Square up with Einstein

"HOW did you square up the product of mass and velocity of light to get E=mc², such a long figure?" Newton asks Einstein.

"It was not easy? I began with elementary numbers." says the great scientist.

"Would you teach me how?" Newton implores.

"Sure," says the old wise man, "Let's begin with numbers 11 to 19. Squares of 11 and 12 should be easy, but beyond 13, persons get unlucky."

lSquare the ones digit. This gives your last digit.

lMultiply the ones digit by 2. This gives your second digit.

lRetain the tens digit (i.e., 1) and append the above results.

Take 17², for example.

l7²=49. The 9 is your last digit. Carry the 4 to the left.

l7 x 2 = 14. The 4 (plus the 4 remaining from the first step) gives your second digit, 8. As before carry the 1 to the left.

lRetain the 1 from "17", add to it the 1 remaining from the previous step, and you get 289.

lSubtract the number you want to square from 100. (100 - x)

lTake this difference and subtract it from the number you want to square. [x - (100 - x)]. This is the first 2 digits of your answer.

lSquare the difference, the answer from step 1. This is the last 2 digits of your answer.

Take 96², for example.

100 - 96 = 4.

96 - 4 = 92. You're halfway there. 92__.

4² = 16. There's your last 2 digits. Thus, your answer: 9,216.

Try 88².

100 - 88 = 12

88 - 12 = 76. 76__.

12² = 144. What now? 88² isn't 76144! 92² was only 9216. Just carry the 1 over the 6.

lSquare the tens digit and add to that the ones digit of your original number. That's your first two numbers. Second, square your ones digit. That's your last two numbers.

Get it? Try it with 54. It should make sense.

5²+ 4 = 29. That's our first two numbers.

4² = 16. That's our last two numbers. Thus, the answer 2,916

Einstein replies:

lSubtract 100 from the number you want to square (x - 100)

lTake the difference and add it to the number you want to square. [x + (x - 100)]. This is the first 3 digits of your answer.

lSquare the difference, the answer from step 1. This is the last 2 digits of your answer.

Take 112², for example.

112 - 100 = 12

112 + 12 = 124. 124__.

12² = 144. Carry the 1 over the 4.

"Thanks professor, an apple struck me and I was worried ever since, trying to square a function in my gravity equation," Newton says.

"Gravity! hmm, attracts me, say do you want to square up with me, then tell me more about it," Einstein says.

— Aditya Rishi