Author 
Topic: How many queens? (Read 11359 times) 

Icarus
wu::riddles Moderator Uberpuzzler
Boldly going where even angels fear to tread.
Gender:
Posts: 4863


How many queens?
« on: Nov 5^{th}, 2003, 7:53pm » 
Quote Modify

(What's this? A chess puzzle from Icarus? Not quite...) How many queens can be placed on a chess board so that each threatens exactly 4 others? Generalize to arbitrary size boards, and attacking n other queens, for n other than 4. (For n>0, the problem is due to Scott Kim. n=0 is a classic problem.)


IP Logged 
"Pi goes on and on and on ... And e is just as cursed. I wonder: Which is larger When their digits are reversed? "  Anonymous



Barukh
Uberpuzzler
Gender:
Posts: 2276


Re: How many queens?
« Reply #1 on: Nov 6^{th}, 2003, 9:17am » 
Quote Modify

A clarification question: If two queens are at the same line, but separated by a 3rd queen, are they at the threat to each other?


IP Logged 



towr
wu::riddles Moderator Uberpuzzler
Some people are average, some are just mean.
Gender:
Posts: 13640


Re: How many queens?
« Reply #2 on: Nov 6^{th}, 2003, 9:28am » 
Quote Modify

I had wondered that myself, but there's a solution in both cases, so I just assumed normal chessrules.. (not a real solution yet though, just a few examples for smaller chessboards)


IP Logged 
Wikipedia, Google, Mathworld, Integer sequence DB



visitor
Guest

Wading in cautiously, my first attempt at a solution came up with 16 queens. I hope the real answer is not an order of magnitude higher than that.


IP Logged 



Icarus
wu::riddles Moderator Uberpuzzler
Boldly going where even angels fear to tread.
Gender:
Posts: 4863


Re: How many queens?
« Reply #4 on: Nov 6^{th}, 2003, 5:23pm » 
Quote Modify

Queens only threaten each other if there are no intervening pieces. (However the other problem may be interesting as well). 16 is a trivially easy solution for the n=3 problem (just place two rows of 8 on opposite sides of the board)  of couse it is not maximum there either. The solution for n=4 is higher than 16, but not by an order of magnitude. After all, there are only 64 available positions, and it is fairly obvious most of them will need to be empty! Some related questions (I know  you haven't got this one, but thinking about these cases might spark some ideas for this one): What happens for n=8? What happens for n=4 if you replace "queen" with "rook" or "bishop"? (or "king" or "knight", but those two are less closely related to the queen problem).


IP Logged 
"Pi goes on and on and on ... And e is just as cursed. I wonder: Which is larger When their digits are reversed? "  Anonymous



towr
wu::riddles Moderator Uberpuzzler
Some people are average, some are just mean.
Gender:
Posts: 13640


Re: How many queens?
« Reply #5 on: Nov 7^{th}, 2003, 3:09am » 
Quote Modify

with n=8 the field has to be empty (every queen needs to be surrounded on all sides, but that's impossible at the edge) I'd say n=5 is allready impossible..


IP Logged 
Wikipedia, Google, Mathworld, Integer sequence DB



visitor
Guest

Okay, I found 17. Are we there yet?


IP Logged 



Icarus
wu::riddles Moderator Uberpuzzler
Boldly going where even angels fear to tread.
Gender:
Posts: 4863


Re: How many queens?
« Reply #7 on: Nov 7^{th}, 2003, 3:59pm » 
Quote Modify

Good, but not yet. (18 isn't the answer either).


IP Logged 
"Pi goes on and on and on ... And e is just as cursed. I wonder: Which is larger When their digits are reversed? "  Anonymous



Barukh
Uberpuzzler
Gender:
Posts: 2276


Re: How many queens?
« Reply #8 on: Nov 8^{th}, 2003, 8:54am » 
Quote Modify

Here's my try: [smiley=blacksquare.gif] 20 queens: a2,a4,a5,a7; b1,b8; c1,c8; d1,d8; e1,e8; f1,f8; g1,g8; h2,h3,h6,h7. [smiley=blacksquare.gif]


IP Logged 



Icarus
wu::riddles Moderator Uberpuzzler
Boldly going where even angels fear to tread.
Gender:
Posts: 4863


Re: How many queens?
« Reply #9 on: Nov 12^{th}, 2003, 7:46pm » 
Quote Modify

That can still be bettered, barely.


