Squaring Two-Digit Numbers

To square 2-digit numbers follow the following algorithm.

Select an arbitrary number, for example 48 to follow:

1. Look for the nearest 10 (ceiling) boundary; for 48 it would be 50.

2. Take d = 50 - 48; so d = 2.

3. Take A = 50 * (50 - 2 * d) = 50 * 46 = 2300.

4. The answer is equal to A + d^2 = 2300 + 2 * 2 = 2304.

48 * 48 = 2304

Another example:

Select an arbitrary number to mentally calculate its square, for example 87.

1. Look for the nearest 10 boundary; for 87 it would be 90.

2. Take d = 90 - 87; so d = 3.

3. Take A = 90 * (90 - 2 * d) = 90 * 84 = 7560.

4. The answer is equal to A + d^2 = 7560 + 3 * 3 = 7569.

87 * 87 = 7569

Now let's try it for 73:

1. Look for the nearest 10 boundary; for 73 it would be 80.

2. Take d = 80 - 73; so d = 7.

3. Take A = 80 * (80 - 2 * 7) = 80 * 66 = 5280.

4. The answer is equal to A + d^2 = 5280 + 7 * 7 = 5329.

73 * 73 = 5329