## Le Monde puzzle [#956]

**A** Le Monde mathematical puzzle with little need of R programming but ending up being rather fascinating:

Does there exist a function f from(NtoNsuch that (i) f is (strictly) increasing, (ii) f(n)≥n, and (iii) f²(n)=ff(n))=3n?

Indeed, the three constraints imply (a) f²(0)=0, hence that that f(0)=0, (b) f(1)=2 as it can be neither 1 (else f²(1) would be equal to 1, not to 3) nor 3 (else f(3)=f²(1)=3 would contradict both f(3)>f(1) and f²(3)=9), and thus (c) f(2)=f²(1)=3. Those ~~three~~ two initial values are actually sufficient to determine all subsequent values of f(n) as there are exactly three possible values between f²(n)=3n and f²(n+1)=3(n+1)=3n+3. (Kind of shaky argument, I must say, but each time an integer n has no antecedent in the previous f(m)’s, there are exactly two possible and successive values for two indeterminate f(n)’s… The resolution in Le Monde [published this afternoon] starts from the observation that f(3^{p})=2 3^{p} and thus that f(2 3^{p})=3^{p+1}, which leaves a single possible image for the values in between 3^{p} and 2 3^{p}.) It however provides no discussion whatsoever on the maths behind this freak function.To plot the above function, I used the R code below:

fillin<-function(N=100){ vals=rep(0,N) vals[1]=2 for (i in 2:N){ #anc for antecedent, f(anc)=i u=0;anc=which(vals==i) #no antecedent if (length(anc)==0){ u=1; anc=which(vals==i-1)} if (length(anc)==0){ u=2; anc=which(vals==i-2)} vals[i]=3*anc*(u==0)+(u>0)*(vals[i-u]+u)} }

The graph has many interesting features, one of which that explains why I did not plot the axes, namely that it is somewhat self-reproducing, in that the plot for N=10³ has the same pattern as the plot for N=10⁵, with a few long linear parts around the line y=√3x (since f is essentially an integer-valued version of √3x). Each linear part is about 3 times longer than the earlier one, with slopes of b=1 and b=3 alternating.

While the resolution is obvious for f²(n)=4n, since f(n)=√4n=2n, there is no single resolution when f²(n)=5n. *(Maybe there is one if instead the third condition is that f⁴(n)=5n… A new puzzle for the reader.)*

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