This will also work for lesser combinations, for example, 11 x 11 = 121;
111 x 111 = 12321 on so forth up to a square of nine ones. It fails for
higher series as the summation of the multiplication rows for one column
requires a carry-over.
11
x11
---
11
11
------
121
When this column summation requires a carryover, the ascending/descending
sequence is broken.
Squaring a series of twelve 1's (111111111111 x 111111111111) yields
12345679012320987654321.
As can be seen, the series breaks down on the column where a carry-over
occurs. Of course, switching to base 16 (hexadecimal) should allow the
sequence to be maintained.
Interesting observation. You should send it in to the author of one of
these books on math shortcuts. One never knows when they will be asked for
the square of 111111 (12345654321)!
Best Regards,
-JBJ-