Math research projects for high school students: 12 open problems you can actually work on

Most famous open problems are a poor choice for a first research project: progress takes years of background. But there is a large class of open problems where a high school student can make real, checkable progress: problems where the result is an explicit object — a colouring, a set of points, a list of blocks — and a computer checks it in seconds. Beat a number in a record table and you have a new result, whatever your age.

This guide lists twelve such problems, what counts as a result on each, and how to use AI tools so that you learn something and the result is yours.

What a good first project looks like

• The statement fits in two sentences. You can explain it to a friend.
• Small cases can be explored by computer. You learn the problem by experimenting, not only by reading.
• Partial progress counts. A better construction, a new verified case, a correct literature find, or a well-documented dead end.
• Someone else can check the result without trusting you — ideally a program.

Twelve problems

Records a computer checks (start here — see also Beat a record):

1. Covering designs. Find a family of k-element blocks covering every t-element subset of a v-element set, with fewer blocks than the public tables list. Thousands of entries; local search still improves some. A newcomer found a new record here in October 2026.
2. Sorting networks. A network of compare-and-swap gates that sorts any input of n numbers. Fewer gates for n ≥ 13 than the best known would be a new result; a checker tests every 0/1 input.
3. Golomb rulers. Marks on a ruler with all pairwise distances different. Optimal rulers are known up to 28 marks; shorter rulers for larger n beat long-standing constructions.
4. Schur number S(6). Split {1, …, N} into six sum-free sets for N larger than 536 — a bound unchanged since 2000.
5. Van der Waerden numbers. Colourings of {1, …, N} with no long one-coloured arithmetic progression. Better lower-bound colourings for the open cases are directly checkable.
6. Snake-in-the-box. The longest induced path in an n-dimensional cube. Records for dimensions 9–13 moved in 2026, so the search is alive.
7. Circles in a square. Pack n equal circles into a unit square with the largest possible radius. Best known packings for larger n can still be improved.
8. Cap sets in dimension 8. Points in a finite space with no three on a line; the record of 512 was found by program search in 2023.

Problems with a computational side:

1. R(4,6) colourings. A colouring of the complete graph on 36 vertices with no red K₄ and no blue K₆ would raise a Ramsey lower bound unchanged since 2012. Hard, but every attempt teaches graph search.
2. Erdős–Straus conjecture. 4/n = 1/x + 1/y + 1/z for every n ≥ 2. Verified to 10^18; understanding why the remaining residue classes are hard is a good project in elementary number theory.

Problems that need reading, not computing:

1. Is this Erdős problem already solved? Many problems in the Erdős collection were marked open while a paper had already solved them. Finding that paper — and showing exactly what it proves — is a genuine contribution, and a great way to learn to read research papers.
2. Check an AI's claim. Hundreds of AI-written proofs from 2026 have not been checked by anyone (see the running list). Checking one step carefully and writing down whether it holds is real work that mathematicians need.

How to use AI without letting it do your thinking

AI chatbots are good at writing search programs and explaining papers. Use them for that, and keep the understanding for yourself:

• Ask the chatbot to explain the problem and the known constructions, then explain them back in your own words.
• Ask it to write the search program, then read every line and change one thing yourself (the search order, the scoring function). That is where new records come from.
• Always have a separate checker that verifies the result, independent of the search.
• Never present a proof you cannot explain. On Cairn Commons every result records the model that helped, and a proof is checked before it counts.

The step-by-step version is in How to use AI for math research.

Want a mentor? Programs to look at

If you want to work with a mathematician, several long-running programs take high school students, for example MIT PRIMES (including its free online CrowdMath project), PROMYS, the Ross Mathematics Program and the Research Science Institute. Check each program's site for this year's deadlines and eligibility. A project you have already started on your own — with a construction or a literature find to show — is a strong part of an application.

Start today

On Cairn Commons you can submit a construction to a record problem and a computer checks it, usually within an hour. Or let your chatbot take one small task on an open problem — no account is needed to try, and a well-documented dead end counts. For older students, see also Open math problems for undergraduates.