Can you place 9 queens on a chessboard?

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Can you place 9 queens on a chessboard?

Postby evert » Fri Jan 27, 2006 12:51 am

The additional information to the following puzzles is that number 1 does not occur twice in any diagonal. For example 1 in R3C5 would eliminate candidates for 1:
Code: Select all
 *-----------*
 |..X|...|X..|
 |...|X.X|...|
 |...|.1.|...|
 |---+---+---|
 |...|X.X|...|
 |..X|...|X..|
 |.X.|...|.X.|
 |---+---+---|
 |X..|...|..X|
 |...|...|...|
 |...|...|...|
 *-----------*


Placement of all 1's is thus a solution to the problem of placing 9 queens on a 9X9 chessboard that do not attack each other.

Code: Select all
 *-----------*
 |...|..9|7..|
 |.3.|...|...|
 |.1.|.73|4..|
 |---+---+---|
 |7.5|...|..9|
 |39.|...|...|
 |46.|...|.3.|
 |---+---+---|
 |...|..7|..1|
 |...|564|.2.|
 |...|...|.8.|
 *-----------*
 
 *-----------*
 |...|..1|27.|
 |...|3..|..4|
 |...|.28|5..|
 |---+---+---|
 |.5.|...|..6|
 |3.7|59.|.4.|
 |...|4..|.9.|
 |---+---+---|
 |...|13.|...|
 |2..|..4|...|
 |5..|...|9..|
 *-----------*

 *-----------*
 |2..|..6|735|
 |3..|...|.9.|
 |...|8..|...|
 |---+---+---|
 |...|.5.|...|
 |7..|.2.|3..|
 |...|...|6.7|
 |---+---+---|
 |..7|5..|8..|
 |..2|7..|..4|
 |..5|...|..2|
 *-----------*

Did I re-invent this?
[edit]replaced the 3d puzzle by another
evert
 
Posts: 186
Joined: 26 August 2005

Postby udosuk » Wed Feb 01, 2006 1:17 pm

Thanks for you 3 puzzles!
The first 2 are relatively simple, but the 3rd is quite a workout and I solved it with a lot of T&E branching... just wondering is there a "purely logical" solving process?
udosuk
 
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Joined: 17 July 2005

Re: Can you place 9 queens on a chessboard?

Postby Darth Tater » Thu Feb 02, 2006 2:48 pm

evert wrote:Placement of all 1's is thus a solution to the problem of placing 9 queens on a 9X9 chessboard that do not attack each other.


I didn't know queens of the same color could "attack" each other.
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Joined: 02 December 2005

Postby evert » Fri Feb 03, 2006 9:01 pm

udosuk wrote:The first 2 are relatively simple, but the 3rd is quite a workout and I solved it with a lot of T&E branching... just wondering is there a "purely logical" solving process?

The 3d puzzle was only tested for unique solution.
Probably one can use all logical steps known from normal sudoku now and then eliminating a "1" on a diagonal.
Next 3 puzzles require locked candidates and naked/hidden subsets:
Code: Select all
000304000
684000070
200000000
000005280
006000030
000700600
570089000
000001850
001000000

005800000
700000002
060015004
400000000
070201006
000034000
000700600
090000080
006050000

000000046
000000030
000900000
003700000
400100680
200004009
005208000
740000058
830000000

Maybe we get to some essentially different logic after studying possibilities for 1s. I didn't find it yet.
evert
 
Posts: 186
Joined: 26 August 2005

Re: Can you place 9 queens on a chessboard?

Postby Chessmaster » Fri Feb 03, 2006 10:29 pm

I didn't know queens of the same color could "attack" each other.


they can't it is just assuming that they can for this situation.

you can not put 9 queens on the board anyways becasue the board is 8*8 and if you have 9 queens more than one would be in the same rank,file, or diaganal.
Last edited by Chessmaster on Sat Feb 04, 2006 11:28 pm, edited 1 time in total.
Chessmaster
 
Posts: 191
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Re: Can you place 9 queens on a chessboard?

Postby evert » Fri Feb 03, 2006 11:29 pm

Chessmaster wrote:they can't it is just assuming that they can for this situation.
:)
Indeed. And for this situation we assume that the chessboard is 9X9.
The classical eight queens problem is often generalized.
My aim was a mixture between these mathematics and sudoku.

