An urn has 4 balls of 4 different colours Red,Blue,Green,Yellow. 
An urn has $4$ balls of $4$ different colours; red, blue, green, and
  yellow. I pick one ball at random at first and if it is red, I paint
  it blue and return it to the urn. If it is blue, I paint it green. If
  it is green, I paint it yellow. If it is yellow, I paint it red. What
  is the expected number of trials to get all $4$ balls of the same
  colour?

Reminder:
$$\color{red}{red}\to \color{blue}{blue}$$
$$\color{blue}{blue}\to \color{green}{green}$$
$$\color{green}{green}\to \color{yellow}{yellow}$$
$$\color{yellow}{yellow}\to \color{red}{red}$$
I am really stuck with this problem. Help!

This is not a solution but rather an empirical simulation using R. Perhaps it could useful as a check of someone's work:
values=c() 
for(i in c(1:(10^5))){
count=0
urn<-c("red","blue","green","yellow")
while(length(unique(urn))!=1){ #while loop ends when the urn contains 1 unique element
u=floor(runif(1,1,5)) #random integer between 1 and 4, inclusive
urn[u]<-ifelse(urn[u]=="red","blue",ifelse(urn[u]=="blue","green",
ifelse(urn[u]=="green","yellow",ifelse(urn[u]=="yellow","red"))))
#changes the value at the randomly selected index to be the newly painted ball color
count=count+1}
i=i+1
values<-c(values,count)} #adds number of iterations needed in that trial to the vector
mean(values)

[1] 69.40828

The expected value should come out to be in the $65-75$ range. Again, this is not in any way a solution. It will be interesting to see how it compares to the analytical solution.
