Wondering about Squares and Cubes

G is my friend who comes home, to get help in mathematics. He is extremely passionate and crazy about cricket. He spends close to 5hrs in the ground with his district team. Well, he is in class 8. When we met for the first time, he said he is not able to follow anything in math class, so he usually shuts off. He would be glad if I can help. "My pleasure". I love one-one time specifically for learning. G doesn’t talk much. I must be very sensitive to his very subtle facial expressions to know what he feels. And most of the times he is tired after his cricket practice.


It was a sunny evening and G walked in at 6pm after his field practice. We sat down and after over a cup of fresh mint juice, started with cubes and cube roots (chapter7 NCERT). I was telling him about cubes what they are… And G gave me this very deep thoughtful look!. “why am I learning this?” he slowly asked, Is it really important to know this???????????. I was not sure how answer that question. To me numbers are always exciting, as exciting as playing cricket was for G.

There are just 4 perfect cubes from 1-100. 
1  =1x1x1
8  =2x2x2
27=3x3x3
64=4x4x4

Guess what? cubes were in news recently with a major breakthrough. Diophantus of Alexandria, a 3rd century mathematician had posed the problem of sum of 3 cubes.

x³+y³+z³=k, with k being all the numbers from one to 100.

Looks simple?, well not really. Enjoy reading
This was in 1994
http://www.asahi-net.or.jp/~KC2H-MSM/mathland/math04/matb0100.htm 

Note the "???" for some numbers 

In March 2019 Andrew Booker solved the sum of 3 cubes for 33In September 2019 Professors Andrew Booker and Andrew Sutherland found the solution for 42 by using Charity Engine
https://phys.org/news/2019-09-sum-cubes-solvedusing-real-life.html

Watch them in action!!!
for 42
for 33
for 74