Vamshi Jandhyala

Erich Friedman asked which directed graphs arise when an arrow runs from each chess piece to every piece it attacks.

How to choose use cases, set thresholds and evaluate results.

N prisoners sit in a circle wearing name-tagged hats, and every night the jailer rotates every hat one cell clockwise.

Two browser pages that draw the Earth, the Moon and the inner planets with nothing exaggerated: true eccentricities, bodies and distances at one scale, and the umbra cone at its real proportions.

On your laptop you answer the question may-the-agent-do-this by being there to click yes.

Four days apart in July 2026, the EU postponed its record-keeping obligation for standalone high-risk AI by sixteen months and a widely adopted tool protocol deprecated its logging feature.

Agents rarely lack judgement because the model is weak.

The most respected LLM benchmarks measure whether a model can do a task in a lab.

A generated query that runs is not one you can trust, and the gap widens as the data outgrows what a human can check by eye.

An agent that works beautifully on your laptop is standing on cushions you stopped noticing: a human watching, a single user, your own credentials, a visible trace, your own judgment, and an undo key.

Vibe coding has not only moved the make-or-buy boundary; it has turned building into a cheap option that reprices the whole decision, even for the core systems the cloud era taught everyone to rent.

A 15-second pedestrian signal whose red/green split is itself uniformly random.

A circular billiard puzzle by Xavier Durawa: Charlotte's strands reflect inside a circular frame; how many earlier strands does the n-th strand cross on average?

An architectural view of how a financial data provider can make its data ready for autonomous and semi-autonomous agents.

On the hunt for beautiful olympiad problems, the indignity of badly typeset mathematics on the web, and a small side project to surface AI training datasets back to the human readers the problems were originally written for.

Closed form for the ratio of inscribed-circle area to polygon area when circles tangent to each side at its midpoint are maximal.

A puzzle by Xavier Durawa: in a regular polygon, place a circle at the midpoint of each side, tangent to the side and as large as possible without overlap.

Constructing the Apollonian gasket fractal from Descartes' Circle Theorem and its complex extension.

A geometric-probability puzzle by Xavier Durawa: a random chord across a circle intersects a diameter; given that intersection, what is the expected ratio of the shorter segment of the diameter to the longer?

A Sudoku variant by Alf Smith with no number clues, only positional gold cells whose values must equal their row, column, or box position.

Lisanne Bainbridge showed in 1983 that automating a task makes the human who remains more important and less capable at the same time.

A puzzle from the Julia Robinson Mathematics Festival: navigate from the top-left corner of a grid to the goal cell at the bottom-right, where each cell's number specifies the exact distance you may jump horizontally or vertically to the next cell.

A Z3 solver for LinkedIn's daily Tango puzzle: fill an n×n grid with suns and moons such that no three adjacent cells in any row or column share a symbol; row and column counts are balanced; and pairs of cells linked by = or × constraints match or oppose.

At what angle should a projectile be launched, under uniform gravity and no air resistance, so that the arc length of its trajectory is maximised?

A generating-function argument in which the count is the convolution of Fibonacci numbers; partial fractions over the golden-ratio roots give a closed form involving Fibonacci and Lucas numbers, with f(10) = 235.

Roll a fair six-sided die until the running total first exceeds 12.

LinkedIn's Queens puzzle: place one queen per row, column, and colour region of an n×n board, with the additional constraint that no two queens touch, not even diagonally.

An essay from 2023 arguing that natural-language queries would replace SQL as the primary interface to enterprise data, with the data catalog as the substrate that makes those queries trustworthy.

An essay from 2023 arguing that LLMs would become primary consumers of APIs, that API documentation would become in-demand training data, and that API product managers would soon be designing for a non-human customer.

Closed form for the expected distance from the centre of a regular n-gon of unit circumradius, via a Jacobian substitution on one fundamental triangle; three triangle-sampling methods for computational verification.

A white knight and a black knight on diagonally opposite corners of a 3×3 square.

Inversion in a circle transforms two hard problems about tangent circles into routine ones: a Pappus chain (prove that the height of the n-th circle equals 2n times its radius) and the distance between the circumscribed and inscribed circles of three mutu…

A computational construction of star patterns from the Islamic tradition: translational units, motifs, rosettes, and the rosette dual, implemented in Python with NumPy.