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Contest: the 9 Queens Problem. Put 9 queens and 1 or 2 pawns such that queens do not see each other. Send your solution before Feb 29, and win a book![All Comments] [Add Comment or Rating]
Kay Dekker wrote on Fri, Apr 23, 2010 03:32 PM UTC:Good ★★★★
Not such an old problem, actually: the earliest reference to the eight queens problem that I have been able to find is by Max Bezzel in 1848.

Tez wrote on Sat, Sep 5, 2009 09:35 AM UTC:
Reply to Andreas:

As for the second archbishop problem (attacking every square), I have found a solution with just eight archbishops. Archbishops on a1, b3, c5, d7, e7, f5, g3 and h1 cover all squares. I believe it is optimal, but I do not know for certain.


David Paulowich wrote on Tue, Apr 24, 2007 10:05 PM UTC:Good ★★★★
The following diagram shows 9 queens on a 9x9 board:

 . . . . . . . . x
 . . . . x . . . .
 . . . . . . . x .
 . . . x . . . . .
 x . . . . . . . .
 . . . . . . x . .
 . x . . . . . . .
 . . . . . x . . .
 . . x . . . . . .

Umer wrote on Tue, Apr 24, 2007 07:42 PM UTC:
Plzz tell me how to arrange 9-Queens on a Nine by Nine Board...

Doug Chatham wrote on Tue, Nov 29, 2005 05:44 PM UTC:
Sorry, I don't recall seeing anything on independence and domination numbers for chancellors or archbishops. However, many recent issues of the Games and Puzzles Journal have discussed arrangements of chess variant pieces. See http://www.gpj.connectfree.co.uk/index.htm.

David Paulowich wrote on Tue, Nov 29, 2005 02:04 PM UTC:Excellent ★★★★★
[Reply to Andreas] I believe that 14 Archbishops is the maximum number that don't attack each other: 8 on the first rank and 6 more on the 8th rank (from b8 to g8). No idea how many Archbishops are needed to attack each square on the board.

Eight Chancellors placed on a long diagonal do not attack each other. Eight Rooks on: a1 c2 e3 g4 b5 d6 f7 h8 attack each EMPTY square on the board. Replacing these pieces with Chancellors covers all 64 squares.


Andreas Kaufmann wrote on Mon, Nov 28, 2005 09:52 PM UTC:
By the way, does anybody tried to solve classical domination and
independence problems for chancellor (rook + knight compound) and
archbishop (bishop + knight). I.e.:

1) Is it possible to setup 8 archbishops on 8x8 board so, that they don't
attack each others?
2) How many archbishops are needed to attack each square on the board?
3) as 1) but for for chancellor (trivial)
4) as 2) but for chancellor (difficult).

Doug Chatham wrote on Mon, Nov 28, 2005 12:04 PM UTC:
For each k>=0, if N is large enough, it is possible to place N+k queens and k pawns on an N x N chessboard so that no queens attack each other. See http://home.earthlink.net/~drrdchat/N+kqueens.pdf and http://home.earthlink.net/~drrdchat/N+khandout.pdf for more details.

Doug Chatham wrote on Sat, Feb 12, 2005 04:28 PM UTC:
GEOFX_BBFC,<br> <p>It looks like the last part of your list was cut off. Your list has only 98 solutions, but there are 128 solutions to the '9 Queens Problem' <p>You may find the following PDF files interesting:<ul><li><a href='http://people.moreheadstate.edu/fs/d.chatham/N+1QP.pdf'>The N+1 Queens Problem</a></li><li><a href='http://people.moreheadstate.edu/fs/d.chatham/queenssep.pdf'>The Queens Separation Problem</a></ul>

GEOFX_BBFC wrote on Sat, Feb 12, 2005 01:20 PM UTC:
Are These All Solutions For Nine-Queen Puzzle?

