Sometime in the upcoming future, I will be doing a presentation as a college alumni to a bunch of undergrads from an organization I was in college. I did a dual major in mathematics and computer science, however the audience that I am presenting to are not necessarily people who enjoy math. So, to get their attention, I was thinking about presenting an interesting problem in math, for example, the birthday problem, to get their attention and have them enjoy the field of math a little.
I feel a question in the field of probability would interest them the most (due to its instinctiveness) , although that's just a personal opinion. The audience studies a variety of majors, from sciences to engineers to literature and the arts.
So here's my question, besides the birthday problem, are there any other interesting problems that would be easy to understand for people who have limited knowledge of calculus and would, hopefully see math as an interesting subject and get their attention? (Doesn't have to be in the field of probability.)
Here is what I think is a very interesting problem. Little children know what "one half" means, and maybe even "one third," etc. When we study fractions, the basic idea is that there are quantities in between the counting quantities. After all, "one half" is really a quantity too. So how can we denote such a quantity? The ingenious scheme is to think of the quantity "one" broken into a counting number of equal parts like, for example, two equal parts. Then we can describe some other number of those parts by another counting number. These TWO counting numbers together then represent a quantity smaller than one, e.g., one out of two equal parts. Next comes the question of how to denote such a thing. So here is the problem: There are different ways we could denote the two counting numbers, e.g., 1|2 meaning "one our of two equal parts" or 1↓2 etc. So why do we denote a fraction by DIVISION of those two counting numbers, with the total number of equal parts as the denominator and the desired number of them as the numerator? In other words, does DIVISION really have anything to do with representing a quantity as a number of equal parts? Is the use of division ingenious? Also, division is the inverse of multiplication. Is the division notation for a fraction the inverse of anything? One purpose of this question is to point out that even though we all think we understand all about fractions, we actually may not. Another is to point out that mathematical definitions are not arbitrary. They are sensible. There are ALWAYS DERIVATIONAL reasons for them. A third is that although we are all comfortable with saying that mathematics is "abstract," quantity is perfectly concrete, as is the mathematical development of its aspects.