Essays
These are our essays. They showcase applied mathematics across a wide span of subjects, from physics and engineering to finance, history and medicine: the models, the methods that solve them, the code that runs them and the algorithms behind them, often with a live model you can try for yourself. Each essay starts from a question and works it through to a number. Nothing here is a product or advice. The code we publish as Jupyter notebooks is under Notebooks, the live boards that each answer one practical question are under Tools, and the research papers are under Research Papers. Every essay belongs to one of our five research desks; the buttons below show one desk's essays at a time.
26. The Needle That Knew Too Much
In 1901 an Italian teacher threw a stick onto a grid of wires 3,408 times and found π to six decimal places. An honest run that size misses by about 0.05. Buffon's needle says how wrong it should have been, a stopping rule says how 355/113 could come out anyway, and the test Fisher used on Mendel's peas says why being too right is itself the evidence.
Read more →27. When to Shoot
Two duellists walk towards each other, one bullet each. Fire too soon and you will probably miss, and a miss is fatal. In 1949 David Blackwell worked out the answer in a day: fire when your two chances add up to one, the good shot and the bad shot at the same instant. Then he made the pistols silent, and the only safe strategy became a roll of the dice.
Read more →28. The Man Who Could Have Won the Nobel Twice
In 1965 Paul Samuelson wrote down an option formula with two unknowns, and admitted he had no theory to pin them down. Set both to the interest rate and it is the Black–Scholes–Merton formula, term for term, eight years early. The missing piece was not mathematics but an argument, and his own student helped supply it.
Read more →29. The Game That Remembered
Roulette cannot be beaten because the wheel forgets every spin. A deck of cards does not forget. In 1961 a young mathematician proved it, took the proof to Reno and won $11,000 in a weekend. Then he found the harder truth: with a real edge you can still go broke, and betting twice the right amount earns you exactly nothing.
Read more →30. The System That Could Not Lose
Dostoevsky wrote home from Wiesbaden that he had found the secret of roulette: you just can't lose. Then he signed away his future, gambled the advance, and dictated a novel in 26 days to save himself. The doubling system every gambler knew wins 39 nights out of 40 — and loses exactly the house edge anyway.
Read more →31. The Tanks That Counted Themselves
Before D-Day the Allies needed to know how many tanks Germany built each month. The spies said about 1,400. A few statisticians read the serial numbers off wrecked gearboxes and said 246. The records, opened after the war, said 245 — and the best estimate throws away every serial number but one.
Read more →32. The Forest That Burns All at Once
Over forty fire seasons in British Columbia, the number of fires varied six-fold and the area burned nearly a thousand-fold. Fewer fires now burn thirty-five times the area they did in the 1990s. A percolation threshold explains why, and a fire PDE on a patchy forest shows fuel 12.5% drier turning the same lightning strike into a fire a hundred times larger.
Read more →33. The Road That Made Everyone Late
Close a busy road and the traffic gets faster. Cities have done it and been surprised. The reason is arithmetic, it needs every driver to be perfectly selfish and perfectly informed, and in the smallest version a free new road costs everybody fifteen minutes. Then it turns out the most destructive road of all is not the free one.
Read more →34. The Urn That Decides Once
An urn with one red ball and one black. Draw one, put it back with a friend of the same colour, forever. The share of red settles — and it settles on a number drawn uniformly from nought to one, so there is no typical outcome at all. Then de Finetti removes the mechanism entirely, and one exponent decides whether the rich getting richer ends in a monopoly or in a permanent accident.
Read more →35. The Bird Does Not Come Home
A drunk man and a drunk pigeon leave the same pub with all the time in the world. The man is certain to find his way back. The pigeon manages it 34% of the time and otherwise never. One extra dimension turns a certainty into a bet you mostly lose — and then the sky turns out to have a ceiling.
Read more →36. Where the Force Vanishes Is Not Where It Rests
Set the arrivals equal to the cancellations, solve for the queue size, and you have found the point where the force on the book vanishes. You have not found where the book sits. The two answers differ by a term nobody writes down, worth 14.2% across three order books that share a balance point exactly.
Read more →37. Everybody Wants the Cheap Plug
Ten thousand vehicles, two cheap places to charge and one hour. A mean field game — Hamilton–Jacobi–Bellman backwards, Fokker–Planck forwards — solved on a torus and animated, and the 1.60% that nobody's mistake accounts for.
Read more →38. One Hundred Prisoners, Fifty Opens
A hundred prisoners, a hundred boxes, fifty opens each, and all must succeed. Opening at random gives one chance in 1.27 x 10^30. One sentence of strategy gives 31 per cent, and it never decays with the size of the room.
Read more →39. A Coin With No Memory Still Looks Cold
For thirty years the hot hand was the textbook case of seeing pattern in noise. The test itself was biased: in four fair flips, the share of heads following a head averages 0.4048, not 0.5.
Read more →40. The Body That Is Always Falling
Stand still for a photograph and you are falling the whole time: an upside-down pendulum whose lean doubles every 0.22 seconds, caught by a brain that sees it late. The equation of standing is a second-order Langevin equation turned upside down, and it has a hard limit: with reflexes slower than 0.46 seconds, nothing can hold you up. Try it yourself.
Read more →41. The Barrier Your Monte Carlo Never Sees
Discrete Monte Carlo monitoring silently under-counts barrier crossings with a stubborn sqrt(dt) bias — a one-line Brownian-bridge correction removes almost all of it.
Read more →42. When Sampling Faster Makes You Wrong
Realized variance should converge to an SDE's quadratic variation as you sample faster — market microstructure noise makes it diverge instead. The volatility signature plot, and the two-scale fix.
Read more →43. The Neuron as a Density: Fokker–Planck with a Reset
The same absorbing barrier, seen forward: the firing rate is a probability flux, the reset kinks the density, and a recurrent network has to solve for its own rate.
Read more →44. The Neuron as an Ornstein–Uhlenbeck Process
A leaky integrate-and-fire neuron's subthreshold voltage is an Ornstein-Uhlenbeck process, and its firing rate is a first-passage time — the same barrier-crossing math that prices a down-and-out option.
Read more →45. Cholesky vs Eigendecomposition: Two Ways to Sample Correlated Gaussians
Two matrix square roots for sampling correlated normals: Cholesky is several times faster, but the eigendecomposition survives a correlation matrix that isn't positive definite.
Read more →46. The 1/√N Wall: Why Monte Carlo Is Slow, and What Beats It
Monte Carlo error falls like 1/sqrt(N) no matter how clever your sampler. Quasi-Monte Carlo bends the exponent, not just the constant.
Read more →47. Euler–Maruyama vs Milstein: The Term That Matters
One extra term from the Itô–Taylor expansion doubles the strong order of an SDE solver — measured directly against an exact solution.
Read more →48. Integrating Oscillations: Filon vs Levin
Two classical methods for oscillatory integrals that beat the O(omega)-points curse — compared against an exact solution, with the log-error plot to prove it.
Read more →49. The Itô Integral: A Running Bet You Fix Before the Flip
Why the naive guess for the Itô integral is wrong, and the correction that makes stochastic calculus honest.
Read more →50. The Average That Drowned a City
In 1997 Grand Forks built its dikes to a forecast crest of 49 feet. The river came in at 54.35. The forecast was a median, the damage was convex, and the city evaluated the damage at the average crest instead of averaging the damage. The forecaster's own record put the chance of topping the dikes at one in four.
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