For example, calculating square of $654$ using Duplex Method as given below
(reference: Speed Maths Square Calculator)
Is duplex method a reliable method to calculate square and can this be proven mathematically? Where can I find all such methods and is there any special area of mathematics which focus on speedy calculations? I am interested in this because if such methods are available, studying them will give an upper edge for my exams.
Yes, it works for squares of three digit numbers of the form $(100a+10b+c)^2$
Multiplying out and then grouping by powers of $10$ gives $10000\times a^2+ 1000 \times 2ab +100 \times (2ac+b^2) +10 \times 2bc + 1 \times c^2$
A more naturally symmetric form might be
$10000\times a^2+ 1000 \times (ab+ba) +100 \times (ac+b^2+ca) +10 \times (bc +cb)+ 1 \times c^2$
There are natural extensions for squares of more digits
Whether this is a speed improvement on other methods or just using a calculator are different issues
Here is a competitive method for the case at hand, that might inspire you for other calculations.
$$\tag{1}654^2=\underbrace{(650+4)^2}_{(a+b)^2}=\underbrace{650^2+8 \times 650+4^2}_{a^2 + 2 a b + b^2}=65^2 \times 100 + 8 \times 650 + 4^2$$
Among the 3 terms above, the last 2 can be easily computed.
The computation of $65^2$ is also easy, knowing the following rule for squaring numbers terminated by 5:
$$\tag{2}(\overline{N5})^2=\overline{(N\times(N+1))25}$$
(examples: $25^2=\overline{(2\times3)25}=625), \ 35^2=\overline{(3\times4)25}=1225$...)
Conclusion: (1) gives $\underbrace{\overline{4225} \times 100}_{422500} \ + \ + \ 5200 \ + \ 16 = 427716.$
Exercice: prove (2).