Squaring, Cubing, and Cube Rooting

I still recall my thrill and disappointment when I read Mathemat-ical Carnival [4], by Martin Gardner. I was thrilled because, as my high school teacher had recommended, mathematics was presented in a playful way that I had never seen before. I was disappointed because it contained a formula that I thought I had “invented ” a few years earlier. I have always had a passion for mental calculation, and the following formula appears in Gardner’s chapter on “Lightning Calcu-lators. ” It was used by the mathematician A. C. Aitken to mentally square large numbers. Squaring Aitken took advantage of the following algebraic identity. A2 = (A − d)(A + d) + d2 Naturally, this formula works for any value of d, but we should choose d to be the distance to a number close to A that is easy to multiply. Examples. To square the number 23, we let d = 3 to get 232 = 20 × 26 + 32 = 520 + 9 = 529. 1 2To square 48, let d = 2 to get 482 = 50 × 46 + 22 = 2300 + 4 = 2304. With just a little practice, it’s possible to square any two-digit num-ber in a matter of seconds. Once you have mastered those, we can quickly square three-digit numbers by rounding up and down to the nearest hundred