cyclic riddle [not in NYC!]

In the riddle of this week on fivethirtyeight, the question is to find the average number of rounds when playing the following game: P=6 players sitting in a circle each have B=3 coins and with probability 3⁻¹ they give one coin to their right or left side neighbour, or dump the coin to the centre. The game ends when all coins are in the centre. Coding this experiment in R is rather easy

situz=rep(B,P)
r=1
while (max(situz)>0){
 unz=runif(P)
 newz=situz-(situz>0)
 for (i in (1:P)[unz<1/3]) 
  newz[i%%P+1]=newz[i%%P+1]+(situz[i]>0)
 for (i in (1:P)[unz>2/3])
  newz[i-1+P*(i==1)]=newz[i-1+P*(i==1)]+(situz[i]>0)
 situz=newz
 r=r+1}

resulting in an average of 15.58, but I cannot figure out (quickly enough) an analytic approach to the problem. (The fit above is to a Gamma(â-1,ĝ) distribution.)

In a completely unrelated aparté (aside), I read earlier this week that New York City had prohibited the use of electric bikes. I was unsure of what this meant (prohibited on sidewalks? expressways? metro carriages?) so rechecked and found that electric bikes are simply not allowed for use anywhere in New York City. Because of the potential for “reckless driving”. The target is apparently the high number of delivery e-bikes, but this ban sounds so absurd that I cannot understand it passed. Especially when De Blasio has committed the city to the Paris climate agreement after Trump moronically dumped it… Banning the cars would sound much more in tune with this commitment! (A further aparté is that I strongly dislike e-bikes, running on nuclear power plants,  especially when they pass me on sharp hills!, but they are clearly a lesser evil when compared with mopeds and cars.)