Please see the attached file for the fully formatted problems.
Problem 3: This problem itself is directly creating perceptual curiosity and at the cognitive level it is against what is known in the mathematics.
The problem is:
13+23+33+.... +n3 = (1+2+3+...+n) 2
Visual representation of problem:
Sum of the cubes equal to square?
13+23+33+...+ n3 = (1+2+3+...+n) 2
+ + +...+ =
Direct proof is required. HOWEVER, Proof by Induction IS NOT wanted as the solutions, that is:
1-For n=1 is true
2- For n = k assumed to be true
6-Show for n = k+1 is true
THIS SOLUTION IS NOT ACCEPTED. WE DON'T WANT THIS SOLUTION
A proof is offered equating a sum fo cubes to an expression of squares.