Solution 0 -(1,2)-(0,2)-(1,0)-(1,7)-(2,3)-(3,6)-(4,2)-(5,5)-(6,1)-(7,4)
/-------------------------------\
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|   |   |   |   | x |   |   |   |
\-------------------------------/
Solution 1 -(1,2)-(0,2)-(1,0)-(1,7)-(2,5)-(3,3)-(4,1)-(5,6)-(6,4)-(7,2)
/-------------------------------\
|   |   | x |   |   |   |   |   |
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| x |   | P |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
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|   |   | x |   |   |   |   |   |
\-------------------------------/
Solution 2 -(1,3)-(0,3)-(1,0)-(1,7)-(2,2)-(3,4)-(4,6)-(5,1)-(6,3)-(7,5)
/-------------------------------\
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   | P |   |   |   | x |
|---|---|---|---|---|---|---|---|
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|   |   |   |   |   |   | x |   |
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|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
\-------------------------------/
Solution 3 -(1,3)-(0,3)-(1,0)-(1,7)-(2,4)-(3,1)-(4,5)-(5,2)-(6,6)-(7,3)
/-------------------------------\
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   | P |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
\-------------------------------/
Solution 4 -(1,3)-(0,3)-(1,1)-(1,6)-(2,4)-(3,7)-(4,0)-(5,3)-(6,5)-(7,2)
/-------------------------------\
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   | P |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
\-------------------------------/
Solution 5 -(1,3)-(0,3)-(1,1)-(1,7)-(2,4)-(3,2)-(4,0)-(5,6)-(6,3)-(7,5)
/-------------------------------\
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   | P |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
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| x |   |   |   |   |   |   |   |
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|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
\-------------------------------/
Solution 6 -(1,4)-(0,4)-(1,0)-(1,6)-(2,3)-(3,5)-(4,7)-(5,1)-(6,4)-(7,2)
/-------------------------------\
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   | P |   | x |   |
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|   |   |   | x |   |   |   |   |
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|   |   |   |   |   | x |   |   |
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|   |   |   |   |   |   |   | x |
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|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
\-------------------------------/
Solution 7 -(1,4)-(0,4)-(1,0)-(1,7)-(2,3)-(3,6)-(4,2)-(5,5)-(6,1)-(7,4)
/-------------------------------\
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   | P |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
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|   |   | x |   |   |   |   |   |
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|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
\-------------------------------/
Solution 8 -(1,4)-(0,4)-(1,0)-(1,7)-(2,5)-(3,3)-(4,1)-(5,6)-(6,4)-(7,2)
/-------------------------------\
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   | P |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
\-------------------------------/
Solution 9 -(1,4)-(0,4)-(1,1)-(1,6)-(2,3)-(3,0)-(4,7)-(5,4)-(6,2)-(7,5)
/-------------------------------\
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   | P |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
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|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
\-------------------------------/
Solution 10 -(1,5)-(0,5)-(1,0)-(1,7)-(2,2)-(3,4)-(4,6)-(5,1)-(6,3)-(7,5)
/-------------------------------\
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   | P |   | x |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
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|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
\-------------------------------/
Solution 11 -(1,5)-(0,5)-(1,0)-(1,7)-(2,4)-(3,1)-(4,5)-(5,2)-(6,6)-(7,3)
/-------------------------------\
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   | P |   | x |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
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|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
\-------------------------------/
Solution 12 -(2,1)-(0,1)-(1,4)-(2,0)-(2,7)-(3,3)-(4,6)-(5,2)-(6,5)-(7,1)
/-------------------------------\
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
| x | P |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
\-------------------------------/
Solution 13 -(2,1)-(0,1)-(1,6)-(2,0)-(2,4)-(3,2)-(4,7)-(5,5)-(6,3)-(7,1)
/-------------------------------\
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
| x | P |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
\-------------------------------/
Solution 14 -(2,2)-(0,2)-(1,4)-(2,1)-(2,6)-(3,3)-(4,0)-(5,2)-(6,7)-(7,5)
/-------------------------------\
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x | P |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
\-------------------------------/
Solution 15 -(2,2)-(0,2)-(1,5)-(2,1)-(2,3)-(3,6)-(4,0)-(5,2)-(6,4)-(7,7)
/-------------------------------\
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   | x | P | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | x |
\-------------------------------/
