The level in this article is for college students familiar with calculus, This material will be also of interest to college professors looking for new material to teach, or for original exam questions, as well as for business data scientists with some spare time, interested in refreshing their math skills. The problems cover real analysis, mathematical algorithms and numerical precision, correct visualizations, as well as geometry. The third problem is the most interesting one in my opinion, and could become a subject of active mathematical research with one new great, unsolved conjecture being proposed, of a probabilistic nature. The last problem has many applications in engineering science.

**This article is structured as follows**:

1. The Simplest Function Defined by an Infinite Product

- Exercise

2. Surprising Series for Powers of Number 2

3. From Continuous Fractions to Nested Square Roots and More

- Algorithm to compute the coefficients
- Problems
- Example: Nested Square Root for the Number Pi
- Conjecture

4. Geometry: Shape Rearrangements and Coverage Problems