12345679*9=111111111
41152263*3=123456789
13717421*9=123456789

http://www.cut-the-knot.org/arithmetic/NumberCuriosities/MoreSquares.shtml

13844502×4

456+654=1110
258+852=1110
159+951=1110
753+357=1110
741+369=1110
147+963=1110
789+321=1110
987+123=1110

If a number ends in 5 then its square always ends in 25. To get the rest of the product take the left digit and multiply it by one more than itself.

35×35 ends in 25. We get the rest of the product by multiplying 3 by one more than 3. So, 3×4 = 12 and that’s the rest of the product. Thus, 35×35 = 1225.

To calculate 535×535, notice that 53×54 = 2862 and write down 286225 as the answer.

To approximate the square root of 5555555555, you can note that 1/9= .111111111…. and thus 5/9=
.555555555…..

Thus, up to small error and a shift of the decimal point
we know that the square root in question is equal
to one third of the square roof of 5.

Now, by standard elementary school methods, or by using a memorized log table, or by imagining a slide rule, you know the square root of five to a bunch of decimal places.

Thus, you can quite a few places in your head, quite quickly.

Used to do this sort of thing for entertainment when I was a child.

p.s.

When you use a slide rule a lot, it becomes pretty easy to see one in your head. And if you took
trig before calculators, there was a lot of solving triangles using nasty arithmetic ( law of sines, tangents etc.) that was done with log tables. After a few weeks, you basically knew the log table (to three
places) by heart, and simple interpolation ( which you did constantly) gave you four places. Or, you could notice that the five place log table was easy to memorize ( though large) by starting with a memorized three place log table.

So, you could do slide rule type stuff in your head.

The absolutely best book on mental calculation
, in a class of its own, is titled ” Dead Reckoning”.

Uparrow means to the power of.

Note that (not counting 1, which today is considered a prime) this is the sum of the squares of the first seven
primes.

Without a telescope, there were seven planets:
back in the day that was: Sun, Moon, Mercury,
Venus, Mars, Jupiter and Saturn.

Which is to say: The true origin of the mystic number 666 is in the numerology cult of Pythagorus,
which predates the Christian Bible.

It is not in greek gematria. And has nothing to do with beasts or devils or antichrists.

p.s. two thirds is NOT = .666
It equals .6666…..

12345679 X 9 = 111111111
so
12345679 X 18 = 222222222
12345679 X 27 = 333333333
…