If you forgot an entry.

Say, you want a square of 13. Do this: add 3 (the last digit) to 13 (the number to be squared) to get 16 = 13 + 3. Square the last digit: 3² = 9. Append the result to the sum: 169.

As another example, find 14². First, as before, add the last digit (4) to the number itself (14) to get 18 = 14 + 4. Next, again as before, square the last digit: 4² = 16. You'd like to append the result (16) to the sum (18) getting 1816 which is clearly too large, for, say, 14 < 20 so that 14² < 20² = 400. What you have to do is append 6 and carry 1 to the previous digit (8) making 14² = 196.

More examples and explanation