Player 1 rolls a die. There are 6 possible rolls.
Player 2 then has to roll a different number. There are 5 possible rolls.
Players 3 and 4 then have to roll such that neither they nor player 4 and 1 have the same roll.
There are 4 rolls for player 3 that lead to 4 possible rolls for player 4 and
There is 1 possible roll (the value that player 1 rolled) that leads to 5 possible rolls for player 4.
Combining all this we have that there are
6 * 5 * (4 * 4 + 1 * 5) = 30*21 = 630 valid rolls among the players.
There are a total of 64 = 1296 possible rolls among the players.
Thus the probability of no consecutive rolls in the ring being the same is
\(p = \dfrac{630}{1296}=\dfrac{35}{72}\)
.