Note:
To find this game in Zillions of Games click the "Square Solitaire"
tile and then choose "8x8 Diagonal Solitaire" from the
Variant menu.
BACKGROUND
In "8x8
Diagonal Solitaire" 48 marbles are initially arranged on an
8x8 grid as folows:
OOOOOOOO
OOOOOOOO
OO::::OO
OO::::OO
OO::::OO
OO::::OO
OOOOOOOO
OOOOOOOO
diagram 1
Marbles
jump diagonally as in Checkers and the jumpedover marble is removed.
The goal is to reduce the position to as few marbles as possible.
In "BASIC
Computer Games" (1978) David H. Ahl writes that "It is
easy to remove 30 to 39 checkers, a challenge to remove 40 to 44,
and a substantial feat to remove 45 to 47." Anthony S. Fipiak
writes in "Mathematical Puzzles and Other Brain Twisters"
(1942) that "To have one peg on board is a very difficult feat."
Neither author furnishes a solution. It turns out that the substantial/difficult
feat they talk about is an impossibility. The best solution possible
is a removal of 44 marbles, leaving 4 marbles remaining on the board.
THE PROOF
If the
board were checkered, then marbles would never leave their own color
since only diagonal jumps are allowed. Thus the problem is really
two instances of the problem of reducing all the marbles on a single
color:
O:O:O:O:
:O:O:O:O
O:::::O:
:O:::::O
O:::::O:
:O:::::O
O:O:O:O:
:O:O:O:O
diagram 2
If you
look at diagram 2 diagonally, tilting it clockwise, you see that
this problem is equivalent to the standard type (orthogonal jump)
solitaire puzzle shown in diagram 3. All the positions of the "other
color" have been removed.
O
OOO
OO:OO
OO:::OO
OO:::OO
OO:OO
OOO
O
diagram 3
Reducing
it to a standard type of puzzle means we can use John D. Beasley's
analysis ("The Ins and Outs of Peg Solitaire", chapter
4) on it. First we label the diagonals as in diagrams 4 and 5:
A
ABC
ABCAB
ABCABCA
BCABCAB
ABCAB
CAB
B
diagram 4
D
EFD
FDEFD
DEFDEFD
FDEFDEF
FDEFD
FDE
F
diagram 5
and count
the squares that have men on them initially:
Diagonal A 10 (even)
Diagonal B 10 (even)
Diagonal C 4 (even)
Diagonal D 10 (even)
Diagonal E 4 (even)
Diagonal F 10 (even)

Total men: 24 (even)
In Beasley's
terminology the position is "in phase" for every diagonal,
since each diagonal has the same parity (even) in the number of
men as the total. As moves are played the parities change, but the
phase relationships do not. Therefore in order for there to be a
single marble solution, diagonals AF would all have to have odd
parity. Since this is not possible, there is no single marble solution.
In practice
it offen occurs that an endgame is reached with three marbles in
a row. For example, a horizontal line just above the center would
have the marbles on CAB and FDE. This is a plausible position since
the total (3) has odd parity as does every diagonal (1). Now if
a jump is made to the right you are left with two marbles. The total
(2) has an even parity as do the resulting diagonals: A,B,D,E (0)
and C,F (2). Since a single marble solution doesn't exist, reducing
down to two marbles like this is the best you can do.
If a
two marble solution is the best that can be done for a single square
color, the minimum solution that can be achieved for the original
puzzle is four marbles (two on each color).
SAMPLE
4MARBLE SOLUTION
This
is a sample 4marble solution. First one color is solved and then
the other color is solved symmetrically. The output is taken from
a "Zillions of Games" saved game file. Algebraic chess
coordinates are used.
1. Blue a8  c6 x b7
2. Blue a6  c4 x b5
3. Blue c8  e6 x d7
4. Blue b1  d3 x c2
5. Blue d3  b5 x c4
6. Blue a2  c4 x b3
7. Blue f7  d5 x e6
8. Blue c6  e4 x d5
9. Blue a4  c6 x b5
10. Blue f1  d3 x e2
11. Blue h1  f3 x g2
12. Blue f3  d5 x e4
13. Blue c6  e4 x d5
14. Blue c4  e2 x d3
15. Blue d1  f3 x e2
16. Blue h5  f7 x g6
17. Blue g8  e6 x f7
18. Blue f3  d5 x e4
19. Blue d5  f7 x e6
20. Blue e8  g6 x f7
21. Blue h7  f5 x g6
22. Blue g4  e6 x f5
23. Blue a1  c3 x b2
24. Blue a3  c5 x b4
25. Blue c1  e3 x d2
26. Blue b8  d6 x c7
27. Blue d6  b4 x c5
28. Blue a7  c5 x b6
29. Blue f2  d4 x e3
30. Blue c3  e5 x d4
31. Blue a5  c3 x b4
32. Blue f8  d6 x e7
33. Blue h8  f6 x g7
34. Blue f6  d4 x e5
35. Blue c3  e5 x d4
36. Blue c5  e7 x d6
37. Blue d8  f6 x e7
38. Blue h4  f2 x g3
39. Blue g1  e3 x f2
40. Blue f6  d4 x e5
41. Blue d4  f2 x e3
42. Blue e1  g3 x f2
43. Blue h2  f4 x g3
44. Blue g5  e3 x f4
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