Solution 16 -(2,2)-(0,2)-(1,7)-(2,1)-(2,5)-(3,3)-(4,0)-(5,6)-(6,4)-(7,2)
/-------------------------------\
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   | x | P |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
\-------------------------------/
Solution 17 -(2,2)-(0,4)-(1,2)-(2,0)-(2,5)-(3,2)-(4,6)-(5,1)-(6,3)-(7,7)
/-------------------------------\
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
| x |   | P |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | x |
\-------------------------------/
Solution 18 -(2,2)-(0,4)-(1,2)-(2,0)-(2,5)-(3,3)-(4,1)-(5,7)-(6,2)-(7,6)
/-------------------------------\
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
| x |   | P |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
\-------------------------------/
Solution 19 -(2,2)-(0,4)-(1,2)-(2,0)-(2,7)-(3,3)-(4,6)-(5,2)-(6,5)-(7,1)
/-------------------------------\
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
| x |   | P |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
\-------------------------------/
Solution 20 -(2,3)-(0,3)-(1,5)-(2,0)-(2,7)-(3,2)-(4,4)-(5,6)-(6,1)-(7,3)
/-------------------------------\
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   | P |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
\-------------------------------/
Solution 21 -(2,3)-(0,5)-(1,3)-(2,1)-(2,6)-(3,4)-(4,2)-(5,0)-(6,3)-(7,7)
/-------------------------------\
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   | P |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | x |
\-------------------------------/
Solution 22 -(2,4)-(0,2)-(1,4)-(2,1)-(2,6)-(3,3)-(4,5)-(5,7)-(6,4)-(7,0)
/-------------------------------\
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   | P |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   |   |   |
\-------------------------------/
Solution 23 -(2,4)-(0,4)-(1,2)-(2,0)-(2,7)-(3,5)-(4,3)-(5,1)-(6,6)-(7,4)
/-------------------------------\
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   | P |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
\-------------------------------/
Solution 24 -(2,5)-(0,3)-(1,5)-(2,0)-(2,7)-(3,4)-(4,1)-(5,5)-(6,2)-(7,6)
/-------------------------------\
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   | P |   | x |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
\-------------------------------/
Solution 25 -(2,5)-(0,3)-(1,5)-(2,2)-(2,7)-(3,4)-(4,6)-(5,0)-(6,5)-(7,1)
/-------------------------------\
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   | P |   | x |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
\-------------------------------/
Solution 26 -(2,5)-(0,3)-(1,5)-(2,2)-(2,7)-(3,5)-(4,1)-(5,6)-(6,4)-(7,0)
/-------------------------------\
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   | P |   | x |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   |   |   |
\-------------------------------/
Solution 27 -(2,5)-(0,5)-(1,0)-(2,2)-(2,6)-(3,4)-(4,7)-(5,1)-(6,3)-(7,5)
/-------------------------------\
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   | P | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
\-------------------------------/
Solution 28 -(2,5)-(0,5)-(1,2)-(2,4)-(2,6)-(3,1)-(4,7)-(5,5)-(6,3)-(7,0)
/-------------------------------\
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x | P | x |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   |   |   |
\-------------------------------/
Solution 29 -(2,5)-(0,5)-(1,3)-(2,1)-(2,6)-(3,4)-(4,7)-(5,5)-(6,0)-(7,2)
/-------------------------------\
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   | P | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
\-------------------------------/
Solution 30 -(2,6)-(0,6)-(1,1)-(2,3)-(2,7)-(3,5)-(4,0)-(5,2)-(6,4)-(7,6)
/-------------------------------\
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   | P | x |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
\-------------------------------/
Solution 31 -(2,6)-(0,6)-(1,3)-(2,0)-(2,7)-(3,4)-(4,1)-(5,5)-(6,2)-(7,6)
/-------------------------------\
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   | P | x |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
\-------------------------------/
Solution 32 -(3,1)-(0,1)-(1,3)-(2,5)-(3,0)-(3,7)-(4,2)-(5,4)-(6,6)-(7,1)
/-------------------------------\
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
| x | P |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
\-------------------------------/
Solution 33 -(3,1)-(0,1)-(1,5)-(2,2)-(3,0)-(3,6)-(4,3)-(5,7)-(6,4)-(7,1)
/-------------------------------\
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
| x | P |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
\-------------------------------/
Solution 34 -(3,1)-(0,4)-(1,1)-(2,3)-(3,0)-(3,6)-(4,2)-(5,7)-(6,5)-(7,1)
/-------------------------------\
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
| x | P |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
\-------------------------------/
Solution 35 -(3,1)-(0,4)-(1,1)-(2,7)-(3,0)-(3,5)-(4,2)-(5,6)-(6,1)-(7,3)
/-------------------------------\
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
| x | P |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
\-------------------------------/
Solution 36 -(3,2)-(0,2)-(1,6)-(2,3)-(3,0)-(3,7)-(4,4)-(5,1)-(6,5)-(7,2)
/-------------------------------\
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
| x |   | P |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
\-------------------------------/
