# Vamshi Jandhyala

> PM and startup advisor at the intersection of AI, APIs, and data — by day. Amateur mathematician by night.

Vamshi Jandhyala is a London-based product and technology leader with 20+ years at Bloomberg, JP Morgan, Fidelity, Schroders, HSBC, BlackRock, and PwC. His work sits at the intersection of AI product management, API design, and enterprise data platforms. He advises early-stage startups in the same space, and publishes mathematical books and handouts under his personal imprint.

## Work

- [Work, career, advisory](https://vamshij.com/work): L8 FAANG-scale product leadership, career table, and outcomes.
- [About](https://vamshij.com/about): bio, education, patents, contact.

## Writing

- [Beautiful problems deserve beautiful typesetting](https://vamshij.com/writing/beautiful-problems): 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.
- [How to protect elderly people from financial scams without taking away their independence](https://vamshij.com/writing/protecting-elderly-financial-security): A product design approach to helping banks protect elderly customers from scams while preserving dignity and autonomy through multi-party authentication and trust networks.
- [Taste in API design: Why developer experience demands aesthetic judgment](https://vamshij.com/writing/taste-in-api-design): APIs aren't just functional contracts, they're user interfaces for developers. Great API design requires the same aesthetic judgment and taste that defines excellent product design.
- [The LLM as an API Designer and Critic](https://vamshij.com/writing/llm-api-designer-critic): When AI builds the API, then evaluates its own work, a dialogue exploring how LLMs design, critique, and consume geospatial APIs.
- [Ironies of AI Automation](https://vamshij.com/writing/ironies-ai-automation): A practitioner's guide to Bainbridge's framework, understanding why more AI automation often means we need more human expertise, not less.
- [The catalog becomes the query interface](https://vamshij.com/writing/catalog-as-query-interface): 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. Republished with a 2026 note.
- [When the customer is an LLM](https://vamshij.com/writing/llm-apis-the-new-customer): 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. Republished with a 2026 note.

## Mathematics

- [Can You Reach the Edge of the Square?](https://vamshij.com/mathematics/fiddler_2025_oct17): Two puzzles from Fiddler on the Proof: starting at the centre of a unit square, move in a uniformly random direction until you hit the boundary — what is the expected distance travelled? Same question in a unit cube.
- [Cozy Circles in Regular Polygons](https://vamshij.com/mathematics/cozy_circles): 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. What fraction of the polygon's area is covered by the n circles?
- [The Apollonian Gasket](https://vamshij.com/mathematics/apollonian_gasket): Constructing the Apollonian gasket fractal from Descartes' Circle Theorem and its complex extension. Includes an elementary algebraic proof, the Lagarias-Mallows-Wilks complex form, and a queue-based Python implementation.
- [Appeasing the Cherry Blossom Horde](https://vamshij.com/mathematics/cherry_blossom_puzzle): 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?
- [Posidoku](https://vamshij.com/mathematics/posidoku): A Sudoku variant by Alf Smith with no number clues, only positional gold cells whose values must equal their row, column, or box position. Solved with Google's CP-SAT.
- [The Monkey Puzzle](https://vamshij.com/mathematics/monkey_puzzle): For which n can the integers 1..n^2 be placed in an n x n grid so that the multiset of row products matches the multiset of column products? Includes a prime-counting impossibility bound and an integer-programming formulation.
- [Figurine Figuring](https://vamshij.com/mathematics/figurine_figuring): A Jane Street holiday puzzle: 78 figurines (12 drummers, 11 pipers, 10 lords, … 1 partridge) are shuffled and drawn without replacement until the partridge emerges. What is the expected value of the maximum count of any one figurine type drawn?
- [Solving the Jumping Julia Maze](https://vamshij.com/mathematics/jumping_julia): 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.
- [Building a LinkedIn Tango Solver with Z3](https://vamshij.com/mathematics/linkedin_tango): 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.
- [Beside the Point](https://vamshij.com/mathematics/beside_the_point): A Jane Street geometric-probability puzzle: two random points (red and blue) are drawn uniformly from a unit square. What is the probability that there is a point on the blue-nearest side of the square equidistant from both?
- [Number Hooks](https://vamshij.com/mathematics/number_hooks): A 9x9 grid puzzle: in each L-shaped hook of size k, place exactly k copies of digit k, with row and column sums fixed. Solved with Z3.
- [Maximising the Length of a Projectile Trajectory](https://vamshij.com/mathematics/projectile_trajectory): 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 standard integral and one implicit equation give the answer.
- [Some Off Square](https://vamshij.com/mathematics/some_off_square): A geometric-probability puzzle: two random points in a unit square define a circle's diameter. What is the probability that part of the circle lies outside the square?
- [Subsets of {1,…,n} with Exactly One Pair of Consecutive Integers](https://vamshij.com/mathematics/subsets_consecutive_pair): 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.
- [Running Total of a Die](https://vamshij.com/mathematics/die_puzzle): Roll a fair six-sided die until the running total first exceeds 12. What is the most likely final total?
- [Block Party](https://vamshij.com/mathematics/block_party): A Jane Street puzzle: fill each region of a 9×9 irregularly-partitioned grid with the numbers 1 through N (where N is the region size), such that for every cell containing K, the nearest matching K (taxicab distance) is exactly K cells away.
- [Block Party 4](https://vamshij.com/mathematics/block_party4): A Jane Street variant of Block Party on a 10×10 grid: fill each region with 1 through N, but with a cleaner taxicab constraint expressed as two implication clauses per cell.
- [Dancer Pairs](https://vamshij.com/mathematics/dancer_pairs): A Jane Street puzzle: fifteen dancers stand in an equilateral-triangle formation, each unit-distant from their nearest neighbours. Each dancer pairs up with one neighbour; all but one are part of a pair. How many distinct sets of seven pairs are possible?
- [Solving the LinkedIn Queens Puzzle with Z3](https://vamshij.com/mathematics/linkedin_queens): 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.
- [Well Well Well](https://vamshij.com/mathematics/well_well_well): A Jane Street puzzle: a 7×7 well with per-cell depths 1 through 49 is filled at 1 cubic foot per minute from the cell marked 1; water disperses evenly among adjacent lower-depth regions. After how many minutes does the water level in cell 43 start to rise?
- [Expected Distance of a Random Point from the Centre of a Regular Polygon](https://vamshij.com/mathematics/geometric_expectation_polygon): 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.
- [Knights on a Chessboard](https://vamshij.com/mathematics/knights-on-a-chessboard): A white knight and a black knight on diagonally opposite corners of a 3×3 square. What is the expected number of moves until the black knight captures the white one? A clean Markov-chain problem with a closed form.
- [Inversion in Geometry](https://vamshij.com/mathematics/inversion): 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 mutually tangent circles of radii 1, 2, 3.
- [Islamic Geometric Patterns](https://vamshij.com/mathematics/islamic_geometric_patterns): A computational construction of star patterns from the Islamic tradition: translational units, motifs, rosettes, and the rosette dual, implemented in Python with NumPy.

