May 2016

[Xyzzy]

https://www.research.ibm.com/haifa/p...s/May2016.html

[R. Gerbicz]

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."

[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)."

[a1call]

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.

[R. Gerbicz]

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).

[Xyzzy]

https://www.research.ibm.com/haifa/p...s/May2016.html

[R. Gerbicz]

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:

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:

==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:

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).