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 2016-05-02, 02:44 #1 Xyzzy     Aug 2002 100001000010002 Posts May 2016
 2016-05-02, 14:05 #2 R. Gerbicz     "Robert Gerbicz" Oct 2005 Hungary 30408 Posts There is an important update: "Update (02/05): You should find two different ways to place chess pieces such that both have the same attack numbers, like the pair =qq= ==== ==== =qq= and ==== q==q q==q ==== in the 4x4 case; not a pattern that creates the same attack number for all the squares (like the trivial empty board). Unlike real chess, the pieces are ignoring others in their way: attacking through them as in the 4x4 example."
 2016-05-03, 17:44 #3 R. Gerbicz     "Robert Gerbicz" Oct 2005 Hungary 25·72 Posts New update for star hunters: "Update (03/05): To earn a '*', find a solution without using pawns; to get '**' find a solution using all other five pieces types (k,q,r,n,b)."
2016-05-03, 18:18   #4
a1call

"Rashid Naimi"
Oct 2015
Remote to Here/There

2·19·59 Posts

Quote:
 Originally Posted by R. Gerbicz New update for star hunters: "Update (03/05): To earn a '*', find a solution without using pawns; to get '**' find a solution using all other five pieces types (k,q,r,n,b)."
Pawns are the only pieces which are unidirectional. Using them is kind of undefined since only one color is used and their attack direction can't be defined and is vague.

2016-05-03, 18:30   #5
R. Gerbicz

"Robert Gerbicz"
Oct 2005
Hungary

25·72 Posts

Quote:
 Originally Posted by a1call Pawns are the only pieces which are unidirectional. Using them is kind of undefined since only one color is used and their attack direction can't be defined and is vague.
To handle this: use a fixed direction of the board. (so fixed attack direction of the pawns).

Last fiddled with by R. Gerbicz on 2016-05-03 at 18:33

 2016-06-05, 11:35 #6 Xyzzy     Aug 2002 204108 Posts
2016-06-06, 19:02   #7
R. Gerbicz

"Robert Gerbicz"
Oct 2005
Hungary

25×72 Posts

An alternate solution: with integer programming a Glpk code to get a rotational+reflectional symmetric pair of boards for ** (see the attachment, notice the compactness of the code). The advantage in this search is that for these boards the threat numbers are also symmetrical. This gives that if the threat numbers are the same in 10 given squares then it will be the same on the remaining squares. Unfortunately this could take a lot of time to find a solution, the easier n=6 case (modify the third line of the code) solved in 2 minutes on my computer:
Code:
Threats:
6 6 7 7 6 6
6 3 8 8 3 6
7 8 11 11 8 7
7 8 11 11 8 7
6 3 8 8 3 6
6 6 7 7 6 6
Boards:
knrrnk
n=bb=n
rbqqbr
rbqqbr
n=bb=n
knrrnk

rbqqbr
bn==nb
q=kk=q
q=kk=q
bn==nb
rbqqbr
(note that n=7 and n=8 has got roughly the same computation complexity, so it makes no sense to run n=7). However even an exhaustive run for symmetric pair of boards is possible for n=8, my c code found for example this pair of boards:
Code:
==krrk==
===bb===
k=q==q=k
rb=nn=br
rb=nn=br
k=q==q=k
===bb===
==krrk==

and

==qkkq==
===nn===
q=b==b=q
kn=rr=nk
kn=rr=nk
q=b==b=q
===nn===
==qkkq==

the threat numbers:
6 3 6 6 6 6 3 6
3 4 4 7 7 4 4 3
6 4 7 7 7 7 4 6
6 7 7 6 6 7 7 6
6 7 7 6 6 7 7 6
6 4 7 7 7 7 4 6
3 4 4 7 7 4 4 3
6 3 6 6 6 6 3 6
ps. First solved a slightly different problem: thought that we need to find two solutions where we see the same threat numbers on every square, such solutions:
Code:
rqqqqqqr
q=rrrr=q
qr=rr=rq
qrr==rrq
qrr==rrq
qr=rr=rq
q=rrrr=q
rqqqqqqr

and

rrrrrrrr
rrrrrrrr
rrrrrrrr
rrrrrrrr
rrrrrrrr
rrrrrrrr
rrrrrrrr
rrrrrrrr
and here the threat numbers on both boards and on every square is 14. (That solution worth a *). And found the first board with a different Glpk code (obviously the second board is a trivial solution).
Attached Files
 chess30.txt (3.6 KB, 158 views)

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