Squaring

First, a nifty shortcut! The square of a number ending in 5 is almost a no-brainer. If n is the number formed by the preceding digit/s (before the 5), get the product of n and n+1. Then just append 25 (i.e. 5 x 5) to this product.

For example, 752:
7 x 8 = 56; therefore solution is 5625.

Another example, 1152:
11 x 12 = 132; therefore solution is 13225

For other cases of squaring, the same shortcut techniques used in multiplication may be utilised. Especially the general-purpose Urdhva Tiryagbhyam (Vertically and Crosswise) formula. To get the square of a number (of two or more digits), simplify by splitting it into at least two parts, a and b.

Thus (a + b)2 = a2 + 2ab + b2

Squaring shortcut technique
The solution comprises three parts, neatly fitting the three boxes shown above. Just adjust for excess carry over.
  1. the head: a2
  2. the middle: crosswise multiplication and doubling a x b x 2
  3. the tail: b2

Here is a simple example to illustrate this technique.
232 = 529

Squaring technique example
The steps are:
  1. tail: 32 = 9, put it down in the rightmost box
  2. middle: 2 x 3 x 2 = 12, put down the 2 in the middle box and carry over the 1
  3. head: 22 = 4, plus the 1 carried over, is 5 in the left box

Another example.
1082 = 11664

Squaring technique example
The steps are:
  1. tail: 82 = 64, put down the 4 and carry over the 6
  2. middle: 10 x 8 x 2 = 160, plus the 6 carried over, is 166; put down the 6 and carry over the 16
  3. head: 10 x 10 = 100, plus the 16 carried over, is 116

The same technique can be expanded upon to handle the squaring of bigger numbers too.