I need some help guys i am currently working on a essay base

I need some help guys, i am currently working on a essay based of mathematical puzzles such as the fibonnaci sequence, pascal triangle, golden ratio,etc and the effectiveness of it and how these puzzles can be implimented. i need help with showing some type of realtions between this puzzles and mathematics. for example the pascal triangle can help with binomial expansion. I really need some more examples in this manner. Thanks in advance!!

Solution

Where we know Pie or p (3.14) can be defined as the ratio of the circumference of a circle to its diameter, Phi or (1.618) can be said Golden Ratio that gives the result when a line is divided in one very special and unique way. For illustration , suppose we were asked to take a string and cut it. There is any number of places that we can cut it in different pieces , and cutting will result in the formation of different ratios formation for the length of the small piece to the large piece, and of the large piece to the entire string. There is always one unique point , at which the ratio of the large piece to the smaller piece is always and exactly the same as compared to the ratio of the whole string to the larger piece, and at this point this Golden Ratio of both is 1.618 to 1, or Phi.