The New Sudoku Players' Forum

[tarek]

Well, then, how about a puzzle which has multiple solutions in each of the following nine modes:
vanilla anti-Knight
cylinder anti-Knight, vertical
cylinder anti-Knight, horizontal
Mobius strip anti-Knight, vertical
Mobius strip anti-Knight, horizontal
torus anti-Knight
projective plane anti-Knight
Klein bottle anti-Knight, Mobius vertical
Klein bottle anti-Knight, Mobius horizontal

-- and, in fact, multiple solutions in any combination of up to eight of the above -- but only one solution that works for all nine.

I said Programmable, I'm not sure that it would give results though. The best chance is to limit it to chess pieces that leap to a maximum of 4 cells i.e. (0,y) or (y,y) ... The ideal startup piece candidate is the (2,2) leaper "Alfil" which I thibnk should be comatible with the Projective plane ....

tarek

[tarek]

The Root50 leaper can mange any (1,7) or (5,5) leap.

Look at the reach of a root50 piece sitting on r1c1 of a sudoku toroid.



Here is the reach of the same piece when on r5c5



Here is an EASY (if you can manage the dizziness) example of an Anti-Root50 Toridal sudoku

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

Solution: Show

[Smythe Dakota]

.... Look at the reach of a root50 piece sitting on r1c1 of a sudoku toroid. ....

.... Here is the reach of the same piece when on r5c5 ....

But in this case it's really the same thing, isn't it? Unlike a projective plane, on a torus any cell is the same as any other.

Whew! I don't know when I'll have time to tackle this one. I'll probably get dizzy.

Bill Smythe

[Smythe Dakota]

I haven't finished yet, but I have quickly discovered that this Root50 piece isn't all it's cracked up to be.

On a torus, the [7,1] part is nothing more than a regular [2,1] knight. 7 cells west is the same thing as 2 cells east.

And the four [5,5] target cells form a nice tight little 2x2 rectangle, the nearest corner of which you can get to by moving four squares diagonally in any direction.

I guess that helps to visualize, at least. It's still not all that easy, though -- at least not with my rather amateurish methods.

Bill Smythe

[Smythe Dakota]

OK, I got it.

It was handy that [7,1] is the same as [2,1], and that [4,4], [4,5],[5,4],[5,5] are all the same. Ah, those good old 9-x equivalences.

Bill Smythe

[tarek]

The toroidal 9x9 sudoku is definitely easy to absorb after practice because your orientation can be visualized easier than with a reverse continuity.

I almost forgot. If the added constraints make it difficult to achieve a 9x9 Anti-kNight sudoku, a better result might be achieved with 16x16 version

Results somtime this week

tarek

[tarek]

As promised, here are the results of the non-extensive search ....

Alfil (The Elephant) is a historic but real chess piece and was the modern Bishop's predecessor. It leaps 2 squares diagonally in any direction.

For an Anti-Alfil sudoku, here is a demo of the grid & the rech of a piece sitting on r5c5



I chose this piece as I think it is the easiest piece compatible with the planes of continuity that have been mentioned in the previous posts.

The following puzzle can be solved as an Anti-Alfil sudoku on a regular sudoku (No continuity) borad but will be difficult. Think of using any continuity plane that Bill mentioned earlier and it will get easier. The difficulty will be reduced significantly (to easy ) if you think of all of them at once (Which is not that easy )

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1.3...9.4.9.....6.8.......3...1.4...............2.5...3.......6.8.....7.2.7...3.5

Solution: Show

The next, however, can't be solved without applying almost all of the continuity planes mentioned before ... I found this tough & had to use some T&E unfortunately

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..........1.7.2.9...........8.....4...........3.....1...........2.9.5.8..........

Solution: Show

I managed to get some Anti-kNight puzzles but they were on a 16x16 board and difficult

tarek

[Smythe Dakota]

I'm not sure if I understand the rules:

.... The following puzzle can be solved as an Anti-Alfil sudoku on a regular sudoku (No continuity) board but will be difficult. Think of using any continuity plane that Bill mentioned earlier and it will get easier. The difficulty will be reduced significantly (to easy ) if you think of all of them at once (Which is not that easy ) ....

This seems to mean that the puzzle is valid (has one and only one solution) on a no-continuity board, but is still valid (one and only one solution) on a pan-continuity board (torus combined with projective plane). Is that correct? If so, that really seems like two separate puzzles to me.

By the way, torus plus projective plane seems to cover all bases -- Klein bottles (both NS and EW), Mobius strips (ditto), and cylinders (ditto).

.... The next, however, can't be solved without applying almost all of the continuity planes mentioned before ....

In other words, the second one is a valid puzzle only if you invoke both torus and projective plane?

Incidentally, what is that peculiar piece you're using to represent the Alfil? It looks like a grotesquely deformed knight crossed with a water pitcher used to water flowers. I hope it doesn't keep staring at me as I attempt to solve the puzzles.

I think we're both going insane, and nobody else is listening.

Bill Smythe

[tarek]

I'm not sure if I understand the rules

The 1st puzzle can be solved as a regular Anti_alfil with no continuity assumed but will be very hard.