IP Logged 
"Pi goes on and on and on ... And e is just as cursed. I wonder: Which is larger When their digits are reversed? "  Anonymous



Icarus
wu::riddles Moderator Uberpuzzler
Boldly going where even angels fear to tread.
Gender:
Posts: 4863


Re: How many queens?
« Reply #11 on: Nov 13^{th}, 2003, 9:39am » 
Quote Modify

Yes  21 is the max  but how do you accomplish it? And how do you prove it is max?


IP Logged 
"Pi goes on and on and on ... And e is just as cursed. I wonder: Which is larger When their digits are reversed? "  Anonymous



Barukh
Uberpuzzler
Gender:
Posts: 2276


Re: How many queens?
« Reply #12 on: Nov 13^{th}, 2003, 10:49am » 
Quote Modify

on Nov 13^{th}, 2003, 9:39am, Icarus wrote:Yes  21 is the max  but how do you accomplish it? And how do you prove it is max? 
 Thanks for clarifying. I would try to build the optimal configuration and look at its properties. BTW, Icarus, can you tell whether the 21queens configuration resembles mine?


IP Logged 



Icarus
wu::riddles Moderator Uberpuzzler
Boldly going where even angels fear to tread.
Gender:
Posts: 4863

According to Martin Gardner, someone name William Marshall did an exhaustive computer search and found 40 solutions  not counting reflections and rotations  for 21 queens. He also provided a nice elementary proof (which Gardner did not reproduce) that 21 was the max for an 8x8 board. Gardner only showed one solution, which shows some similarities to yours, but also some differences. In particular, a 21 queen position obviously cannot be symmetric. He also gave a 20 queen solution which differs from yours in the positions of two queens (after a rotation  his was open to the top).


IP Logged 
"Pi goes on and on and on ... And e is just as cursed. I wonder: Which is larger When their digits are reversed? "  Anonymous



SWF
Uberpuzzler
Posts: 879


Re: How many queens?
« Reply #14 on: Nov 17^{th}, 2003, 7:08pm » 
Quote Modify

on Nov 7^{th}, 2003, 3:09am, towr wrote:I'd say n=5 is allready impossible.. 
 I agree than n>4 is not possible: A queen in a corner can attack no more than 3 other queens, and a queen on an edge can attack no more than 5. If n>3 the corner cannot have a queen on it. Since the corner is is vacant, the edge square next to can only attack 4 queens, so if n>4 that square must be empty. Similarly for the edge square next to it and so on to the end of the row. Now repeat from first square on next row. A queen there can attack a maximum of 3 queens since the row above it is vacant... Therefore, for n>4 zero queens can be placed on the board satisfying the conditions.


IP Logged 



towr
wu::riddles Moderator Uberpuzzler
Some people are average, some are just mean.
Gender:
Posts: 13640


Re: How many queens?
« Reply #15 on: Nov 24^{th}, 2009, 7:38am » 
Quote Modify

on Nov 24^{th}, 2009, 6:57am, Hippo wrote:BTW: Why was this topic on the top? Someone added and deleted his post? 
 Probably spam that was promptly delete by a moderator.


IP Logged 
Wikipedia, Google, Mathworld, Integer sequence DB



Vondell
Junior Member
Gender:
Posts: 72


Re: How many queens?
« Reply #16 on: Nov 24^{th}, 2009, 4:29pm » 
Quote Modify

Nah, I had posted a question which I answered myself not much longer afterward, so I deleted it.


IP Logged 
Why is it that people ask for my two cents worth, and then only offer me a penny for my thoughts? That deal sucks!



Grimbal
wu::riddles Moderator Uberpuzzler
Gender:
Posts: 7408


Re: How many queens?
« Reply #17 on: Nov 25^{th}, 2009, 2:05am » 
Quote Modify

on Nov 17^{th}, 2003, 7:08pm, SWF wrote: I agree than n>4 is not possible: A queen in a corner can attack no more than 3 other queens, and a queen on an edge can attack no more than 5. If n>3 the corner cannot have a queen on it. Since the corner is is vacant, the edge square next to can only attack 4 queens, so if n>4 that square must be empty. Similarly for the edge square next to it and so on to the end of the row. Now repeat from first square on next row. A queen there can attack a maximum of 3 queens since the row above it is vacant... Therefore, for n>4 zero queens can be placed on the board satisfying the conditions. 
 Or more directly: Consider the topmost row with a queen on it. Consider the leftmost queen on that row. That queen can attack only in 4 directions. So n>4 cannot have a queen on the board.