As a next step:
Code: Select all
090000020
018000000
700000000
080000000
200060400
007000180
006090230
000000000
000500009

The extra diagonal rule holds for 1, 2 and 3!

I'm curious about a puzzle (or even a completed grid) with this rule holding for even more digits, but computing time becomes worse.:(
evert
 
Posts: 186
Joined: 26 August 2005

Postby Chessmaster » Fri Feb 03, 2006 11:58 pm

i tried to do it but had no success. i will report back if i make a puzzle like that.
Chessmaster
 
Posts: 191
Joined: 21 December 2005

Postby Chessmaster » Sat Feb 04, 2006 12:10 am

got it partial i will try to finish the puzzle
Chessmaster
 
Posts: 191
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Postby udosuk » Sat Feb 04, 2006 6:21 pm

Great! Thanks for another four!

I remember last year Condor had done a lot of fantastic work on this area and he made some marvellous puzzles. Try to search his posts and you might find something special!
udosuk
 
Posts: 2698
Joined: 17 July 2005

Postby evert » Sun Feb 05, 2006 3:56 pm

I found a completed grid with the diagonal rule holding for 1 through 4:
Code: Select all
925418367
876395412
134726895
582167943
769534128
341289756
697851234
258943671
413672589

Even for 1 through 5:
Code: Select all
318547296
269381457
547269381
891452673
736918542
452736918
674125839
185693724
923874165

A grid for 1 through 6 exists, but I used a counting function that doesn't show the grid.
Any further steps seem to require hours of computing time.
evert
 
Posts: 186
Joined: 26 August 2005

Postby evert » Sun Feb 05, 2006 11:49 pm

udosuk wrote:I remember last year Condor had done a lot of fantastic work on this area and he made some marvellous puzzles. Try to search his posts and you might find something special!
Found it.
evert
 
Posts: 186
Joined: 26 August 2005

Postby evert » Tue Feb 07, 2006 10:50 pm

This 12 clue puzzle has Condors grid (see link) as unique solution if the diagonal rule is assumed for 1, 3, 5, 6, 7, 8 and 9:
Code: Select all
 *-----------*
 |..3|..1|.2.|
 |...|...|...|
 |8.6|..9|...|
 |---+---+---|
 |...|...|..7|
 |.62|...|...|
 |...|...|...|
 |---+---+---|
 |...|9..|...|
 |...|7..|5..|
 |...|...|...|
 *-----------*

Only naked/hidden subsets and/or locked candidates are required here. However some kind of coloring might become very prominent:
Code: Select all
 *-----------*
 |.1.|...|1..|
 |...|...|...|
 |...|...|...|
 |---+---+---|
 |...|...|...|
 |...|...|...|
 |.X.|...|X..|
 |---+---+---|
 |...|...|...|
 |...|...|...|
 |...|...|...|
 *-----------*
2 candidates in R1 -> 1s eliminated in R6

 *-----------*
 |..1|X..|...|
 |...|...|...|
 |.1.|.X.|...|
 |---+---+---|
 |..X|...|...|
 |...|...|...|
 |...|...|...|
 |---+---+---|
 |...|...|...|
 |...|...|...|
 |...|...|...|
 *-----------*
2 candidates in B1 -> 1s eliminated in R3C5 and R4C3

 *-----------*
 |1..|...|...|
 |...|...|...|
 |..1|...|...|
 |---+---+---|
 |...|X..|...|
 |...|.X.|...|
 |...|..X|...|
 |---+---+---|
 |...|...|X..|
 |...|...|.X.|
 |...|...|..X|
 *-----------*

2 candidates in B1 -> 1s eliminated on the diagonal

Hope I can implement this in a generator some time.:)
evert
 
Posts: 186
Joined: 26 August 2005

Postby udosuk » Thu Feb 09, 2006 5:32 pm

Pretty good effort for the 12-clue puzzle!

However, I myself created this one with 8 clues only (which is the fewest possible for "sudoku with extra feature" when all symbols used are independent to each other):

Code: Select all
.........
.........
.........
...123...
.....6...
...789...
....4....
.........
.........

8 of the 9 digits must each occur 5 times on a short diagonal.


This is the source.

Of course, if the symbols are related to each other (e.g. sortable) then you can generate sudoku puzzles with an empty grid.
See this.
And this.
udosuk
 
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