Solution 37 -(3,2)-(0,5)-(1,2)-(2,4)-(3,1)-(3,6)-(4,3)-(5,0)-(6,2)-(7,7)
/-------------------------------\
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x | P |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | x |
\-------------------------------/
Solution 38 -(3,3)-(0,0)-(1,3)-(2,6)-(3,2)-(3,4)-(4,7)-(5,1)-(6,3)-(7,5)
/-------------------------------\
| x |   |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   | x | P | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
\-------------------------------/
Solution 39 -(3,3)-(0,0)-(1,5)-(2,3)-(3,1)-(3,6)-(4,3)-(5,7)-(6,2)-(7,4)
/-------------------------------\
| x |   |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   | P |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
\-------------------------------/
Solution 40 -(3,3)-(0,2)-(1,5)-(2,3)-(3,0)-(3,6)-(4,3)-(5,1)-(6,4)-(7,7)
/-------------------------------\
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   | P |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | x |
\-------------------------------/
Solution 41 -(3,3)-(0,2)-(1,7)-(2,3)-(3,1)-(3,6)-(4,3)-(5,5)-(6,0)-(7,4)
/-------------------------------\
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   | P |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
\-------------------------------/
Solution 42 -(3,3)-(0,3)-(1,1)-(2,4)-(3,2)-(3,7)-(4,0)-(5,6)-(6,3)-(7,5)
/-------------------------------\
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x | P |   |   |   | x |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
\-------------------------------/
Solution 43 -(3,3)-(0,3)-(1,1)-(2,4)-(3,2)-(3,7)-(4,5)-(5,3)-(6,6)-(7,0)
/-------------------------------\
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x | P |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   |   |   |
\-------------------------------/
Solution 44 -(3,3)-(0,3)-(1,5)-(2,0)-(3,2)-(3,4)-(4,6)-(5,1)-(6,3)-(7,7)
/-------------------------------\
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x | P | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | x |
\-------------------------------/
Solution 45 -(3,3)-(0,4)-(1,1)-(2,3)-(3,0)-(3,6)-(4,2)-(5,7)-(6,5)-(7,3)
/-------------------------------\
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   | P |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
\-------------------------------/
Solution 46 -(3,3)-(0,6)-(1,3)-(2,0)-(3,2)-(3,4)-(4,7)-(5,5)-(6,3)-(7,1)
/-------------------------------\
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x | P | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
\-------------------------------/
Solution 47 -(3,3)-(0,7)-(1,1)-(2,3)-(3,0)-(3,5)-(4,2)-(5,4)-(6,6)-(7,3)
/-------------------------------\
|   |   |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   | P |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
\-------------------------------/
Solution 48 -(3,4)-(0,0)-(1,6)-(2,4)-(3,2)-(3,7)-(4,5)-(5,3)-(6,1)-(7,4)
/-------------------------------\
| x |   |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   | P |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
\-------------------------------/
Solution 49 -(3,4)-(0,1)-(1,4)-(2,7)-(3,3)-(3,5)-(4,0)-(5,2)-(6,4)-(7,6)
/-------------------------------\
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   |   | x | P | x |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
\-------------------------------/
Solution 50 -(3,4)-(0,3)-(1,6)-(2,4)-(3,1)-(3,7)-(4,5)-(5,0)-(6,2)-(7,4)
/-------------------------------\
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   | P |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
\-------------------------------/
Solution 51 -(3,4)-(0,4)-(1,2)-(2,7)-(3,3)-(3,5)-(4,1)-(5,6)-(6,4)-(7,0)
/-------------------------------\
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   |   | x | P | x |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   |   |   |
\-------------------------------/
Solution 52 -(3,4)-(0,4)-(1,6)-(2,3)-(3,0)-(3,5)-(4,2)-(5,4)-(6,1)-(7,7)
/-------------------------------\
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   | P | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | x |
\-------------------------------/
Solution 53 -(3,4)-(0,4)-(1,6)-(2,3)-(3,0)-(3,5)-(4,7)-(5,1)-(6,4)-(7,2)
/-------------------------------\
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   | P | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
\-------------------------------/
Solution 54 -(3,4)-(0,5)-(1,0)-(2,4)-(3,1)-(3,6)-(4,4)-(5,2)-(6,7)-(7,3)
/-------------------------------\
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   | P |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
\-------------------------------/
Solution 55 -(3,4)-(0,5)-(1,2)-(2,4)-(3,1)-(3,7)-(4,4)-(5,6)-(6,3)-(7,0)
/-------------------------------\
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   | P |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   |   |   |
\-------------------------------/
Solution 56 -(3,4)-(0,7)-(1,2)-(2,4)-(3,1)-(3,6)-(4,4)-(5,0)-(6,5)-(7,3)
/-------------------------------\
|   |   |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   | P |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
\-------------------------------/
Solution 57 -(3,4)-(0,7)-(1,4)-(2,1)-(3,3)-(3,5)-(4,0)-(5,6)-(6,4)-(7,2)
/-------------------------------\
|   |   |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x | P | x |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
