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As a coincidence, while I was waiting for the solution to puzzle #737 published this Friday in Le Monde, the delivery (wo)man forgot to include the weekend magazine and I had to buy it this morning with my baguette (as if anyone cares!). The solution is (y_{0},z_{0},w_{0})=(38,40,46) and…it does not work! The value of (x_{1},y_{1},z_{1},w_{1}) is indeed (19,39,43,8.5). Now, it does not work because the update mechanism used by the writers of the Le Monde column is to move all positions simultaneously, rather than one at a time: the update thus reads as

if u^{n}=(x_{n},y_{n},z_{n},w_{n}), for i=1,…,4,

if u_{i}^{n-1} is not the maximum of u^{n-1} and

otherwise.

(to compare with my earlier proposal). In this new configuration, the core of the R code I use is

while (nodd){
nsuite=suite=start=c(0,sort(2*sample(1:23,3)))
for (t in 1:20){
#some values of start never produce non-integer values
#of suite and never visit suite[1]=0
nsuite[-4]=(suite[-4]+suite[-1]+48*(suite[-1]-suite[-4]<0))/2
nsuite[4]=(suite[4]+suite[1]+48*(suite[1]-suite[4]<0))/2
suite=nsuite%%48
if ((suite[1]==0)||(max(is.decimal(suite))==1)){
nodd=(max(is.decimal(suite))==1)
print(t)
break()}
}}

This new version leads to solutions (unsurprisingly!), returning