Le Monde puzzle [#737]

The puzzle in the weekend edition of Le Monde this week can be expressed as follows:

Consider four integer sequences (xn), (yn), (zn), and (wn), such that

x_0=0 < y_0 < z_0 < w_0 <48

and, if u=(xn,yn,zn,wn), for i=1,…,4,


if ui is not the maximum of u and

u_{i}=\frac{1}{2}\left(u_{i}+48+\min_{j\ne i}u_{j}\right)\,\text{mod}\,48

otherwise. Find the first return time n (if any) such that xn=0. Find the value of (y0,z0,w0) that minimises this return time.

The difficulty stands with the constraint that the sequences only take integer values, which eliminates a lot of starting values. I wrote an R code that corresponds to this puzzle:

while (nodd){

plot(clock[1,],clock[2,],type="l", axes=F,xlab="",ylab="")

for (t in 1:10^5){

  for (j in 1:4){

   if (suite[j]<max(suite)){

  plot(clock[1,],clock[2,],type="l", axes=F,xlab="",ylab="")
  for (j in 1:4)

  if ((suite[1]==0)||(max(is.decimal(suite))==1)){

but it fails to produce a result, always bumping into unacceptable starting values after one or two (rarely three) iterations! So either the starting conditions are very special out of the 23*22*21/6=1771 possible values of sort(2*sample(1:23,3)) or…I missed a point in the original puzzle.

2 Responses to “Le Monde puzzle [#737]”

  1. […] weekend puzzle in Le Monde this week is again about a clock.  Now, the clock has one hand and x ticks where a lamp is either on or off. The hand moves from […]

  2. […] 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 […]

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