Riddle of the lanes

An express riddle from the Riddler about reopening pools, where lanes are allowed provided there is no swimmer in the lane or in any of the adjacent lanes. If swimmers pick their lane at random (while they can), what is the average number of occupied lanes?

If there are n lanes and E(n) is the expected number of swimmers, E(n) satisfies a recurrence relation determined by the location of the first swimmer:

$E(n)=1+\frac{1}{n}[2E(n-2)+\sum_{i=2}^{n-1}\{E(i-2)+E(n-i-1)\}]$

with E(0)=0, E(1)=E(2)=1. The above can be checked with a quick R experiment:

en=0
for(t in 1:T){
   la=rep(u<-0,N)
   while(sum(la)<N){
     i=sample(rep((1:N)[!la],2),1)
     la[max(1,i-1):min(N,i+1)]=1
     u=u+1}
   en=en+u}

This entry was posted on July 13, 2020 at 12:20 am and is filed under Books, Kids, R with tags COVID-19, lane, R, recursion, swimming pool, The Riddler. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

2 Responses to “Riddle of the lanes”

Riddle of the lanes | R-bloggers Says:
July 13, 2020 at 11:13 am

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Riddle of the lanes – Data Science Austria Says:
July 13, 2020 at 5:17 am

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