Math lesson for secondary school [closed]

I have to deliver a lecture for secondary school, about one of this topic: integers, Euclid's Elements, polyhedra, prime numbers, non-Euclidean geometry, arithmetic functions, graphs.
It should be something interesting and easy to understand for high-school students, and 45 minutes long.
My ideas are: platonic solids, something about prime numbers (but don't know what exactly), golden ratio. They should be good but they seem to me not very original.
Do you have any suggestion? Maybe an interesting problem solved using one these mathematical tools? Or a particular aspect of one of these that could catch the eye? Thank you

closed as too broad by Daniel W. Farlow, Namaste, Siong Thye Goh, JonMark Perry, user91500Aug 31 '17 at 8:47

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• May be better to ask the question at matheducators.stackexchange.com – Claude Leibovici Aug 30 '17 at 9:39
• @Pickle The question, as it stands, is perfectly consistent with user xyzt being competent in those subjects (in fact, he plans to give a lecture about them), but unsure about his choice being sound for the audience. For instance: "Would those things go over their heads? Would they deem them interesting? Should I do something else entirely? Are 45 minutes enough time to explain them properly?". If anything, this seems quite a scrupulous attitude to me. – user228113 Aug 30 '17 at 10:10
• How about give a proof that $p_n\le 2^{2^n}$, where $p_n$ is the $n^{th}$ prime, to begin with? It involves mathematical induction and prime numbers, which the former gives a good strategy of proofs, while the latter along with the result gives a cool results that the prime numbers can actually be bounded. – BAI Aug 30 '17 at 10:48
• @xyzt the bound $2^{2^n}$ as said above could just be proved by induction with a trick inspired by Euclid's proof that there are infinitely many primes (i.e. $p_n\le 1+p_1p_2\ldots p_{n-1}$). However, I'm happy to say that, there is also an elementary proof of the bound $p_n\le 4^n$ using combinatorics. If your lecture would focus on number theory, it would also be a good example to introduce the students the relation between number theory and combinatorics. – BAI Aug 30 '17 at 15:08
• @xyzt plus a well-done Paper on elementary methods in the study of the distribution of prime numbers. – BAI Aug 30 '17 at 15:14