There is just one way to go from XXXXOOOO to XOXOXOXO in five moves if *three* adjacent coins are moved at each turn. How?
My solutions
3
OOOXXX_ _ _
O_ _ _XXOOX
OXOOX_ _ _X
_ _ _OXOXOX
4
OOOOXXXX _ _ _
OO_ _ _ XXXOOX
OOXXOX_ _ _ OX
O_ _ _ OXOXXOX or-->_ _ _ XOXOOXOX N-1 turns
OXOXOXOX _ _ _
5
OOOOOXXXXX
OOO_ _ _ XXXXOOX
OOOXXOXX_ _ _ OX
O_ _ _ XOXXOOXOX
OXOOXOX_ _ _ XOX
_ _ _ OXOXOXOXOX
6
OOOOOOXXXXXX
OOOO_ _ _ XXXXXOOX
OOOOXXOXXX_ _ _ OX
OO_ _ _ XOXXXOOXOX
OOXXOXOX_ _ _ OXOX
O_ _ _ OXOXOXXOXOX N-1 turns
OXOXOXOXOX_ _ _ OX
remain a space of 3 coins
7
OOOOOOOXXXXXXX
OOOOO_ _ _ XXXXXXOOX
OOOOOXXOXXXX_ _ _ OX
OOO_ _ _ XOXXXXOOXOX
OOOXXOXOXX_ _ _ OXOX
O_ _ _ XOXOXXOOXOXOX
OXOOXOXOX_ _ _ XOXOX
_ _ _ OXOXOXOXOXOXOX
8
OOOOOOOOXXXXXXXX
OOOOOO_ _ _ XXXXXXXOOX
OOOOOOXXOXXXXX_ _ _ OX
OOOO_ _ _ XOXXXXXOOXOX
OOOOXXOXOXXX_ _ _ OXOX
OO_ _ _ XOXOXXXOOXOXOX
OOXXOXOXOX_ _ _ OXOXOX
_ _ _ XOXOXOXOOXOXOXOX N-1 turns
OXOXOXOXOXO_ _ _ XOXOX
remain a space of 3 coins
9
OOOOOOOOOXXXXXXXXX
OOOOOOO_ _ _ XXXXXXXXOOX
OOOOOOOXXOXXXXXX_ _ _ OX
OOOOO_ _ _ XOXXXXXXOOXOX
OOOOOXXOXOXXXX_ _ _ OXOX
OOO_ _ _ XOXOXXXXOOXOXOX
OOOXXOXOXOXX_ _ _ OXOXOX
O_ _ _ XOXOXOXXOOXOXOXOX
OXOOXOXOXOX_ _ _ XOXOXOX
_ _ _ OXOXOXOXOXOXOXOXOX
10
OOOOOOOOOOXXXXXXXXXX
OOOOOOOO_ _ _ XXXXXXXXXOOX
OOOOOOOOXXOXXXXXXX_ _ _ OX
OOOOOO_ _ _ XOXXXXXXXOOXOX
OOOOOOXXOXOXXXXX_ _ _ OXOX
OOOO_ _ _ XOXOXXXXXOOXOXOX
OOOOXXOXOXOXXX_ _ _ OXOXOX
OO_ _ _ XOXOXOXXXOOXOXOXOX
OOXXOXOXOXOX_ _ _ OXOXOXOX
_ _ _ XOXOXOXOXOOXOXOXOXOX N-1 turns
OXOXOXOXOXOXO_ _ _ XOXOXOX
remain a space of 3 coins
11
OOOOOOOOOOOXXXXXXXXXXX
OOOOOOOOO_ _ _ XXXXXXXXXXOOX
OOOOOOOOOXXOXXXXXXXX_ _ _ OX
OOOOOOO_ _ _ XOXXXXXXXXOOXOX
OOOOOOOXXOXOXXXXXX_ _ _ OXOX
OOOOO_ _ _ XOXOXXXXXXOOXOXOX
OOOOOXXOXOXOXXXX_ _ _ OXOXOX
OOO_ _ _ XOXOXOXXXXOOXOXOXOX
OOOXXOXOXOXOXX_ _ _ OXOXOXOX
O_ _ _ XOXOXOXOXXOOXOXOXOXOX
OXOOXOXOXOXOX_ _ _ XOXOXOXOX
_ _ _ OXOXOXOXOXOXOXOXOXOXOX
12
OOOOOOOOOOOOXXXXXXXXXXXX
OOOOOOOOOO_ _ _ XXXXXXXXXXXOOX
OOOOOOOOOOXXOXXXXXXXXX_ _ _ OX
OOOOOOOO_ _ _ XOXXXXXXXXXOOXOX
OOOOOOOOXXOXOXXXXXXX_ _ _ OXOX
OOOOOO_ _ _ XOXOXXXXXXXOOXOXOX
OOOOOOXXOXOXOXXXXX_ _ _ OXOXOX
OOOO_ _ _ XOXOXOXXXXXOOXOXOXOX
OOOOXXOXOXOXOXXX_ _ _ OXOXOXOX
OO_ _ _ XOXOXOXOXXXOOXOXOXOXOX
OOXXOXOXOXOXOX_ _ _ OXOXOXOXOX
_ _ _ XOXOXOXOXOXOOXOXOXOXOXOX N-1 turns
OXOXOXOXOXOXOXO_ _ _ XOXOXOXOX
remain a space of 3 coins
for n=odd integer and n>=3 OK
for n=even and > 4 remain a space of 3 coins in N turn, in N-1 turn, we can made as:
XOXOXOXOOXOXOXOX for n=6
XOXOXOXOXOOXOXOXOXOX for n=10
n=6 for 8 turns
OOOOOOXXXXXX
_ _ _ OOOXXXXXXOOO
XXOOOOXXXX_ _ _ OO
XXOO_ _ _ XXXOOXOO
XXOOOXOXXXO_ _ _ O
XXOOO_ _ _ XXOXOXO
X_ _ _ OXOOXXOXOXO
XOXXOXO_ _ _ OXOXO
_ _ _ XOXOXOXOXOXO
My solutions
