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 Post subject: kNight Rules
PostPosted: Sat May 31, 2008 9:36 am 
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Grand Master
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Firstly a description of the pieces:

A kNight is a (1,2) leaper which features in Chess. It can move 1 cell horizontally + 2 cells vertically, or 1 cell vertically + 2 cells horizontally. A kNight @ r5c5 can move to 8 cells: r37c46+r46c37.

A Quad-kNight is a (4,8) leaper, i.e. move 4 times the distance as a kNight. A Quad-kNight @ r1c1 can move to 2 cells: r5c9+r9c5. There exists only one Quad-kNight's circuit on a 9x9 board: r1c1-r5c9-r9c1-r1c5-r9c9-r5c1-r1c9-r9c5-r1c1.

With Non-Consecutiveness based on these 2 pieces here is a valid puzzle with one single given clue:

Image

(Without any given clue there would be 8 solutions, all equivalent to each other via reflection/rotation.)

But the above puzzle is (probably) solvable only by computer programs using brute force, so here is a human-solvable version with 3 more given clues and the QNNC property replaced by the AC (Anti-Camel) property:

(A Camel is a (1,3) leaper. It can move 1 cell horizontally + 3 cells vertically, or 1 cell vertically + 3 cells horizontally. A Camel @ r5c5 can move to 8 cells: r28c46+r46c28.)

Image

:ugeek:

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 Post subject: Re: kNight Rules
PostPosted: Tue Jun 03, 2008 11:23 pm 
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OK Matt: You Win

I was just about to post a Ferz NC & Camel NC which has a classic 72 solutions.

But with 8 you've got it dead-on; and unbeatable.

When I came up with the kNight NC idea (boast), I new it would very "restrained" - but the idea of doubling it is magic!!!


HATMAN

{Restrained - copyright Ruud: for sudoku puzzle structures with very limited solution sets (i.e. before the givens/cages are introduced}


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 Post subject: Re: kNight Rules
PostPosted: Wed Jun 04, 2008 4:52 am 
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To get 8 solutions one would need the NC property with {19} not considered consecutive. Also one probably need an end-to-end leaper (one leaping across opposite edges, e.g. Ostrich on a 6x6 board). Note with 9x9 NC AK X you can add the NC property of an (8,8) leaper (i.e. opposite corner leaper) to get 8 solutions.

Interesting find in FNC CNC having 9x8 solutions. Will explore that! :ugeek:

PS: Note NNC+AC also has 9x8 solutions. NNC with 15x8 solutions is probably the best you can do in 9x9 with one variant rule. :geek:

PPS: Your FNC CNC has a one-to-one mapping from FNC AK AN with the following transformations:

If FNC AK AN has r123456789, then r123897456 is FNC CNC.
-or-
If FNC AK AN has n123456789, then n123897456 is FNC CNC.

:idea:

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