\-------------------------------/
Solution 58 -(3,5)-(0,2)-(1,5)-(2,3)-(3,1)-(3,6)-(4,4)-(5,7)-(6,5)-(7,0)
/-------------------------------\
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   | P | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   |   |   |
\-------------------------------/
Solution 59 -(3,5)-(0,5)-(1,1)-(2,4)-(3,0)-(3,7)-(4,3)-(5,6)-(6,2)-(7,5)
/-------------------------------\
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   | P |   | x |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
\-------------------------------/
Solution 60 -(3,6)-(0,3)-(1,6)-(2,0)-(3,2)-(3,7)-(4,5)-(5,1)-(6,6)-(7,4)
/-------------------------------\
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   | P | x |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
\-------------------------------/
Solution 61 -(3,6)-(0,3)-(1,6)-(2,4)-(3,1)-(3,7)-(4,5)-(5,0)-(6,2)-(7,6)
/-------------------------------\
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   | P | x |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
\-------------------------------/
Solution 62 -(3,6)-(0,6)-(1,2)-(2,5)-(3,1)-(3,7)-(4,4)-(5,0)-(6,3)-(7,6)
/-------------------------------\
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   | P | x |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
\-------------------------------/
Solution 63 -(3,6)-(0,6)-(1,4)-(2,2)-(3,0)-(3,7)-(4,5)-(5,3)-(6,1)-(7,6)
/-------------------------------\
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   | P | x |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
\-------------------------------/
Solution 64 -(4,1)-(0,1)-(1,4)-(2,7)-(3,3)-(4,0)-(4,6)-(5,2)-(6,5)-(7,1)
/-------------------------------\
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
| x | P |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
\-------------------------------/
Solution 65 -(4,1)-(0,1)-(1,5)-(2,7)-(3,2)-(4,0)-(4,6)-(5,3)-(6,1)-(7,4)
/-------------------------------\
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
| x | P |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
\-------------------------------/
Solution 66 -(4,1)-(0,1)-(1,6)-(2,4)-(3,2)-(4,0)-(4,7)-(5,5)-(6,3)-(7,1)
/-------------------------------\
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
| x | P |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
\-------------------------------/
Solution 67 -(4,1)-(0,3)-(1,1)-(2,6)-(3,2)-(4,0)-(4,5)-(5,7)-(6,1)-(7,4)
/-------------------------------\
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
| x | P |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
\-------------------------------/
Solution 68 -(4,2)-(0,2)-(1,5)-(2,1)-(3,4)-(4,0)-(4,7)-(5,3)-(6,6)-(7,2)
/-------------------------------\
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
| x |   | P |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
\-------------------------------/
Solution 69 -(4,2)-(0,7)-(1,2)-(2,0)-(3,3)-(4,1)-(4,6)-(5,4)-(6,2)-(7,5)
/-------------------------------\
|   |   |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x | P |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
\-------------------------------/
Solution 70 -(4,3)-(0,0)-(1,6)-(2,3)-(3,5)-(4,2)-(4,7)-(5,4)-(6,1)-(7,3)
/-------------------------------\
| x |   |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x | P |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
\-------------------------------/
Solution 71 -(4,3)-(0,1)-(1,3)-(2,5)-(3,7)-(4,2)-(4,4)-(5,0)-(6,3)-(7,6)
/-------------------------------\
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   | x | P | x |   |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
\-------------------------------/
Solution 72 -(4,3)-(0,3)-(1,5)-(2,7)-(3,2)-(4,0)-(4,6)-(5,3)-(6,1)-(7,4)
/-------------------------------\
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   | P |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
\-------------------------------/
Solution 73 -(4,3)-(0,3)-(1,6)-(2,4)-(3,2)-(4,0)-(4,5)-(5,3)-(6,1)-(7,7)
/-------------------------------\
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   | P |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | x |
\-------------------------------/
Solution 74 -(4,3)-(0,4)-(1,0)-(2,5)-(3,3)-(4,1)-(4,6)-(5,3)-(6,7)-(7,2)
/-------------------------------\
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   | P |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
\-------------------------------/
Solution 75 -(4,3)-(0,4)-(1,2)-(2,7)-(3,3)-(4,1)-(4,6)-(5,3)-(6,5)-(7,0)
/-------------------------------\
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   | P |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   |   |   |
\-------------------------------/
Solution 76 -(4,3)-(0,5)-(1,3)-(2,1)-(3,7)-(4,2)-(4,4)-(5,6)-(6,3)-(7,0)
/-------------------------------\
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   | x | P | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   |   |   |
\-------------------------------/
Solution 77 -(4,3)-(0,5)-(1,3)-(2,6)-(3,0)-(4,2)-(4,7)-(5,4)-(6,1)-(7,3)
/-------------------------------\
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x | P |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
\-------------------------------/
Solution 78 -(4,3)-(0,7)-(1,3)-(2,1)-(3,6)-(4,2)-(4,4)-(5,0)-(6,5)-(7,3)
/-------------------------------\
|   |   |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   | x | P | x |   |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
