The Magic of Cube Roots & trail to Ramanujan’s Number

A tale of mental math for the perfect cube roots. And a surprise awaits at the end of the “cube” trail.

I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavourable omen. “No,” he replied, “it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.”

It was a Sunday evening in Trichy. The kids were playing a game of stones. Every time she visited them, she had something interesting to show. Today, it was about “cube roots” of six-digit numbers. With the aid of a calculator, the kids were trying their best to ask her really tough questions she was mentally figuring it out it in a jiffy.

Rama was watching all of this with his mouth wide open. “Wow! This is amazing. She is so fast. How does she do it? It must be very hard. Is she a genius?”, were his thoughts.

Slowly, Rama approaches Isha. “That was just brilliant and cool. Do you think I could do it too?”

Isha could see the gleam of curiosity in Rama’s eyes. “Of course, you can do it, Rama! With practice, you will be able to do it with lightning speed.”

“And numbers are like lego bricks. Make them your friends. Just play with them and you never know, what you will build out of them one day.”

Not a hard problem

“Rama, Let us break the problem into smaller steps.”

“The 6 digit numbers like 941,192 might look big.. Don’t get scared by them. Their cube roots are just 2 digits. 100³ is 1,000,000 which is a 7 digit number. What does that mean? Our codeword AB has to be a 2 digit number. It is all about finding these 2 magical digits A and B. Like the pile of 2 pebbles, it is about quickly assessing the count in each.”

Rama was pleased to hear this.

The first 10 cubes

“Let’s examine the cubes of the first 10 numbers. They have to be 3 digit numbers all less than 10³.”

0  000
1  001
2  008
3  027
4  064
5  125
6  216
7  343
8  512
9  729

“There is something very very unique and interesting about the last digit of the cubes. It is either the number itself or the 10’s complement.”

“The numbers 0, 1,9 and the numbers 4,5,6 have their cubes end in the same number.”

Same Number
0 000
1 001
9 729
4 064
5 125
6 216

“And for the remaining numbers 2,3,7,8, their cubes end in their 10’s complement (ie 2 + 8 = 10, 3 + 7 = 10)”

10’s Complement
2  008
3  027
7  343
8  512

Rama was thrilled to observe this. “Oh wow, such a unique and nice mapping. I tried it for a few numbers in my head and it is actually true.”

The unit digit B

“From what we just saw, we can conclude that in order to find the unit digit B, we only care about the last digit of the cube. I told you, it is very easy. Try the unit’s digit B for these numbers.”

064,000   (0)
001,331   (1)
042,875   (5)
970,299   (9)
238,328   (2)
389,017   (3)
658,503   (7)
941,192   (8)

Rama was excited that he was able to get 50% of the problem.

The ten’s digit A

“Now that we know the unit digit, let’s see how we can predict the ten’s digit. If I were to ask you to guess the cube root of 68,921 what would it be?”, asked Esha.

“My guess would be 4. The number is greater than 64,000 which is 40³ and lesser than 125000 which is 50³”, replied Rama.

“You got it right Rama. For deducing the ten’s digit, we only care about the higher 3 digits of the cube number. And we check which is the highest cube lesser than this. While checking A for 042,875, we know that 3³ < 42 < 4³and hence A should be 3. Similarly for 068,921, we know 4³ < 68 <5³ and hence it should be 4.”

042,875   (35)
068,921   (41) 64 < 68 < 125

Rama was a very curious child, but he took time in understanding and appreciating things. And memorising numbers was not exactly easy for him.

“Isha, this is so cool. And not too hard as I had earlier imagined. All it needs is to remember the first 10 cubes. I am fine with 1, 8, 27, 64, 125.”

“However, I need to make friends with these big numbers. Can you help me? I need help with 216, 343, 512, 729.”

6  216
7  343
8  512
9  729

“Sure Rama, let’s dig open these big numbers and search for some hints. We may find something that we like and make friends with them. We already have an idea of their unit digits.

“8³ is 2⁹ which is half of the famous 1024. That is how I remember 512.”

“9³ has 9s embedded deeply. 729 has 72 and 9 both divisible by 9 which is 81 which is 9². That is how I remember 729.”

“6³ & 7³ are in the 200 and 300 range. Their endings have to be 6 and 3. 216 sums up to 9 and 343 is a palindrome with 3+4 as 7. Not a great story, but if it helps.”

“I get it.. 216, 343, 512, 729.. My cube mates..”

“Let try out a few. As you keep doing them, you will get better at it.”

Summary,
For the unit’s digit, check the last digit.
0, 1, 4, 5, 6, 9 yields in the same number, and 2, 3, 7, 8 yields in 10’s complement.

For the ten’s digit, range check the upper 3 digits.
1, 8, 27, 64, 125, 216, 343, 512, 729, 1000.

238,328   (62) 216 < 238 < 389
389,017   (73) 343 < 389 < 512
658,503   (87) 512 < 658 < 729
941,192   (98) 729 < 941 < 1000

Esha was happy to find little Rama solving the cube roots mentally. He was a little slow at first, but with Esha’s help, he became good at it.“Bravo, you have mastered the cubes well. They are like weapons cutting through the woods of numbers. Did you see how easily you could solve the cube roots in your mind?”

Next visit, Esha has another topic. “Let’s step up to higher numbers. Number 11 is like a mirror staircase going up and down. Looks at its powers. 11, 121, 1331.”

“12² is 144, 12³ is 1728.”

“Isha, I think my number-friends are telling me something. Did you just say 1728?”

1728 is 12³ and 729 is 9³. There is a difference of order 1000 and 1. Can they be related? Rama was elated at his findings.
1729 = 1728 + 1 = 12³ + 1³
1729 = 729 + 1000 = 9³ + 10³

“1729 can be put as a sum of 2 cubes in 2 different ways. Am I right Isha?”

One of the Infinite ways to come up with a story. I have talked to so many kids about the magic of cube roots. I don't even recollect where I have first heard about it.

He had actually discovered this property long before this incident. He had discovered this number when he was working with Diophantine equations of a particular form. This is taken from one of his notebooks:

https://www.quora.com/What-is-unique-about-the-Ramanujan-number