by 6), its square ends in 0, and its preceding digit must be 0 or 3;
* if a number is divisible neither by 2 nor by 3, its square ends in 1, and its preceding digit must be even;
* if a number is divisible by 2, but not by 3, its square ends in 4, and its preceding digit must be 0, 1, 4, 5, 8, or 9; and
* if a number is not divisible by 2, but by 3, its square ends in 9, and its preceding digit must be 0 or 6.

Similar rules can be given for other bases, or for earlier digits (the tens instead of the units digit, for example).