The Solution grid is however compatible with all other combinations. So essentially this bit of information will help you make the puzzle easier ... only the full range will allow you to make it very easy.

By the way, torus plus projective plane seems to cover all bases -- Klein bottles (both NS and EW), Mobius strips (ditto), and cylinders (ditto).

Not really ... The main difference is when you need to cross 2 grids in 1 move.

If the piece has a range of <=3 then for the projective plane jump will land in the same box ... Only with pieces of range > 3 will show the difference The duality of the klein bottle has more cells exposed than the Projective plane because the cover different cells if a piece can make jump over 2 grids ... This mostly happens closer to the corners.

I'm not sure if I understand the rules:

Incidentally, what is that peculiar piece you're using to represent the Alfil? It looks like a grotesquely deformed knight crossed with a water pitcher used to water flowers. I hope it doesn't keep staring at me as I attempt to solve the puzzles.

It was a botched work after a sleepless night ... it should be an elephant

tarek

[Smythe Dakota]

By the way, torus plus projective plane seems to cover all bases -- Klein bottles (both NS and EW), Mobius strips (ditto), and cylinders (ditto).

Not really ... The main difference is when you need to cross 2 grids in 1 move. ....

What I meant was, if any proposed solution works with both the torus and the projective plane, then it also will work with all versions of Mobius strips, cylinders, and Klein bottles.

Bill Smythe

[tarek]

By the way, torus plus projective plane seems to cover all bases -- Klein bottles (both NS and EW), Mobius strips (ditto), and cylinders (ditto).

Not really ... The main difference is when you need to cross 2 grids in 1 move. ....

What I meant was, if any proposed solution works with both the torus and the projective plane, then it also will work with all versions of Mobius strips, cylinders, and Klein bottles.

I'm trying to tell you that they're not. Remember that a chess piece that decides to leap in the projective plan can't change it's mind mid flight and choose the toroidal plane. Simply look at the reach of Alfil at r1c1 if it was on a toroid and then it's reach when on the projective plane. Compare to a jump done in a Klein bottle plane. There will be different cells on both sides.

[Smythe Dakota]

But if you can make a given jump on a Mobius strip, you can always make that same jump on a projective plane (but not conversely, in general).

Bill Smythe

[tarek]

But if you can make a given jump on a Mobius strip, you can always make that same jump on a projective plane (but not conversely, in general).

Yes

Returning to the differences between these continuity planes, here is how it looks for this Projective Plain Grid ... The Centre grid (Grid 5) is your starting Grid



A leap that would take a fairy chess piece to Grid 2 (Top Middle) will be equivalent to a vertical mobius strip.

If a chess piece however can leap into grid 1 from grid 5 ... then as you said it wouldn't be possible to replicate in the Mobius strip, Klein bottle, Cylinder Or toroidal plains.

Because the chess piece will leap in a certain continuity plain then the projective plain grid is not equivalent to a vertical mobius grid combined with a horizontal mobius grid.

Grids 1,3,7,9 are different if the grid is assumed Toroidal, Klein bottle (V or H) or Projective plain.

When I say that the grid can be solved in any combination that means that you can solve it assuming any continuity plain from the list you mentiond.

Horizontal Cylinder
Vertical Cylinder
Horizontal Mobius
Vertical Mobius
Klein Bottle (Horizontal Mobius Vertical Cylinder)
Klein Bottle (Vertical Mobius Horizontal Cylinder)
Toroidal
Projective

Now,

Depending on the chess pieces ... The destination cells possible might be similar in some combinations:

Looking at the same grid ... If the chess piece has a reach of less than 4 squares then the only way it can cross into grids 1,3,7, or 9 from grid 5 in the projective plain is when it starts from the corner boxes ... It will then land in the same box that it started & therefore it will not have any more cells to cover than a Dual Klein Bottle (V & H) grid. With the dual Klein bottle grid the chess piece in this case will have more destinations possible. In this case if the grid is solved assuming Projective plain, it will be solved assuming Klein bottles in both directions.

Now I really think we are insane & that there is nobody listening

[Smythe Dakota]

.... Now I really think we are insane & that there is nobody listening

I think my ability to listen (or understand) is rapidly waning. But I guess I started it, didn't I?

I just had a horrible thought. The entire concepts of projective plane, Mobius strip, and Klein bottle don't work at all with Sudoku -- at least, not without a modification of the rules. Only cylinders and the torus still work.

With a vertically wrapped Mobius strip, for example, column 2 continues into column 8. This column now has 18 cells. So there will always be duplicate digits in columns.

The rules must state that the wraps apply only to the leapers, not to the straight rows or columns. I suspect both of us were subconsciouly invoking that rule without realizing it. But it just dawned on me that we were doing it.

Bill Smythe

[tarek]

The rules must state that the wraps apply only to the leapers, not to the straight rows or columns. I suspect both of us were subconsciouly invoking that rule without realizing it. But it just dawned on me that we were doing it.

I guess so, no harm in making the the description clearer from now on ... BTW, my head now started to hurt from twisting my neck in the mobius planes I'll post one more in this series before giving it a rest.