\-------------------------------/
Solution 79 -(4,3)-(0,7)-(1,4)-(2,1)-(3,3)-(4,0)-(4,6)-(5,3)-(6,5)-(7,2)
/-------------------------------\
|   |   |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   | P |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
\-------------------------------/
Solution 80 -(4,4)-(0,0)-(1,3)-(2,6)-(3,4)-(4,1)-(4,7)-(5,4)-(6,2)-(7,5)
/-------------------------------\
| x |   |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   | P |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
\-------------------------------/
Solution 81 -(4,4)-(0,0)-(1,4)-(2,6)-(3,1)-(4,3)-(4,5)-(5,7)-(6,2)-(7,4)
/-------------------------------\
| x |   |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x | P | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
\-------------------------------/
Solution 82 -(4,4)-(0,2)-(1,4)-(2,1)-(3,7)-(4,0)-(4,5)-(5,3)-(6,6)-(7,4)
/-------------------------------\
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
| x |   |   |   | P | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
\-------------------------------/
Solution 83 -(4,4)-(0,2)-(1,4)-(2,6)-(3,0)-(4,3)-(4,5)-(5,1)-(6,4)-(7,7)
/-------------------------------\
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x | P | x |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | x |
\-------------------------------/
Solution 84 -(4,4)-(0,3)-(1,5)-(2,0)-(3,4)-(4,1)-(4,6)-(5,4)-(6,2)-(7,7)
/-------------------------------\
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   | P |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | x |
\-------------------------------/
Solution 85 -(4,4)-(0,3)-(1,7)-(2,2)-(3,4)-(4,1)-(4,6)-(5,4)-(6,0)-(7,5)
/-------------------------------\
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   | P |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
\-------------------------------/
Solution 86 -(4,4)-(0,4)-(1,1)-(2,3)-(3,5)-(4,2)-(4,7)-(5,4)-(6,6)-(7,0)
/-------------------------------\
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   | P |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   |   |   |
\-------------------------------/
Solution 87 -(4,4)-(0,4)-(1,2)-(2,0)-(3,5)-(4,1)-(4,7)-(5,4)-(6,6)-(7,3)
/-------------------------------\
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   | P |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
\-------------------------------/
Solution 88 -(4,4)-(0,6)-(1,4)-(2,2)-(3,0)-(4,3)-(4,5)-(5,7)-(6,4)-(7,1)
/-------------------------------\
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x | P | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
\-------------------------------/
Solution 89 -(4,4)-(0,7)-(1,1)-(2,4)-(3,2)-(4,0)-(4,5)-(5,3)-(6,6)-(7,4)
/-------------------------------\
|   |   |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   | P | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
\-------------------------------/
Solution 90 -(4,5)-(0,0)-(1,5)-(2,7)-(3,4)-(4,1)-(4,6)-(5,3)-(6,5)-(7,2)
/-------------------------------\
| x |   |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   | P | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
\-------------------------------/
Solution 91 -(4,5)-(0,5)-(1,2)-(2,6)-(3,3)-(4,0)-(4,7)-(5,4)-(6,1)-(7,5)
/-------------------------------\
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   | P |   | x |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
\-------------------------------/
Solution 92 -(4,6)-(0,4)-(1,6)-(2,1)-(3,5)-(4,2)-(4,7)-(5,0)-(6,6)-(7,3)
/-------------------------------\
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   | P | x |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
\-------------------------------/
Solution 93 -(4,6)-(0,6)-(1,1)-(2,3)-(3,5)-(4,0)-(4,7)-(5,2)-(6,4)-(7,6)
/-------------------------------\
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   | P | x |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
\-------------------------------/
Solution 94 -(4,6)-(0,6)-(1,2)-(2,0)-(3,5)-(4,1)-(4,7)-(5,4)-(6,6)-(7,3)
/-------------------------------\
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   | P | x |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
\-------------------------------/
Solution 95 -(4,6)-(0,6)-(1,3)-(2,0)-(3,4)-(4,1)-(4,7)-(5,5)-(6,2)-(7,6)
/-------------------------------\
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
| x |   |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   | P | x |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
\-------------------------------/
Solution 96 -(5,1)-(0,1)-(1,3)-(2,5)-(3,7)-(4,2)-(5,0)-(5,4)-(6,6)-(7,1)
/-------------------------------\
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
| x | P |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   | x |   |   |   |   |   |   |
\-------------------------------/
Solution 97 -(5,1)-(0,1)-(1,5)-(2,2)-(3,6)-(4,3)-(5,0)-(5,7)-(6,4)-(7,1)
/-------------------------------\
|   | x |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   | x |   |   |
|---|---|---|---|---|---|---|---|
|   |   | x |   |   |   |   |   |
|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   | x |   |
|---|---|---|---|---|---|---|---|
|   |   |   | x |   |   |   |   |
|---|---|---|---|---|---|---|---|
| x | P |   |   |   |   |   | x |
|---|---|---|---|---|---|---|---|
|   |   |   |   | x |   |   |   |
|---|---|---|---|---|---|---|---|
|

