(111111111 111111111) ^2 are
012345679012345679*999999999999999999
012345679012345678<--- End digit 9 is reduced by a-unit as 8
987654320987654321<---- All digits are vertically made equal to 9
Merging of equal start-digits and end digits from two halves makes (more than 9 times 1...1) ^2. All 'step of 9'-pairs will work like this!

Let us extend said application to (222222222 222222222) ^2
049382716049382716*999999999 999999999 is above square
049382716049382715<--- End digit 6 is reduced by a-unit as 5
950617283950617284<--- All digits are vertically made equal to 9.
Merging of equal start-digits and end digits of two halves makes an (more than 9 times 2...2) ^2
Let us further extend it to (333333333 333333333) ^2 ...
111111111111111111*999999999 999999999 is above square.
111111111111111110<---- End digit 1 is reduced by a-unit as 0
888888888888888889<---- All digits are vertically made equal to 9.
111111111111111110 888888888888888889. Merging of equal 'start digits' and 'end digits' of two halves makes a (more than 9 times 3...3) ^2. A 1111^2 is (11108889), which is effectively 1111*9999

We can endlessly expand similar applications by linking higher order matrixes 00...99, 000...999, 0000...9999 so on, which is so simple! Related principle is "each said answer relates a concerned matrix"!

A sub-sutra (sishyate seshasamjna-->remainder) hints at remainders
(10- 3^2), (1100-33^2), (111000-333^2) ... has 1, 11, 111...so on!
(40- 6^2), (4400-66^2), (444000-666^2) ... has 4, 44, 444...so on!
(90- 9^2), (9900-99^2), (999000-999^2) ... has 9, 99, 999...so on!

Matrixes 0...9, 00...99, 000...999or a higher order have great inter-relations. A sutra relates computing in unrelated manners, which is not entirely paradoxical! Ekadhikena is a very good example to it. Various utilities actually help good learners to memorize variations by a language-sutra! Modern computing principles are already huge and ever expanding, which is unlikely to remain as a public-awareness!