IP Logged 



rmsgrey
Uberpuzzler
Gender:
Posts: 2818


Re: How many queens?
« Reply #18 on: Nov 25^{th}, 2009, 5:54am » 
Quote Modify

on Nov 25^{th}, 2009, 2:05am, Grimbal wrote: Or more directly: Consider the topmost row with a queen on it. Consider the leftmost queen on that row. That queen can attack only in 4 directions. So n>4 cannot have a queen on the board. 
 ...assuming that the topmost row and leftmost queen on that row are identifiable  an infinite number of queens can each threaten 5 others on a board that extends infinitely east and west  and a board without boundaries can support infinitely many each attacking 8 others.


IP Logged 



Grimbal
wu::riddles Moderator Uberpuzzler
Gender:
Posts: 7408


Re: How many queens?
« Reply #19 on: Nov 29^{th}, 2009, 7:55am » 
Quote Modify

Well, yes... the question was about a regular chess board.


IP Logged 



rmsgrey
Uberpuzzler
Gender:
Posts: 2818


Re: How many queens?
« Reply #20 on: Jan 6^{th}, 2010, 7:57am » 
Quote Modify

on Nov 29^{th}, 2009, 7:55am, Grimbal wrote:Well, yes... the question was about a regular chess board. 
 The question was about a regular chess board and n=4 on Nov 5^{th}, 2003, 7:53pm, Icarus wrote:Generalize to arbitrary size boards, and attacking n other queens, for n other than 4. 
 My comment was for the generalised version where the arbitrary size board need not be bounded.


IP Logged 



ThudnBlunder
wu::riddles Moderator Uberpuzzler
The dewdrop slides into the shining Sea
Gender:
Posts: 4489


Re: How many queens?
« Reply #21 on: Jan 9^{th}, 2010, 7:55am » 
Quote Modify

on Nov 5^{th}, 2003, 7:53pm, Icarus wrote: Generalize to arbitrary size boards, and attacking n other other queens, for n than 4.) 
 Letting order of the board be k For n= 2, max = 2k  2 for k > 2 For n = 3, max = largest even number less than or equal to (12k  4)/5 For n = 4, max = 3k  3 for k > 5 For n = 1, k = 9 n = 2, k = 5 n = 2, k = 6 n = 4, k = 6 there are unique solutions.

« Last Edit: Jan 9^{th}, 2010, 7:56am by ThudnBlunder » 
IP Logged 
THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you.



Grimbal
wu::riddles Moderator Uberpuzzler
Gender:
Posts: 7408


Re: How many queens?
« Reply #22 on: Jan 9^{th}, 2010, 10:45am » 
Quote Modify

on Jan 6^{th}, 2010, 7:57am, rmsgrey wrote:My comment was for the generalised version where the arbitrary size board need not be bounded 
 OK, sorry. Indeed, my answer was meant for the case of a regular (or at least finite) board.


IP Logged 



lopez
Newbie
Posts: 10


Re: How many queens?
« Reply #23 on: Jul 12^{th}, 2012, 3:07am » 
Quote Modify

There are only two queens in a single game


IP Logged 



rmsgrey
Uberpuzzler
Gender:
Posts: 2818


Re: How many queens?
« Reply #24 on: Jul 12^{th}, 2012, 4:59am » 
Quote Modify

on Jul 12^{th}, 2012, 3:07am, lopez wrote:There are only two queens in a single game 
 There can be up to 18 queens in a single game. Disclaimer: I have not evolved a line of play to demonstrate this, nor proved that such a line must exist  it's possible to promote all sixteen pawns during the course of a game (requiring at least eight captures of pieces by pawns) but it may not be possible to promote all sixteen pawns and juggle all eighteen queens without producing a stalemate or checkmate position on the way.


IP Logged 



