01NUMBERS
Can you spot the pattern?
Look at the gaps. What would you put in the last space?
2468?
A pattern is a guess that is waiting for another test.
GIFT HORSE MOUTH INSPECTIONS / A CURIOUS LITTLE LAB
Come see what a question can do.
We follow ideas through numbers, substances, words, and movement. Pick a door, pull a lever, and see what changes.
LOOKCLOSER
There is no wrong first question.Only interesting next ones.
01PLAN
Pick a thing you want to understand.
02GOAL
Decide what would count as a clue.
03CREATIVITY
Change one part. Watch what follows.
01 / THE PLAYGROUNDPick a door.
Pick a door.
See what opens.
Four little experiments for people who like to find out. Nothing is precious here: change it and see what happens.
02SUBSTANCES
Build a tiny model.
Add a few atoms and see how a model keeps track of its parts.
C
A model is not the thing itself. It helps us ask what could fit together.
03WORDS
Change one word. Change the picture.
A sentence is a little machine. Swap a part and listen for the difference.
Questionscangrow.
A question can be a seed.
04MOTION
Make an idea move.
Drag the slider. The rule stays simple; the behaviour does not.
42% along the path.
YOUR TURN
Try changing one thing at a time. Then ask the useful question: what stayed the same?
Bring your own idea02 / THE PEOPLEThe lab is made
The lab is made
of question-askers.
There is no single way to belong here. Some people calculate. Some mix, write, draw, build, listen, or make things move.
Finds the rule hiding in the noise.
They notice when a number, a shape, or a strange result is trying to tell us something.
Asks what happens when tiny parts meet.
They build models, test a hunch, and remember that a model is a question with handles.
Turns an idea until it fits.
They listen for the word that changes the picture, then leave the door open for another one.
Gives a thought a body and lets it move.
They make invisible things visible: a path, a rhythm, a wobble, or a machine that learns.
FROM YOUR QUESTIONS TO OURSSame curiosity. Further to go.
03 / INSIDE THE RESEARCHHere, we don’t
Here, we don’t
know the ending.
The playground asks you to try a rule. In the lab, we ask what a rule can explain — and where it stops working. These are some of the questions we take to work.
R.01 / STRUCTURE & LIFEIn progress
Different structures.
A family resemblance.
Two organisms can be built differently and still share an interesting trait. How do we describe that resemblance without pretending they are the same?
We are exploring second- and higher-order abstract syntax trees: structured descriptions that can themselves describe other structures. We want to organise number-theoretic features, investigate conditions such as surjectivity, and ask how these descriptions might reveal relationships between biological organisms with a shared phenotype.
A little deeper
Surjectivity asks whether a map reaches every element of its target. The research aims to taxonomise arithmetic features that can help determine this and other properties. A connected biological question concerns organisms whose chosen structural representations are non-isomorphic, yet whose observable traits — their phenotypes — are meaningfully analogous. Which representation, arithmetic features and notion of analogy make that comparison useful? That is part of the investigation.
The drawing is a conceptual comparison, not a specimen analysis or a validated classification.
STRUCTURE → DESCRIPTION → COMPARISON01
Biological structures. Conceptual sketch; no specimen data shown.
R.02 / COMPUTATIONAL MATHEMATICSIn progress
Start somewhere easy.
Follow it somewhere hard.
Imagine changing a puzzle very gently, while keeping hold of its answers. That is the idea behind homotopy continuation.
We start with a polynomial system whose roots we know, deform it towards the system we want to solve, and follow the solution paths. The challenge is keeping hold of every path as the geometry gets awkward.
Why a GPU makes this interesting
A GPU can work on many paths at once. But paths need different step sizes, different numbers of corrections, and sometimes more numerical precision. Some finish quickly while others need care near singularities. That uneven workload makes a general-purpose implementation difficult.
Our research explores how symmetry in dessins d’enfants and in the starting systems can help organise path tracking. Exploiting a symmetry requires checking that the relevant homotopy preserves it; symmetry of the start alone does not guarantee symmetry of every later path.
Background: Glabe & McCarthy, GPU continuation (2020) ↗FOLLOW THE ROOTSt = 0.55
An exact four-root example, projected into a path diagram. Drag to explore.
TWO TOOLS IN THE NOTEBOOKFind the symmetry. Recover the exact structure.
01 / DESSINS D’ENFANTSA drawing with
A drawing with
arithmetic inside.
Black vertices, white vertices, and edges joining them. These drawings encode maps with remarkable arithmetic structure. A symmetry can turn one piece of work into a clue about another.
What this drawing says
For the Belyi map β(z) = z⁴, the inverse image of the interval [0,1] is this four-armed star. Multiplying z by i rotates it by 90°. Our toy continuation also keeps a fourfold symmetry about its moving centre.
Rotation 0° · same connections
02 / LATTICE REDUCTIONFrom almost right
From almost right
to exactly what?
A numerical root gives us digits. Lattice reduction helps search for small integer relations that could reveal an exact polynomial. Shorter basis vectors make that search more manageable.
α ≈ 1.41421356237…α² − 2 = 0
Why the last step needs a check
The lattice picture is a two-dimensional basis-reduction example. Actual algebraic recognition uses a lattice built from high-precision approximations to powers of a number. A small relation suggests a candidate polynomial; it needs further precision and exact verification. The square-root example is separate from the four-root animation.
Long vectors. Same lattice.
The equation behind the moving paths
The illustration follows all four roots of (z − 0.6t)⁴ − (1 + t) exp(5iπt) = 0, for 0 ≤ t ≤ 1. At t = 0, the roots are 1, i, −1, −i. At t = 1, they solve (z − 0.6)⁴ + 2 = 0. The diagram projects real part, imaginary part and time onto the page; crossings in the drawing need not be collisions of roots. This is a small exact illustration, not a GPU solver.
THE ANSWER CAN BE TINY.
yes▌
THE THINKING DOESN’T HAVE TO BE.R.03 / QUESTIONS FOR AIDataset contribution
A short answer.
A very long way to find it.
Humanity’s Last Exam brought together expert questions that challenged leading AI models. We contributed mathematical questions to that effort.
Our emphasis was on compact problems with very short answers — sometimes a number, sometimes simply yes or no. The answer might fit in about ten characters. Finding and justifying it was the hard part.
~50mathematical questions accepted
Our contributor’s approximate count.
Our contributor’s approximate count.
About the project
HLE spans many disciplines and includes both multiple-choice and short-answer questions with known, verifiable solutions. Questions were screened against contemporary models and reviewed by experts. The original public benchmark was finalised at 2,500 questions; later releases have evolved. A private test set helps assess overfitting.
Unlike our open research questions above, these questions have known answers. A short answer is a way to check the result, not a measure of how much thought it takes.
THE THREAD THAT CONNECTS IT ALL
You change a rule. We change a representation.
Both begin with: what happens if…?
What do you want
to make clearer?
Choose a starting point. Then make it stranger, simpler, louder, smaller, or more useful. That is how a good question earns its next question.
YOUR FIRST MOVE
Start with something surprising. Make one small change and write down what stayed the same.
Try an experiment