## Books

- [Geometric Probability](https://vamshij.com/books/geometric_probability): Forty-eight problems in classical geometric probability, the broken stick, Buffon's needle, Bertrand's paradox, Sylvester's four-point problem, Wendel's theorem, Cauchy's shadow formula, each with an analytic solution and a Monte Carlo verification box.
- [Loren Larson](https://vamshij.com/books/lorenlarson): A re-typeset edition of Loren C. Larson's *Problem-Solving Through Problems*, paired with full worked solutions and commentary. Each problem is stated cleanly in modern typography and worked through in the spirit of the original — heuristic first, technique second, finished proof last.
- [Nikoli](https://vamshij.com/books/nikoli): Twenty-five Japanese pencil puzzles, Kakuro, Sudoku, Skyscrapers, Kakurasu, Takuzu, KenKen, Flow Free, Slitherlink, Hashi, Akari, Shikaku, Nurikabe, Masyu, Hitori, Fillomino, Heyawake, Shakashaka, Marupeke, Walls, L-Panel, BlockNumber, Searchlights, Numbrix, and Three-in-a-Row, each modelled as a constraint-satisfaction problem and solved by Google's CP-SAT.
- [Problems in Classical and Contemporary Mathematics](https://vamshij.com/books/problem_portfolio): Eighty-two problems across series, integrals, probability, algebra, number theory, geometry, combinatorics, and inequalities, organised by the technique that unlocks them.
- [Solving Puzzles through Mathematical Programming](https://vamshij.com/books/puzzles_mp): Fifteen classical and contemporary puzzles, Fish, Calendar, Praxis Rhombus, Bedlam Cube, Instant Insanity, Drive Ya Nuts, Ostomachion, Langford, Quintomino on the dodecahedron, Dobble, Bug Byte, Monkey Cat Dog, Prime Circle, Rolling Cubes, and Pilgrims, modelled as constraint-satisfaction problems and solved by Z3, CP-SAT, and NetworkX.
- [The Riddler](https://vamshij.com/books/riddler): Selected problems from Oliver Roeder's *Riddler* column at FiveThirtyEight, set out cleanly and solved with a mix of paper-and-pencil reasoning, generating-function techniques, and small Python simulations where the closed form is out of reach.
- [GCSE Physics](https://vamshij.com/books/gcse_physics): A problem book covering the GCSE Physics syllabus, with full worked solutions to selected problems from the *Isaac Physics* online platform. Solutions are written in the same expository style as the rest of the imprint - statement, set-up, working, answer - with the physics drawn out clearly at every step rather than reduced to formula plug-and-chug.

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