Goutham wrote: A positive integer is called monotonic if when written in base $10$, the digits are weakly increasing. Thus $12226778$ is monotonic. Note that a positive integer cannot have first digit which is $0$. Prove that for every positive integer $n$, there is an $n$-digit monotonic number which is a perfect square. This one becomes from a BaMO or JBaMO: Numbers of the form $333\cdots 3 5$ satisfy for an even number of digits for the monotonic number $(\ge4).$ $9,16,169$ for $1,2,3.$ ANd $166666\cdots67$ satisfy for odd numbers.

I don't understand, what do you mean numbers of the form 33335? neither of 3335,333335 and 33333335 are perfect squares. Or perhaps I'm not understanding correctly.

$33\ldots 35^2=11\ldots 122\ldots 225$. So the square with $n$ 1's, $n+1$ 2's and one 5 is monotonic. That's what he means by that. And similarly $166\ldots 67^2=277\ldots 78\ldots 889$, so this is also a monotonic square: one 2, $n$ 7's, $n+1$ 8's, and one $9$.