# Le Monde puzzle [#738]

September 1, 2011
By

(This article was first published on Xi'an's Og » R, and kindly contributed to R-bloggers)

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 2:6){
pfctsq=pdfct[(pdfct>=10^t)&(pdfct0)
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!)

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