Xi'an's Og

An arithmetic Le Monde mathematical puzzle (or two independent ones, again!):

1. Take the integers from 1 to 19, pick two of them with identical parity at random and replace the pair with their average. Repeat 17 times to obtain a single integer. What are the values between 1 and 19 that cannot be achieved?
2.  Take the integers from 1 to 19, pick four of them at random so that the average is an integer and replace the quadruplet with their average. Repeat 5 times to obtain a single integer. What are the values between 1 and 19 that can be achieved?

The first question seems pretty open to brute force simulation. Here is an R code I wrote

numbz=1:M
for (t in 2:M){
 numbz=sample(numbz);count=0
 while((count<100)&(sum(numbz[1:2])%%2>0)){
   numbz=sample(numbz);count=count+1}
if (count==100) break()
 numbz[1]=as.integer(mean(numbz[1:2]))
 numbz=numbz[-2]}

with the stopping rule resulting from the fact that the remaining two digits may sometimes be of opposite parity (a possibility omitted in the wording of the puzzle, along with a mistake in the number of repetitions). However, the outcome of this random exploration misses the extreme possible values. For instance, 10⁶ attempts produce the range

4 5 6 7 8 9 10 11 12 13 14 15 16 17

while the extremes should be 2 and 18 according to this scratch computation:

which appears to have too low a probability of occurring for being part of the 10⁶ instances. Running the code a mere (!) 10⁷ iterations managed to reach 3 as well. (Interestingly, the above sequence uses 2 the most and 19 the least, but weights 19 the most and 2 the least!)

The second puzzle is also open to random exploration with a very similar R code:

utcome=NULL
for (z in 1:1e6){
numbz=1:19
for (t in 1:6){
  numbz=sample(numbz);count=0
  while ((sum(numbz[1:4])%%4>0)&(count<100)){
    numbz=sample(numbz);count=count+1}
  if (count==100) break()
  numbz[1]=as.integer(mean(numbz[1:4]))
  numbz=numbz[-(2:4)]}
if (count<100) utcome=c(utcome,numbz)}

returning the values

4 7 10 13 16

This entry was posted on June 2, 2017 at 12:17 am and is filed under Books, Kids with tags competition, Le Monde, mathematical puzzle, R, rank statistics. 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.