the riddle of the stands

May 10, 2018
By

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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 $latexN+2^{\lceil log_2(N-1) \rceil}$ 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.

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