## Can you place 9 queens on a chessboard?

For fans of Killer Sudoku, Samurai Sudoku and other variants

### Can you place 9 queens on a chessboard?

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?
replaced the 3d puzzle by another
evert

Posts: 186
Joined: 26 August 2005

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

Posts: 2698
Joined: 17 July 2005

### Re: Can you place 9 queens on a chessboard?

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.
Darth Tater

Posts: 6
Joined: 02 December 2005

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
`000304000684000070200000000000005280006000030000700600570089000000001850001000000005800000700000002060015004400000000070201006000034000000700600090000080006050000000000046000000030000900000003700000400100680200004009005208000740000058830000000`

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?

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
Joined: 21 December 2005

### Re: Can you place 9 queens on a chessboard?

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
`090000020018000000700000000080000000200060400007000180006090230000000000000500009`

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

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

got it partial i will try to finish the puzzle
Chessmaster

Posts: 191
Joined: 21 December 2005

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

I found a completed grid with the diagonal rule holding for 1 through 4:
Code: Select all
`925418367876395412134726895582167943769534128341289756697851234258943671413672589`

Even for 1 through 5:
Code: Select all
`318547296269381457547269381891452673736918542452736918674125839185693724923874165`

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

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

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

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

Posts: 2698
Joined: 17 July 2005