not, with the additional feature in mind, you can work your way into the unique solution. Enjoy...

[udosuk]

Okay, I intended to make the 8-clue puzzle more elegant by changing it to a "?" sign, but the corrections were lost:

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

And yes, it's solvable logically towards a unique solution with the extra feature that "8 of the short diagonals must each have a digit occuring 5 times". You don't even need to apply X-wing or Colors, etc on it. This must be the puzzle with the fewest possible clues using a general additional property (i.e. one that doesn't relate to any particular digit/letter/symbol such as a word on the long diagonal)...

And I made a mistake in the discrepancy between the first 2 grids. If we norm them such that the 1st row reads "123456789", and the 2nd row has the minimal value, then there are 16 differences. However, if we rotate the 2nd grid 90 deg clockwise and then exchange digits to make the 1st row "123456789", then the number of differences drops to 14! All other rotations and reflections result in a higher number, but I'm still not sure if 14 is the smallest possible number here, because there could be another substitution that works better (and you need a program to check).

I think the major factor lies in the 2 "skeletons": the 1379 one, when you mirror horizontally and then rotate clockwise, and then exchange digits, will result in its original form. So the normed 2nd grid is just the mirrored 1379 of the 1st grid+the same 2458, while the rotated 2nd grid is the same 1379+the rotated 2458, which results in a smaller number of discrepancies...

I mean, gurus like dukuso(Guenter) must have some programs which can verify if 2 grids are essentially identical. But if you modify it a little bit could you find the smallest possible number of discrepancies between 2 different grids... Not that this has any significancy to major projects like the quest for the 16-hinter...

[Condor]

And the 8-clue puzzle requested by dukuso:

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

Believe it or not, with the additional feature in mind, you can work your way into the unique solution. Enjoy...

I like it. You have made a nice discovery yourself.

That must be the sudoku with the fewest clues that has a unique solution with just 1 extra stipulation.

Just seen your other one too. Will have to try it as well.

[udosuk]

Thanks Condor. Here are the things I'd like to create/see:

1. Your "Sudoku Queen Twins" grids reduced to a 7-clue puzzle (and remains logically solvable for those 7 digits)... Couldn't make it by hand, too hard... And maybe impossible as that feature does not fix digits on certain positions like the one you have here...

2. A sudoku grid where you have fewer than 8 diagonal duplications in total... Need a computer program to search...

3. A sudoku grid where no digits are diagonally adjacent, and "knightly" adjacent... I've tried to make grids that each digit cannot have a duplicate in its 5x5 neighbourhood, and it's easy to prove the impossibility... So this is the next possibility, the "Sudoku King-Knight" grid...

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Normal Sudoku:      Sudoku Queen:      Sudoku 5x5:         Sudoku King-Knight:

. . . x . . . . .   x . . x . . x . .   . . . x . . . . .   . . . x . . . . .
. . . x . . . . .   . x . x . x . . .   . x x x x x . . .   . . x x x . . . .
. . . x . . . . .   . . x x x . . . .   . x x x x x . . .   . x x x x x . . .
x x x o x x x x x   x x x o x x x x x   x x x o x x x x x   x x x o x x x x x
. . . x x x . . .   . . x x x x . . .   . x x x x x . . .   . x x x x x . . .
. . . x x x . . .   . x . x x x . . .   . x x x x x . . .   . . x x x x . . .
. . . x . . . . .   x . . x . . x . .   . . . x . . . . .   . . . x . . . . .
. . . x . . . . .   . . . x . . . x .   . . . x . . . . .   . . . x . . . . .
. . . x . . . . .   . . . x . . . . x   . . . x . . . . .   . . . x . . . . .

(x meaning no duplication of o allowed)

And as a footnote, I'm proposing to name your discovery in this thread as the "Sudoku 8-fiver Triples" (very unimaginative)... which ranks highly in my collection of unique or rare elegant suduku grids, aka the "Sudoku Royalties"...

[tso]

That must be the sudoku with the fewest clues that has a unique solution with just 1 extra stipulation.

That would be this one:

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. . . | . . . | . . .
. . . | . . . | . . .
. . . | . . . | . . .
------+-------+------
. . . | . . . | . . .
. . . | . . . | . . .
. . . | . . . | . . .
------+-------+------
. . . | . . . | . . .
. . . | . . . | . . .
. . . | . . . | . . .


Place the digits in this Sudoku such that the resulting 81 digit number formed by stringing the rows one after the other in order from top to bottom is minimum.

[Condor]

I'm proposing to name your discovery in this thread as the "Sudoku 8-fiver Triples" (very unimaginative)

I had not even thought of naming them until someone, dukuso I think, called the first one the sudoku queen. I did think of the 'Pentagonals' for these ones.

btw, after putting just 3 numbers into the second sudoku you gave, I realised it was exaltly the same so did not need to continue to know it had a unique solution.

