Mathematics
A working library.
- Can You Reach the Edge of the Square?
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.
Download PDF - Cozy Circles in Regular Polygons
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?
Download PDF - The 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.
Download PDF - Appeasing the Cherry Blossom Horde
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?
Download PDF - 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.
Download PDF - The 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.
Download PDF - 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?
Download PDF - Solving the Jumping Julia Maze
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.
Download PDF - Building a LinkedIn Tango Solver with Z3
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.
Download PDF - 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?
Download PDF - 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.
Download PDF - Maximising the Length of a 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.
Download PDF - 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?
Download PDF - Subsets of {1,…,n} with Exactly One Pair of Consecutive Integers
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.
Download PDF - Running Total of a Die
Roll a fair six-sided die until the running total first exceeds 12. What is the most likely final total?
Download PDF - 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.
Download PDF - Block Party 4
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.
Download PDF - 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?
Download PDF - Solving the LinkedIn Queens Puzzle with Z3
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.
Download PDF - 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?
Download PDF - Expected Distance of a Random Point from the Centre of a Regular 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.
Download PDF - 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.
Download PDF - Inversion in Geometry
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.
Download PDF - 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.
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