A solution by Paulo S.L.M. Barreto {pbarreto@uninet.com.br} for
the "Kutu of Ekakin" puzzle, proposed in Discis Entertainment's
"Jewels of the Oracle" game.

N.B.  Although the final position is not as symmetrical as that
of the related Chinese Checkers puzzle, it is accepted as valid
by the Jewels of the Oracle (note, however, that both initial
and final positions are bilaterally symmetrical).

NOTATION:

We describe moves in the form "r-c X", where r is the row number
(1 to 7), c is the column number (1 to 9) and X is the direction of
the move (L for left, R for right, U for up, D for down).

Each step consists of moving a marble P1 two rows or two columns over
another marble P2 onto a hole; P1 occupies that hole and P2 is removed.

For instance, "3-5 D" (the first move) means "move down the marble in
row 3, column 5; put it on the hole in row 5, column 5; then remove
the marble in row 4, column 5".

1: Initial position

        o o . o o
        o o o o o
    o o o o o o o o o
    o o o o o o o o o
    o o o o . o o o o
    o o o o o o o o o
    o o . o o o . o o

First, let's eliminate the two upper rows:

2: After 3-5 D

        o o . o o
        o o o o o
    o o o o . o o o o
    o o o o . o o o o
    o o o o o o o o o
    o o o o o o o o o
    o o . o o o . o o

3: After 1-3 R

        . . o o o
        o o o o o
    o o o o . o o o o
    o o o o . o o o o
    o o o o o o o o o
    o o o o o o o o o
    o o . o o o . o o

4: After 1-5 D

        . . . o o
        o o . o o
    o o o o o o o o o
    o o o o . o o o o
    o o o o o o o o o
    o o o o o o o o o
    o o . o o o . o o

5: After 2-3 R

        . . . o o
        . . o o o
    o o o o o o o o o
    o o o o . o o o o
    o o o o o o o o o
    o o o o o o o o o
    o o . o o o . o o

6: After 2-5 D

        . . . o o
        . . . o o
    o o o o . o o o o
    o o o o o o o o o
    o o o o o o o o o
    o o o o o o o o o
    o o . o o o . o o

7: After 1-7 L

        . . o . .
        . . . o o
    o o o o . o o o o
    o o o o o o o o o
    o o o o o o o o o
    o o o o o o o o o
    o o . o o o . o o

8: After 2-7 L

        . . o . .
        . . o . .
    o o o o . o o o o
    o o o o o o o o o
    o o o o o o o o o
    o o o o o o o o o
    o o . o o o . o o

9: After 1-5 D

        . . . . .
        . . . . .
    o o o o o o o o o
    o o o o o o o o o
    o o o o o o o o o
    o o o o o o o o o
    o o . o o o . o o

Now let's shrink the left and right sides:

10: After 5-3 D, 5-7 D

        . . . . .
        . . . . .
    o o o o o o o o o
    o o o o o o o o o
    o o . o o o . o o
    o o . o o o . o o
    o o o o o o o o o

11: After 3-3 U, 3-5 U, 3-7 U

        . . . . .
        o . o . o
    o o . o . o . o o
    o o . o . o . o o
    o o . o o o . o o
    o o . o o o . o o
    o o o o o o o o o

12: After 3-1 R, 4-1 R, 5-1 R, 6-1 R, 3-9 L, 4-9 L, 5-9 L, 6-9 L

        . . . . .
        o . o . o
    . . o o . o o . .
    . . o o . o o . .
    . . o o o o o . .
    . . o o o o o . .
    o o o o o o o o o

Next, we eliminate lines 2 and 3:

13: After 3-3 R (see step 17)

        . . . . .
        o . o . o
    . . . . o o o . .
    . . o o . o o . .
    . . o o o o o . .
    . . o o o o o . .
    o o o o o o o o o

14: After 5-3 U (see step 18)

        . . . . .
        o . o . o
    . . o . o o o . .
    . . . o . o o . .
    . . . o o o o . .
    . . o o o o o . .
    o o o o o o o o o

15: After 2-3 D (see step 19)

        . . . . .
        . . o . o
    . . . . o o o . .
    . . o o . o o . .
    . . . o o o o . .
    . . o o o o o . .
    o o o o o o o o o

16: After 2-5 D

        . . . . .
        . . . . o
    . . . . . o o . .
    . . o o o o o . .
    . . . o o o o . .
    . . o o o o o . .
    o o o o o o o o o

17: After 3-7 L (see step 13)

        . . . . .
        . . . . o
    . . . . o . . . .
    . . o o o o o . .
    . . . o o o o . .
    . . o o o o o . .
    o o o o o o o o o

