## Le Monde puzzle [#738]

The Friday puzzle in Le Monde this week is about “friendly perfect squares”, namely perfect squares x2>10 and y2>10 with the same number of digits and such that, when drifting all digits of x2 by the same value a (modulo 10), one recovers y2. For instance, 121 is “friend” with 676. Here is my R code:

```xtrct=function(x){
x=as.integer(x)
digs=NULL
for (i in 0:trunc(log(x,10))){
digs[i+1]=trunc((x-sum(digs[1:i]*10^(trunc(log(x,10)):(trunc(log(x,10))-
i+1))))/10^(trunc(log(x,10))-i))}
return(digs)
}

pdfct=(4:999)^2
for (t in 1:5){
pfctsq=pdfct[(pdfct>=10^t)&(pdfct<10^(t+1))]
rstrct=apply(as.matrix(pfctsq),1,xtrct)

for (i in 1:(dim(rstrct)-2)){

dive=apply(matrix(rstrct[,(i+1):dim(rstrct)]-
rstrct[,i],nrow=t+1),2,unique)
if (is.matrix(dive))
dive=lapply(seq_len(ncol(dive)), function(i) dive[,i])
dive=as.integer(lapply(dive,length))
if (sum(dive==1)>0)
print(c(pfctsq[i],pfctsq[
((i+1):dim(rstrct))[(dive==1)]]))
}
}
```

which returns

``` 121 676
 1156 4489
 2025 3136
 13225 24336
 111556 444889
```

namely the pairs (121,676), (1156,4489), (2025,3136), (13225,24336), and (111556,444889) as the solutions. The strange line of R code

```    if (is.matrix(dive))
dive=lapply(seq_len(ncol(dive)), function(i) dive[,i])
```

is due to the fact that, when the above result is a matrix, turning it into a list means each entry of the matrix is an entry of the list. After trying to solve the problem on my own for a long while (!), I found the above trick on stackoverflow. (As usual, the puzzle is used as an exercise in [basic] R programming. There always exists a neat mathematical solution!)

### 6 Responses to “Le Monde puzzle [#738]”

1. Rick Says:

Hi,

I”m very interested to see if I can get this to work and understand the script. Unofrtunately I get the error message that object ‘rstrct’ is not found. Do you have some idea why that is?

Thanks!

• xi'an Says:

Rick: Sorry about this. The cut&paste from my vi window to the wordpress editor window is often missing bits, for no clear reason. I hope the code is now complete… (I checked it and it works on my machine!)

2. Le Monde puzzle [website] « Xi'an's Og Says:

[…] last, the mathematical puzzles proposed by Buser and Cohen in the weekend edition of Le Monde are available on a dedicated […]

3. Alan Parker Says:

I ALMOST tried to beat you to this, as an exercise for my elementary R. Finally, I didn’t make it, this time. However, I did notice that the puzzle starts from 2 figures, so doesn’t [16,49] count?

• xi'an Says:

Yes, indeed, I noticed in Le Monde tonight that they had included two figure numbers… Will need to check now!

• xi'an Says:

Easy fix: my loop starts with t=2, it should start with t=1! So the first results are
 16 49
 25 36

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