Hans Bodlaender wrote on Wed, Mar 10, 2004 01:45 PM UTC:
Gary Gifford is the winner. I did a fair draw from the seven correct solutions that were send by email. I'll post some more on what people wrote about solutions etc. later. <p> Are you interested in another such contest? If so, email me ideas about the possible format! <p> Best regards, Hans

Anonymous wrote on Thu, Mar 4, 2004 05:42 PM UTC:
So who won?

Hans Bodlaender wrote on Mon, Jan 12, 2004 08:43 AM UTC:
The rules are modified as follows:
<p>
Solutions that arrive from now on should not have a pawn on a centre
square, that is: d4, d5, e4, and e5.
<p>
The solutions that arrived earlier (three so far) keep running for the
prize; new entries hence have to be essentially different from the posted
one. I hope this is fair enough.
<p>
For those interested in this type of puzzles, I would be willing to run a
similar contest later (still have about two books which I can give away
for prizes.)
<p>
Please try to refrain posting hints or solutions to this puzzle before
the
deadline. In the meantime, you may try to find out <em>how many</em>
essentially different solutions there are. I've read that the original 8
queens problem has 12 essentially different solutions. (Solutions are
essentially different if you cannot get one from another by mirroring
and/or rotating the board.)

Jared McComb wrote on Sun, Jan 11, 2004 01:12 AM UTC:
Actually, I was thinking something along the lines of 'maximize the number of queens on a standard 91-cell hex chessboard,' because nobody's done that yet.

