Consider the first 10 square numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.

Now look just at the units digits: 1, 4, 9, 6, 5, 6, 9, 4, 1, 0. Notice the reverse symmetry with the group 1, 4, 9, 6 repeated after the 5.

In those 10 units digits four integers are repeated twice therefore four of the integers 0 - 9 must be missing. These are of course 2, 3, 7, 8.

Now consider the squares of numbers greater than 10.

*The units digit of the product of two numbers is the product of the units digits of the two numbers.*

Consider 237^{2} . The units digit of the answer just comes from 7*7 and we know the units digit of the answer is 9.

We only need consider the units digit of the number to be squared to determine the units digit of the answer. We already know the results for the first 10 numbers and therefore the pattern will be repeated infinitely.

*The characteristic of the square numbers is that the units digit will never be 2, 3, 7 or 8.*