High School Mathematical Research Project Ideas

Question

i am currently looking out for some possible topics i could study for my research project in high school. Algebra, trigonometry, Pythagoras’ theorem, geometry, circles and their properties, etc. and perhaps combined with a little knowledge from Physics i.e. Kinematics, Gravity, etc. could interest me.

Another particularly interesting one i stumbled upon was the Rubik’s cube since i myself do enjoy solving it. I am not very sure what type of research topics i could come up with that could be geared towards and tied with the math behind solving Rubik’s cube? Any suggestions or advices are greatly appreciated :)

Answer 1

What are the requirements of your research project? I'm guessing they can't expect a high schooler to discover something new; they probably just want you to work through a non-syllabus proof, possibly after you conjecture the result from special cases.

If you want to do something with a Rubik cube you could discuss its group theory, or number of configurations using combinatorics. Physics is harder because you might not have enough calculus, but 3blue1brown's video on why orbits are ellipses might give you some ideas for a more geometry-based analysis of gravity.

I live in the UK, so what you're trying to do is probably similar to our GCSE maths coursework. Based on my memories of that, possible projects include:

• Given a 2D grid of numbers in which each row and column is an arithmetic progression, what is the difference between the products of endpoints of a rectangle's diagonals?
• How many diagonals does a polygon have?
• How is the sum of a positive integer's positive factors related to its prime factorisation? (Or you may find some other multiplicative function in number theory more interesting.)
• What is the sum of the first $n$ positive cubes? Is there anything interesting about its square root?
• In how many ways can you get from a point on a 2D grid to another point neither below not to the left of it, in unit steps that only go up or to the right?

Answer 2

Welcome to MSE! The rubic's cube is an interesting object from the group theory point of view. Below you will find the enumeration of the faces of the cube. The central faces remain stationary. There six basic rotations of 90 degrees as shown: Left, Right, Up, Down, Front, Back.

These rotations can be described by the elements (called permutations) of the symmetric group $S_{48}$. The group consists of all bijections $[48]\rightarrow[48]$, where $[48]=\{1,\ldots,48\}$.

The basic rotations are

F = (17, 19, 24, 22)(18, 21, 23, 20)(06, 25, 43, 16) (07, 28, 42, 13)(08, 30, 41, 11)

B = (33, 35, 40, 38)(34, 37, 39, 36)(03, 09, 46, 32) (02, 12, 47, 29)(01, 14, 48, 27)

L = (09, 11, 16, 14)(10, 13, 15, 12)(01, 17, 41, 40) (04, 20, 44, 37)(06, 22, 46, 35)

R = (25, 27, 32, 30)(26, 29, 31, 28)(03, 38, 43, 19) (05, 36, 45, 21)(08, 33, 48, 24)

U = (01, 03, 08, 06)(02, 05, 07, 04)(09, 33, 25, 17) (10, 34, 26, 18)(11, 35, 27, 19)

D = (41, 43, 48, 46)(42, 45, 47, 44)(14, 22, 30, 38) (15, 23, 31, 39)(16, 24, 32, 40).

(All these permutations have order 4.)

For instance, in view of F, the cycle (17, 19, 24, 22) says that face 17 mapsto 19, 19 mapsto 24, 24 mapsto 22, and 22 maps to 17.

A basic question would be how to obtain a certain constellation from the unscrambled cube, such as the superflip (all corner stones are at the right position but the edge stones are flipped).

The answer is $UR^2FBRB^2RU^2LB^2RU^{−1}D^{−1}R^2FR^{−1}LB^2U^2F^2$