Ps. It would be nice if people put a little bit more information about themselves. At least city and country/state.

[Condor]

That would be this one:

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. . . | . . . | . . .
. . . | . . . | . . .
. . . | . . . | . . .
------+-------+------
. . . | . . . | . . .
. . . | . . . | . . .
. . . | . . . | . . .
------+-------+------
. . . | . . . | . . .
. . . | . . . | . . .
. . . | . . . | . . .


Place the digits in this Sudoku such that the resulting 81 digit number formed by stringing the rows one after the other in order from top to bottom is minimum.

I was thinking of a grid with clues in it. Did not think of an empty grid. Well done!.

Suggest this as a solution.

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

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

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

I just used my grey-matter computer for that one. Unfortunately it is known to suffer from frequent brown-outs, consumes too much energy, requires too much matainance, and is pre world war two technology.

[udosuk]

tso's idea of the sudoku grid with no clue looks simple at the start, but is in fact very intriguing that I am starting a new thread about it's class...

However those features turn the puzzle into a mathematical one instead of a pure logical one...

"Sudoku Pentagonal Triples" sounds very nice, though it will make people think there are some interesting "pentagons" inside the grid. Well, anything to catch their attention...

You're right my 2nd 8-clue grid is almost the same as the 1st. Just the shape looks much more interesting...

And I can ascertain you've find the correct solution of tso's puzzle.

[dukuso]

pentadiagonal ? or do you mean pandiagonal ?

the sudoku-pieces are not so nice, because their movement depends
on their position.

how about a modular (=pan) sudoku - rook-king or "crowned rook" ?
It can move like a chess-king or a chess-rook and
the board-ends are folded like a torus.

This way any 3*3 subsquare would hold 1,2,3,4,5,6,7,8,9,
not only the 9 aligned squares.

Is it possible ?

[udosuk]

Of course... "pentadiagonal" makes much more sense than "pentagonal". But doesn't it mean "5 diagonals"? In that case "8-fiver" or "8-penta" are more accurate... or perhaps "octopus of quins"? Sorry I'm no expert of nomenclature here...

how about a modular (=pan) sudoku - rook-king or "crowned rook" ?
It can move like a chess-king or a chess-rook and
the board-ends are folded like a torus.

By normal sudoku restrictions the rook movements are automatically prohibited... So "Sudoku Queen" is actually same as "Sudoku Bishop"...
So what you're looking for here is only "Sudoku King"... Which if you don't need the modular/torus property is satisfied by the completed "Sudoku Queen Twins"... I think grids with the torus property exist but I don't know how to search except manually...

This way any 3*3 subsquare would hold 1,2,3,4,5,6,7,8,9,
not only the 9 aligned squares. Is it possible ?

Believe me I've checked that already. Again without the torus property I look for grids with 7x7=49 3x3 squares of 1-9. The following simple proof demonstrates the impossibility:

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. . .  . . .  . . .
. . .  . . .  . . .
. . z  x x o  . . .

. . x  1 2 3  . . .
. . x  4 5 6  . . .
. . o  7 8 9  . . .

. . .  . . .  . . .
. . .  . . .  . . .
. . .  . . .  . . .

One of the 2 top x's must be a 9, as well as one of the 2 left x's... So the 3x3 square with z at top-left must have 2 9s.

[udosuk]

Turns out that some of the puzzles I sought are easily created using dukuso's "Gangster 1" band:

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Sudoku King-Knight ("cylinder" but not "torus")

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

Of course this is also a "Soduku Knight" as a torus, but the "king" property is seriously violated on the last row. This grid was once claimed by dukuso as the "most canonical grid" (1,1,1-42,42,42) but was quickly dethroned by his (1,1,1-1,1,1) grid.

The following is produced by rearranging rows on the (1,1,1-1,1,1) grid:

Code: Select all

Sudoku King (torus)

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

When we exchange the 2nd & 3rd bands of the (1,1,1-1,1,1) grid:

Code: Select all

Sudoku 3-step Bishop (torus)

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

Which is such named because no digit is duplicated within 3 steps diagonally. The "king" property is automatically implied of course, but as a compensation the digits are very heavily knight-connected...
There is a total of 16 diagonal duplications, a double of the Queens' 8, but still a low figure compare to average.

So the question remained is to find the torus version of the "Sudoku King-Knight", which may be quite rare... [edited: just used a simple program to verify that the torus Sudoku King-Knight doesn't exist... but will try to find a 7 or 8 digits version like the Queens.]

Although I haven't really think about it, I anticipate that the normal "Sudoku King-Knight" and "Sudoku 3-step Bishop" would not be rare at all... which I need programming experts like dukuso or Condor to verify...