<s>[INST] Does the sum of digits squared of a number when we keep on doing it lead to a single digit number? When we add the digits of number for eg $2478125$
$$2+4+7+8+1+2+5=29;\\
2+9=11;\\
1+1=2$$
and when we square the digits and add them following this rule
$$2^2+4^2+7^2+8^2+1^2+2^2+5^2=163;\\
1^2+6^2+3^2=46; \\
4^2+6^2=52;\\
5^2+2^2=29;\\
2^2+9^2=85;$$and this goes on, so the question is 
will it ever reach a single digit.And what is the proof of it.
 [/INST] The answer is yes, it's the subject of this year's french olympiads : for one or two digits numbers, you prove by hand (or computer) verification that the resulting sequence is either stationnary at $1$, or loops indefinitely from $4$ to $16$, $37$, $58$, $89$, $145$, $42$, $20$, $4$ and so on.
Then you prove it for 3 digits numbers (again by hand verification), and finally, prove that if $x$ has $p\ge4$ digits, then $f(x)$ has at most $p-1$ digits.
There may be smarter solutions for $2$ and $3$ digits numbers, for example construct a graph, because the order of digits is irrelevant.
</s>