Trying to make a knight's-move puzzle with minimal rules/hints

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Firlz
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Trying to make a knight's-move puzzle with minimal rules/hints

Post by Firlz » Sat 05 Dec, 2020 8:49 am

After seeing some of the Cracking The Cryptid videos that involve knight's move restrictions I have been mulling over how to try and use just normal sudoku and knight's move rules to drastically reduce the minimum number of required hints.

So far this has been my best attempt at getting a single knight's move rule and managing what I believe is a unique solution with just 8 hints. I have not yet figured out a method with just a single rule regarding knight's moves that permits fewer hints.

Any advice regarding how to more clearly word the rule, or whether I messed up and there is a second possible solution would be appreciated!

Alternatively any suggestions for an alternative knight's rule that would reduce the required clue count would be appreciated!
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Forward Slashing Knights.pdf
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detuned
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Re: Trying to make a knight's-move puzzle with minimal rules/hints

Post by detuned » Sat 05 Dec, 2020 10:46 am

I’m not sure I understand the constraint properly (perhaps you could draw a picture to demonstrate what you mean?) - but I would add that unless the constraint gives some information about digits relative to one another (like consecutive) then 8 will always be the lower bound for clues.

To see this, suppose you had a puzzle with 7 clues, which for the sake of argument were 1 each of the numbers 1-7. How are you going to get the 18 cells needing to contain either an 8 or a 9 to resolve? The answer is that you can’t.

Firlz
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Re: Trying to make a knight's-move puzzle with minimal rules/hints

Post by Firlz » Sat 05 Dec, 2020 6:36 pm

The constraint works in the following way (I don't know how to draw images directly in here, so please reference attached image).

Yellow is the square we want to check the constraints on.

Because we are only constrained by knights-moves in a down-right direction, the blue highlighted squares are unconstrained with relation to the yellow square.

Because the constraint only applies to cases where you cross exactly one boundary, the orange and green squares are unconstrained. Green is not constrained because it crosses two boundaries (no matter how you make your knight's move, you either go through the box on the left and THEN cross into the upper left box, or you go up into the middle-top box and then left into the upper-left box). Orange is not constrained because it crosses no boundaries (it's in the same box).

The red and purple squares must be the same as yellow. The red square is a knight's move in the relevant direction (right right down/down right right) so it's the same via the rule.

Purple must be the same by corollary. Since it's relationship from yellow is up up left/left up up, this means when looking at purple you find that yellow is down and to the right (right right down/ down right right) and thus the rule tells us yellow must match purple. Since yellow must match purple, purple must also match yellow. (the rule works implicitly in reverse- the formal constraint is that you only care down and to the right, but the corollary implied restraint actually also forces you up and left, because those are in fact the same thing).


re lower bound: Thanks, that is good to know. I really struggled for a good half-hour trying to figure out how to get down to seven and I just couldn't do it no matter how I tried. It is good to know it wasn't me missing something clever and the fact that the remaining digits were transposable is actually a known thing.

Let me know if this helps!

Firlz
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Re: Trying to make a knight's-move puzzle with minimal rules/hints

Post by Firlz » Sat 05 Dec, 2020 6:37 pm

I forgot to attach the pdf :oops:
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knight's restriction example.pdf
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jupilogy
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Re: Trying to make a knight's-move puzzle with minimal rules/hints

Post by jupilogy » Sat 05 Dec, 2020 7:12 pm

I’m not clear on the rules. Does any down to the right knight’s move which crosses exactly one boundary have the same digit at both sides of the knight's move? If so, consider the attached edit to your example. The blue circles must have the same digit as the blue square. But they are in the same 3x3 grid.
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knight's restriction example.pdf
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Firlz
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Re: Trying to make a knight's-move puzzle with minimal rules/hints

Post by Firlz » Sat 05 Dec, 2020 7:26 pm

ooooooh thank you for pointing that one out. That IS a problem. The rule doesn't state it clearly, I may have to think about how to rephrase.

The moves are "vector-tied" to the boundary they are crossing, and the "long" part of the knight's move should always go through the boundary if it applies.

So the Square square is only supposed to be tied to the square below the yellow, not to the one to the right of the yellow (as that is a horizontal border, the "long" part of the knight's move must go through that border)

So a

_
_
_ _

Knight's move should always go through a horizontal border, and a

_
_ _ _

Knight's move should always go through a vertical border.

The "exactly one" rule was intended to make this clear (since that prevents you from crossing both a horizontal and vertical border), but does not appear to fully capture this intent.

Firlz
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Re: Trying to make a knight's-move puzzle with minimal rules/hints

Post by Firlz » Sat 05 Dec, 2020 7:40 pm

"Every square on either side of a knight's move in the shape of a forward slash (down and to the right) which crosses the boundary of exactly one pair of boxes must be the same if the boundary is perpendicular to the long leg of the knight's move."

Does that sufficiently describe a limitation that a

_
_ _ _

move only cares about vertical borders and

_
_
_ _

only cares about horizontal borders?

Firlz
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Re: Trying to make a knight's-move puzzle with minimal rules/hints

Post by Firlz » Sat 05 Dec, 2020 8:11 pm

Actually, noticed something after looking at it. I think the logic might mean that when you cross a single border in multiple ways you can choose which of the two (but not both) has to be the copy, and it turns out the sudoku rules combine to get a discrete "path" that matches "pick the one going long." So it may not be necessary to explicitly state that that is the rule being followed (attempting to go the other way breaks fairly quickly).

This issue also only comes up with centers and bottom lefts as well, which makes it fairly quick to prove which choices cannot be made if you have the "freedom" to choose which one in such cases.

Thus

"For each square, when at least one knight's move across the border of two orthogonally adjacent boxes is in the shape of a forward slash (down and to the right), at least one of them must be the same."

Is this language adequate to capture "pick one" in the case you described? If so then following through re-applying and checking sudoku rules will fine there is only one legal choice.

Using the attached, if Orange is the square to examine, we find two choices going down and right from it. Purple and yellow. You can't pick yellow, because if you do it forces a choice between light and dark green. If you dark green, then black is forced and illegal by sudoku.

If you pick light green it is less immediately obvious but it ends up forcing (by the corollary up-left = down right rule) light blue then dark blue then purple, which is illegal as well.

However if you choose purple instead of yellow, you don't have such a problem, dark green light blue and dark blue are forced, but legal.

The same properties then force the red squares via the purple square, disallowing the relationship between orange and grey.

Am I missing something or does this logic hold that "you can pick one if two choices" rule actually still just force a single actual choice?

edit: forgot to attach file
Attachments
example of relationship breaking.pdf
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