The Number Theory Of '1'

From the hint, we note that:

$\begin{array} {rcl} \underbrace{1}_\color{#D61F06}{1} \times 1 & =& \color{#D61F06}{1} \\ \underbrace{11}_\color{#D61F06}{2} \times 11 & = & 1\color{#D61F06}{2}1 \\ \underbrace{111}_\color{#D61F06}{3} \times 111 & =& 12\color{#D61F06}{3}21 \\ \underbrace{1111}_\color{#D61F06}{4} \times 1111 & =& 123\color{#D61F06}{4}321 \\ \implies \underbrace{1111111}_\color{#D61F06}{7} \times 1111111 & = & \boxed{123456\color{#D61F06}{7}654321} \end {array}$

Not a solution, but why is none of the above at the top?

If: x is any digit from 1 to 9

Then: ( x ones)×( x ones)=123... x ...321

So: 1111111×1111111=(7 ones)×(7 ones)=1234567654321

The final answer is: 1234567654321

Count The Number Of One(s) In The Number. That Is The Middle Number

For The Question, There Is 7 One(s) In Each Multiplication. So The Middle Number Is 7

And The Numbers Of 123... Will Increase Until It Reaches 6.

Then Add The Number 7 Behind It And Decrease The Number Like This :1234567654...

And You Will Get Your Answer 1234567654321