riddles on a line

August 21, 2018
By

(This article was first published on R – Xi'an's Og, and kindly contributed to R-bloggers)

In the Riddler of August 18, two riddles connected with the integer set Ð={2,3,…,10}:

  1. Given a permutation of Ð, what is the probability that the most likely variation (up or down) occurs at each term?
  2. Given three players choosing three different integers in Ð sequentially, and rewards in Ð allocated to the closest of the three players (with splits in case of equal distance), what is the reward for each given an optimal strategy?

For the first question, a simple code returns 0.17…

winofail<-function(permz){ 
  if (length(permz)>1){
    lenoperm=length(permz[-1])/2
    win=(permz[1]permz[1])>lenoperm)+
     (permz[1]>permz[2])*(sum(permz[-1]lenoperm)+
      (runif(1)<.5)*(sum(permz[-1]>permz[1])==lenoperm)
    win=win&&winofail(permz[-1])
  }else win=TRUE
  return(win)}

(but the analytic solution exhibits a cool Pascal triangular relation!) and for the second question, a quick recursion or dynamic programming produces 20, 19, 15 as the rewards (and 5, 9, 8 as the locations)

gainz<-function(seqz){
  difz=t(abs(outer(2:10,seqz,"-")))
  cloz=apply(difz,2,rank)
  return(apply(rbind(2:10,2:10,2:10)*
   ((cloz==1)+.5*(cloz==1.5)),1,sum))}

timeline<-function(prev){ 
  if (length(prev)==0){ 
   sol=timeline(10);bez=gainz(sol)[1] 
   for (i in 2:9){ 
    bol=timeline(i);comp=gainz(bol)[1] 
    if (comp>bez){
        bez=comp;sol=bol}}}
  if (length(prev)==1){
    bez=-10
    for (i in (2:10)[-(prev-1)]){
      bol=timeline(c(prev,i))
      comp=gainz(bol)[2]
      if (comp>bez){
        bez=comp;sol=bol}}}
  if (length(prev)==2){
    bez=-10
    for (i in (2:10)[-(prev-1)]){
      bol=c(prev,i)
      comp=gainz(bol)[3]
      if (comp>bez){
        bez=comp;sol=bol}}}
  return(sol)}

After reading the solution on the Riddler, I realised I had misunderstood the line as starting at 2 when it was actually starting from 1. Correcting for this leads to the same 5, 9, 8 locations of picks, with rewards of 21, 19, 15.

To leave a comment for the author, please follow the link and comment on their blog: R – Xi'an's Og.

R-bloggers.com offers daily e-mail updates about R news and tutorials on topics such as: Data science, Big Data, R jobs, visualization (ggplot2, Boxplots, maps, animation), programming (RStudio, Sweave, LaTeX, SQL, Eclipse, git, hadoop, Web Scraping) statistics (regression, PCA, time series, trading) and more...



If you got this far, why not subscribe for updates from the site? Choose your flavor: e-mail, twitter, RSS, or facebook...

Comments are closed.

Search R-bloggers

Sponsors

Never miss an update!
Subscribe to R-bloggers to receive
e-mails with the latest R posts.
(You will not see this message again.)

Click here to close (This popup will not appear again)