The sum from 1 to n is easy various ways, a matter of personal preference. One can "make rainbows", pairing 1 with n, 2 with n-1, ending up with n/2 copies of n+1. I prefer to tear an n by n+1 rectangle of squares in half down its diagonal, to create two copies of 1 + 2 + ... + n.

Gauss famously summed the integers to 100 in school as a kid. It's a litmus test for mathematicians, ask them how they think he did it.

The sum of cubes 1 + 8 + 27 + ... + n^3 happens to be the square of this formula, making it easy from here. Arrange cubes in 4-dimensional space filling out a shape which is the product of two copies of the above triangle of squares illustrating 1 + 2 + ... + n. In other words, working in wxyz space, the shadow in both the wx and yz planes is this triangle. Now rearrange these cubes to form a pyramid with an n by n by n base directly illustrating 1 + 8 + 27 + ... + n^3.

Mathematics is an extraordinary technical accomplishment allowing people to instrument-fly airplanes with the windows painted over. Some of us, most famously Bill Thurston, prefer to look out the window. This exercise is a great example for "feeling" mathematics rather than working around a failure of cognition using formulas.

AI is putting mathematics into crisis. Soon we'll all have AI technical superpowers, and the only meaningful way any of us can excel is by having actual creative vision.

https://m.youtube.com/watch?v=4AuV93LOPcE&pp=ygURbWF0aG9sb2d...

Mathologer has an amazing video on this technic.