## the riddle of the stands

Posted in Books, Kids, R with tags , , , , , on May 11, 2018 by xi'an

The simple riddle of last week on The Riddler, about the minimum number of urinals needed for n men to pee if the occupation rule is to stay as far as possible from anyone there and never to stand next to another man,  is quickly solved by an R code:

```ocupee=function(M){
ok=rep(0,M)
ok[1]=ok[M]=1
ok[trunc((1+M/2))]=1
while (max(diff((1:M)[ok!=0])>2)){
i=order(-diff((1:M)[ok!=0]))[1]
ok[(1:M)[ok!=0][i]+trunc((diff((1:M)[ok!=0])[i]/2))]=1
}
return(sum(ok>0))
}
```

with maximal occupation illustrated by the graph below:

Meaning that the efficiency of the positioning scheme is not optimal when following the sequential positioning, requiring $N+2^{\lceil log_2(N-1) \rceil}$ urinals. Rather than one out of two, requiring 2N-1 urinals. What is most funny in this simple exercise is the connection exposed in the Riddler with an Xkcd blag written a few years go about the topic.