Doug Chatham wrote on Sat, Jan 10, 2004 10:22 PM UTC:
Well, there are <i>many</i> alternative problems we could look at. For example, replace the Queens with Amazons. (See <a href='http://www.durangobill.com/N_Queens.html'>http://www.durangobill.com/N_Queens.html </a>for some results on putting n Amazons --he calls them Superqueens -- on an nxn board, with no pawns.)<p> For more problems involving combinations on chessboards, see <blockquote>S.M. Hedetniemi, S.T. Hedetniemi and R. Reynolds, Combinatorial problems on chessboards: II, Chapter 6 in Domination in Graphs: Advanced Topics, T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Eds., Marcel Dekker, New York, 133-162 (1998). </blockquote>

Jared McComb wrote on Sat, Jan 10, 2004 03:23 AM UTC:Excellent ★★★★★
If anyone is interested, I have a problem of my own that could replace this one for the book prize.

Gary Gifford wrote on Sat, Jan 10, 2004 02:26 AM UTC:
Roberto: Thank you for explaining your mathematical excitement over the very interesting problem. I solved the problem in a matter of a minute or two. But I did not use math (at least not a a concious level). I had a strong hunch that a central pawn would be best to isolate 2 pawns on the rank, and then 2 on the file... I believed that the 'knight move' pattern optimized Queen placements for this problem... the Knight pattern is certainly abundant in the solution and I think that a mathematical solution should somehow incorporate the 'knight move' aspect of Queen placement. Best regards, Gary

Roberto Lavieri wrote on Sat, Jan 10, 2004 02:08 AM UTC:
Conjecture:
It exists n ( n <= 8 ) such that, if N >= n, and if M is a number such
that N < M <= 2N, then it is possible to put M Queens and M-N Pawns on the
NxN board ia a manner that the Queens are safe each other

Roberto Lavieri wrote on Sat, Jan 10, 2004 02:00 AM UTC:
Gary, excuse it. I agree with your dissapointment. I were effusive because
I enjoyed the problem and the way I attacked the solution, trying to give
a proof to my wrong hypothesis of impossibility, it conduced me to find
the solution. Of course, I´m not the winner of the book, it was almost
clear in the Andreas mail that he found the solution, and he did it before
me; and now I know you did it days before. But I am more interested in the
mathematical problem than in the prize. I´m going to think in the general
problem, but it is a very difficult one, I conjecture that if N>=n
(certain n<=8), and if N<M<2N, it is possible to put M Queens and M-N Paws
on the board in such way the Queens are safe each other.

Andreas Kaufmann wrote on Fri, Jan 9, 2004 11:14 PM UTC:Excellent ★★★★★
Roberto, the solution for ten queens and two pawns is very elegant! I
failed to find it, also I easily found the soultion for nine queens and a
pawn. 
 
How about general problem? Namely, given board of size NxN, M queens (M>N)
find the minumum number of pawns, so that it is possible to setup queens
and pawns in such a way that queens don't attack each other. Looks like a
quite difficult mathematical problem.

Gary Gifford wrote on Fri, Jan 9, 2004 10:09 PM UTC:
I am disappointed to see that a solution was posted as it ruins the challenge for problem solvers that have not yet solved it or were still trying for 1 pawn solutions. I sent in a solution several days ago with 9 Queens and 1 pawn and was looking forward to winning a book. Now I see these answers here... they should have waited until after the contest ended before posting them.

Roberto Lavieri wrote on Fri, Jan 9, 2004 09:25 PM UTC:Excellent ★★★★★
HERE IS THE ANSWER: YES!!!, it is possible to put 9 Queens and one Pawn in
an 8x8 Board, every Queen unattacked by other Queen.
POSITION:
QUEENS: a8, b5, c2, d4, d6, e1, f7, g5, h3
PAWN: d4
Verify!!!

Roberto Lavieri wrote on Fri, Jan 9, 2004 08:54 PM UTC:Excellent ★★★★★
This position shows TEN QUEENS and TWO PAWNS, Queens safe each other:
QUEENS: a8, b5, c3, d1, d6, e4, f2, f7, g5, h3
PAWNS: d5, f3

Is it possible with NINE QUEENS and ONE PAWN?. I think the answer is NO,
but it is not easy to give a proof!.

Roberto Lavieri wrote on Fri, Jan 9, 2004 05:46 PM UTC:Excellent ★★★★★
Nice problem!. I have not a proof yet, but I think it is not possible a
solution with only one Pawn.
With two Pawns I have constructed this one:
QUEENS in  a1, b3, c5, d3, d8, e6, f4, g2, h5.
PAWNS in  c3, b5.

Andreas Kaufmann wrote on Fri, Jan 9, 2004 09:50 AM UTC:Excellent ★★★★★
Very interesting variation of the famous problem!

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