Squares Can Be Computed Squentially

In case A is a successor of a number with a known square, you find A⊃ by adding to the latter itself and then A. For example, A = 111 is a successor of a = 110 whose square is 12100. Added to this 110 and then 111 to get A²:

Why does this work?

If A = a + 1, then A² = (a + 1)² = a² + a + (a + 1).

Another example. Let A = 46. Then a = 45. It's easy to find 45² = 2025. It then follows that A² = 2025 + 45 + 46 = 2116.