ROHIT SINGH, a Class IX student who lives in Amardass Nagar of Jalandhar, sent us a letter when he was in the previous class. "I want a method to find the cube root of any number without using log tables, a calculator, a computer or the conventional procedure," he says. The answer comes with a statutory warning that martial-arts star Bruce Lee had received from his teacher: "Use this knowledge only in emergency — for self defence or protecting others."

Armed with this knowledge, you can find the cube root of any number N between -1,000,000 and +1,000,000, the cube root of which is an integer. Suppose N = 778,688.

Step 1: The Sign: N has the same sign as its root. Step 2: Tens Digit: Mostly estimation. Divide N by 1,000 and take the quotient. (the hundred-thousandths digit, ten-thousandths digit, and thousands digit) In this case, it's 778. The tens digit is the largest number between 0 and 9 whose cube is less than this number. Since 9^3 = 729 < 778; 9 is the tens digit.

Step 3: The Ones Digit: If we presume the ones digit of the root to be 'x', then, there is only one x between 0 and 9 such that the ones digit x^3 is the same as the ones digit of N.

Find the number between 0 and 9 which, when cubed, has the same ones digit as N.

0^3 = 0; 1^3 = 1; 2^3 = 8; 3^3 = 27; 4^3 = 64; 5^3 = 125; 6^3 = 216; 7^3 = 343; 8^3 = 512; 9^3 = 729. The ones digits of the cube are the same as that of the root, except that 3 and 7 have 7 and 3 as ones digits, and that 2 and 8 have 8 and 2 as ones digits, respectively. Since N ends in 8, the ones digit is 2. Therefore the cube root of N = 92.

The last digit of 2^3 is 8, 3^3 is 7 and vice-versa. Everything else repeats itself.

Procedure II for finding a cube: Start from right and put a comma when three digits are over: 9,261, 1,728, 32,768, 175,616. After putting the comma, see the last digit of the number and compare it with the list of last digits given earlier. You get the last digit.

After this, see the first group of numbers and ascertain cube of which number is less than the group. That number is your first digit. You have, thus, found the first digit and the last digit. Take 9261: Counting from last, we put a comma after 9. The last digit gives the last digit of cube root to be 1. Step 2: we see 9 and deduce that 2^3 = 8, is less than 9 and 3^3 = 27 is more than this. Our first digit, thus, comes to 2 and the answer is 21. Take 32,768. The last digit gives the last digit of cube root to be 2. By seeing 32, we put 3, as our first digit (3^3 = 27) is less than 32 and 4^3 = 64 is more than this. The answer, thus, is 32. You will prefer the conventional long Newtonian method to the short cut in advanced mathematics and engineering courses. Write at The Tribune or adityarishi99@yahoo.co.in.

— Aditya Rishi