18: After 5-7 U (see step 14)

        . . . . .
        . . . . o
    . . . . o . o . .
    . . o o o o . . .
    . . . o o o . . .
    . . o o o o o . .
    o o o o o o o o o

19: After 2-7 D (see step 15)

        . . . . .
        . . . . .
    . . . . o . . . .
    . . o o o o o . .
    . . . o o o . . .
    . . o o o o o . .
    o o o o o o o o o

Next, we shrink the structure towards the bottom center:

20: After 7-3 U, 7-7 U

        . . . . .
        . . . . .
    . . . . o . . . .
    . . o o o o o . .
    . . o o o o o . .
    . . . o o o . . .
    o o . o o o . o o

21: After 7-1 R, 7-9 L

        . . . . .
        . . . . .
    . . . . o . . . .
    . . o o o o o . .
    . . o o o o o . .
    . . . o o o . . .
    . . o o o o o . .

22: After 4-3 D, 4-7 D

        . . . . .
        . . . . .
    . . . . o . . . .
    . . . o o o . . .
    . . . o o o . . .
    . . o o o o o . .
    . . o o o o o . .

23: After 7-3 U, 7-7 U

        . . . . .
        . . . . .
    . . . . o . . . .
    . . . o o o . . .
    . . o o o o o . .
    . . . o o o . . .
    . . . o o o . . .

This diagram is very interesting.  It contains a structure that appears
(though perhaps inverted) in the solution of the Chinese Checkers puzzle
after the four symmetrical patterns have been applied, namely:

        . . . . .
        . . . . .
    . . . . o . . . .
    . . . o o o . . .
    . . o o o o o . .
    . . . o . o . . .
    . . . . . . . . .

Hence, we follow similar steps to solve this substructure, obtaining
another structure that appears in the Chinese Checkers puzzle (namely,
an inverted "T").

24: After 6-5 L

        . . . . .
        . . . . .
    . . . . o . . . .
    . . . o o o . . .
    . . o o o o o . .
    . . o . . o . . .
    . . . o o o . . .

25: After 4-5 D

        . . . . .
        . . . . .
    . . . . o . . . .
    . . . o . o . . .
    . . o o . o o . .
    . . o . o o . . .
    . . . o o o . . .

26: After 6-6 L

        . . . . .
        . . . . .
    . . . . o . . . .
    . . . o . o . . .
    . . o o . o o . .
    . . o o . . . . .
    . . . o o o . . .

27: After 6-3 R

        . . . . .
        . . . . .
    . . . . o . . . .
    . . . o . o . . .
    . . o o . o o . .
    . . . . o . . . .
    . . . o o o . . .

28: After 5-4 U

        . . . . .
        . . . . .
    . . . o o . . . .
    . . . . . o . . .
    . . o . . o o . .
    . . . . o . . . .
    . . . o o o . . .

29: After 3-4 R

        . . . . .
        . . . . .
    . . . . . o . . .
    . . . . . o . . .
    . . o . . o o . .
    . . . . o . . . .
    . . . o o o . . .

30: After 5-7 L

        . . . . .
        . . . . .
    . . . . . o . . .
    . . . . . o . . .
    . . o . o . . . .
    . . . . o . . . .
    . . . o o o . . .

31: After 3-6 D

        . . . . .
        . . . . .
    . . . . . . . . .
    . . . . . . . . .
    . . o . o o . . .
    . . . . o . . . .
    . . . o o o . . .

32: After 5-6 L

        . . . . .
        . . . . .
    . . . . . . . . .
    . . . . . . . . .
    . . o o . . . . .
    . . . . o . . . .
    . . . o o o . . .

33: After 5-3 R

        . . . . .
        . . . . .
    . . . . . . . . .
    . . . . . . . . .
    . . . . o . . . .
    . . . . o . . . .
    . . . o o o . . .

Finally we solve the "T":

34: After 7-5 R

        . . . . .
        . . . . .
    . . . . . . . . .
    . . . . . . . . .
    . . . . o . . . .
    . . . . o . . . .
    . . . o . . o . .

35: After 5-5 D

        . . . . .
        . . . . .
    . . . . . . . . .
    . . . . . . . . .
    . . . . . . . . .
    . . . . . . . . .
    . . . o o . o . .

36: After 7-4 R

        . . . . .
        . . . . .
    . . . . . . . . .
    . . . . . . . . .
    . . . . . . . . .
    . . . . . . . . .
    . . . . . o o . .

37: After 7-7 L

        . . . . .
        . . . . .
    . . . . . . . . .
    . . . . . . . . .
    . . . . . . . . .
    . . . . . . . . .
    . . . . o . . . .

That's it!


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