Please refer to the Chinese webpage

Given any regular k-gon, what is the smallest regular n-gon that encloses it?  This natural problem has been solved by Dilworth and Mane in “On a problem of Croft on optimally nested regular polygons” (2010), and our project has re-proved the following results by more elementary methods:
(i) n = 3
(ii) n = 4
(iii) Both n and k are even.
Our methodology may not be generalizable to tackle the other cases where at least one of n and k is odd.

As a lonely introvert, it is important that I should always keep distant from strangers while I’m in public, to avoid potential awkwardness. What if, everyone in public is an introvert, and they all want to keep distant from others? Where would they choose to stand, to keep themselves far from other people? And this is the purpose of our project. If we have n introverts together in the same room, we want to investigate about how we can arrange their positions, to maximize the minimum pairwise distance.

In the project, we explored some basic properties of integer triangles and Heronian triangles. In part 1 and 2, we explored the general form of integer triangles given an angle and generated the lists of integer and Heronian triangles with special angles with the help of programming. In part 3 and 4, we investigated the combinations of putting two integer triangles together into a larger integer triangle or a cyclic quadrilateral.

Please refer to the Chinese webpage

Chess is a complicated game consisting of the opening, middle, and endgame. There are certain patterns to remember that can guarantee victory over others, but variations are limited in endgames. With patterns arising, an interesting question arises – is there a way to connect math equations with endgame patterns? In this project, we found out how many moves it would take for a knight to capture a pawn without looking at the amount of squares between them. Also, we have tried to figure out the general equation for an infinite chessboard that can calculate the amount of moves.

Please refer to the Chinese webpage

Through the project, we hope to explore the general solution to an engaging IQ question, leveraging fundamental concepts from secondary school mathematics. The question includes application of the mathematical topic “Solving Equations” (introduced in S.1), “Simultaneous Equations in Two Unknowns” (introduced in S.2), and “Gaussian Elimination” (introduced in M2) to devise an effective approach to the problem. By blending these techniques, our goal is to not only provide a practical solution but also demonstrate the connection of how these methods address complex mathematical challenges. This project showcases how foundational tools can offers both clarity and insight into the problem-solving process.