If you can make xxxxoooo to oxoxoxox or xoxoxoxo, you absolutely can make

oxoxoxox to xxxxoooo or ooooxxxx.

For any positive integer "n" greater than 2 (n>2), we can easily let "n" of "o" and "n" of "x"

, (for example, oooooxxxxx, ooooooxxxxxx) each time change 2 adjacent within "n" steps

change to "n" pairs of "ox" or "xo" (oxoxoxoxox, xoxoxoxoxoxo, ...).

The key is you should know the rules.

3(0), 4(1,2), 5(1,1), 6(1,3,0), 7(1,2,1), 8(1,2,2,2), 9(1,2,2,1), 10(1,2,2,3,0)

Did you find the similarity of above rules?

1. There are three groups: The first is 3,5,7,9,..(odd), the second is 6,10,14,...(single even)

, and the third is 4, 8,12,...(double even).

2. Odd:(1,...(m of 2),1) m=n/2-2.5 ; n>3; when n=5, m=0(none),n=7, m=1

3. Single even: (1, ...(m of 2),3,0) m=n/2-3; n>=6,; when n=6, m=0, n=10, m=2

4. Double even: (1,...(m of 2),2) m=n/2-2; n>=4; when n=4, m=0, n=8, m=2

5. in (), show the steps you should move "oo" or "xx".

3(0)

OOOXXX

_ _ OXXXOO

_ _ OXX_ _ OXO

_ _ _ _ XOXOXO

(o) means take the most outside two adjacent in the first step.

4(1,2)

OOOOXXXX

O_ _ OXXXXOO

OXXO_ _ XXOO

OXXOXOX_ _ O

_ _ XOXOXOXO

(1,2) means the first step you should remain 1 in the most outside position, and the second

step remain 2 in the other side (from the original most outside).

5(1,1)

OOOOOXXXXX

O_ _ OOXXXXXOO

OXXOOXX_ _ XOO

OXXO_ _ XOXXOO

OXXOXOXOX_ _ O

_ _ XOXOXOXOXO

(1,1) means the first step you should remain 1 in the most outside position, and the second

step remain 1 in the other side (from the original most outside).

6

OOOOOOXXXXXX

O_ _ OOOXXXXXXOO

OXXOOOX_ _ XXXOO

OXX_ _ OXOOXXXOO

OXXOXOXO_ _ XXOO

OXXOXOXOXOX_ _ O

_ _ XOXOXOXOXOXO

(1,3,0) means the first step you should remain 1 in the most outside position, the second step

remain 3 in the other side (from the original most outside), and the third time you should not

remain any in the previous (and back to the side) added adjacent position.

7 (1,2,1)

OOOOOOOXXXXXXX

O_ _ OOOOXXXXXXXOO

OXXOOOOXXX_ _ XXOO

OXXO_ _ OXXXOOXXOO

OXXOXOOXX_ _ OXXOO

OXXOXO_ _ XOXOXXOO

OXXOXOXOXOXOX_ _ O

_ _ XOXOXOXOXOXOXO

8 (1,2,2,2)

OOOOOOOOXXXXXXXX

O_ _ OOOOOXXXXXXXXOO

OXXOOOOOXXXX_ _ XXOO

OXXOO_ _ OXXXXOOXXOO

OXXOOXXO_ _ XXOOXXOO

OXXOOXXOXOX_ _ OXXOO

OXXO_ _ XOXOXOXOXXOO

OXXOXOXOXOXOXOX_ _ O

_ _ XOXOXOXOXOXOXOXO

9 (1,2,2,1)

OOOOOOOOOXXXXXXXXX

O_ _ OOOOOOXXXXXXXXXOO

OXXOOOOOOXXXXX_ _ XXOO

OXXOO_ _ OOXXXXXOOXXOO

OXXOOXXOOXX_ _ XOOXXOO

OXXOOXXO_ _ XOXXOOXXOO

OXXOOXXOXOXOX_ _ OXXOO

OXXO_ _ XOXOXOXOXOXXOO

OXXOXOXOXOXOXOXOX_ _ O

_ _ XOXOXOXOXOXOXOXOXO

14 (1,2,2,2,2,3,0)

OOOOOOOOOOOOOOXXXXXXXXXXXXXX

O_ _ OOOOOOOOOOOXXXXXXXXXXXXXXOO

OXXOOOOOOOOOOOXXXXXXXXXX_ _ XXOO

OXXOO_ _ OOOOOOOXXXXXXXXXXOOXXOO

OXXOOXXOOOOOOOXXXXXX_ _ XXOOXXOO

OXXOOXXOO_ _ OOOXXXXXXOOXXOOXXOO

OXXOOXXOOXXOOOX_ _ XXXOOXXOOXXOO

OXXOOXXOOXX_ _ OXOOXXXOOXXOOXXOO

OXXOOXXO_ _ XOXOXOOXXXOOXXOOXXOO

OXXOOXXOXOXOXOXOOXXXOOXXOOX_ _ O

OXXO_ _ XOXOXOXOXOOXXXOOXXOOXOXO

OXXOXOXOXOXOXOXOOXXXOOX_ _ OXOXO

OXXOXOXOXOXOXOXO_ _ XXOOXOXOXOXO

OXXOXOXOXOXOXOXOXOX_ _ OXOXOXOXO

_ _ XOXOXOXOXOXOXOXOXOXOXOXOXOXO

You can just reverse them to get the answer from

xoxoxoxo to ooooxxxx or problems of any n>2 positive integer.

An example:

10+10(1,2,2,3,0)

OOOOOOOOOOXXXXXXXXXX

O_ _ OOOOOOOXXXXXXXXXXOO

OXXOOOOOOOXXXXXX_ _ XXOO

OXXOO_ _ OOOXXXXXXOOXXOO

OXXOOXXOOOX_ _ XXXOOXXOO

OXXOOXX_ _ OXOOXXXOOXXOO

OXXOOXXOXOXO_ _ XXOOXXOO

OXXOOXXOXOXOXOX_ _ OXXOO

OXXO_ _ XOXOXOXOXOXOXXOO

OXXOXOXOXOXOXOXOXOX_ _ O

_ _ XOXOXOXOXOXOXOXOXOXO

_ _ XOXOXOXOXOXOXOXOXOXO

OXXOXOXOXOXOXOXOXOX_ _ O

OXXO_ _ XOXOXOXOXOXOXXOO

OXXOOXXOXOXOXOX_ _ OXXOO

OXXOOXXOXOXO_ _ XXOOXXOO

OXXOOXX_ _ OXOOXXXOOXXOO

OXXOOXXOOOX_ _ XXXOOXXOO

OXXOO_ _ OOOXXXXXXOOXXOO

OXXOOOOOOOXXXXXX_ _ XXOO

O_ _ OOOOOOOXXXXXXXXXXOO

OOOOOOOOOOXXXXXXXXXX