## Le Monde puzzle [#818]

**T**he current puzzle is as follows:

Define thesymmetricof an integer as the integer obtained by inverting the order of its digits, eg4321is the symmetric of1234. What are the numbers for which the square is equal to the symmetric of the square of the symmetric?

**I** first consulted stackexchange to find a convenient R function to create the symmetric:

int2digit=function(x){ as.numeric(sapply(sequence(nchar(x)), function(y) substr(x, y, y)))} digit2int=function(a){ as.numeric(paste(a,collapse=""))} flip=function(x){ digit2int(rev(int2digit(x)))}

and then found that ~~all integers but the multiples of 10 are symmetric!~~ only some integers involving 1,2,3 and sometimes zero would work:

> for (i in 1:1000){ + if (i^2==flip(flip(i)^2)) print(i)} [1] 1 [1] 2 [1] 3 [1] 11 [1] 12 [1] 13 [1] 21 [1] 22 [1] 31 [1] 101 [1] 102 [1] 103 [1] 111 [1] 112 [1] 113 [1] 121 [1] 122 [1] 201 [1] 202 [1] 211 [1] 212 [1] 221 [1] 301 [1] 311

**I**f I am not (again) confused, the symmetric integers would be those (a) not ending with zero and (b) only involving digits whose products are all less than 10.

May 9, 2013 at 2:35 pm

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May 2, 2013 at 7:02 pm

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May 2, 2013 at 1:13 am

What am I missing?

14*14 = 196

41*41 = 1681

Isn’t this a counterexample to your claim?

May 2, 2013 at 6:40 am

thanks, I indeed made a (sickbed) mistake in checking for those symmetric solutions…

May 1, 2013 at 11:38 pm

‘all integers but the multiples of 10 are symmetric!’

? This is not true!

May 2, 2013 at 6:39 am

yes, thanks!, this is a lesson: never try R coding while under (hospital) drugs!