Compute 1030 * 970 in one step and state the identity that makes it possible. What condition on the two numbers has to hold before you can use it?

Compute 1030 * 970 in one step and state the identity that makes it possible. What condition on the two numbers has to hold before you can use it?

Approach: Check whether the two numbers are equally spaced around a round midpoint, then replace the product with one square and one subtraction.

999100. Both numbers sit 30 from 1000, so the difference of squares applies: (1000 + 30)(1000 - 30) = 1000^2 - 30^2 = 1000000 - 900 = 999100. Any pair with midpoint m and half-gap d multiplies to m^2 - d^2, which replaces a three-digit product with one square and one subtraction. The condition is that the midpoint (a + b)/2 has to be a number you can square instantly, so the trick pays only when the two numbers straddle a round value. Here the midpoint is 1000 and the half-gap is 30. The same identity handles 47 * 53 as 2500 - 9 and 88 * 92 as 8100 - 4.

Follow-up: Two numbers average 1000 and differ by 74. What is their product, and how much harder does the half-gap of 37 make it?

Key concepts: difference of squares, midpoint, half-gap.