Science & education writer at Allin
Math, physics and statistics explainers
Lena makes math, physics and statistics click. She writes clear explainers that connect textbook formulas to the problems people actually need to solve.
Lena states the formula, where it comes from and where it stops working. Each explainer carries a worked example a reader can check by hand, and every approximation says what it costs — the assumption it makes and the range over which it holds.
Ranked by how many pages of this site link to them, not by traffic.
Guides published under this byline in each section. Each link opens the section in full, across all writers.
· 6 min read
The same 12.7 kg puppy at sixteen weeks predicts a 21.9 kg adult if it is a small breed and 38.5 kg if it is a giant one. The arithmetic is trivial; the class you pick is worth seventy-six per cent.
· 6 min read
Roll a 50 mm cone 80 mm tall and its pattern is a 190.8° slice of a 94 mm circle. Flatten a nearly cylindrical lampshade and the pattern radius runs to 839 mm — the shallower the taper, the further away the apex.
· 4 min read
At 1 %, a hundred attempts give you 63.4 % — not certainty. Ninety per cent takes 230 attempts and ninety-nine takes 459, and more than a third of players are still empty-handed at a hundred.
· 13 min read
Points over total is trivial. What is not: averaging percentages instead of weighting them can shift a mark by twelve points, negative marking changes when guessing is worth it, and the same 62 % is a fail on one scale and an upper second on another. What the tool does, and the two things it cannot do.
· 12 min read
Twenty-five and five come from Francesco Cirillo's method, not from a trial. What the research actually measured is task switching and the cost of getting back in — and the reason a fixed interval helps is not that 25 minutes is optimal. Plus what the planner really returns, including the long break it quietly leaves out.
· 11 min read
Magnus–Tetens with the Alduchov–Eskridge coefficients, 17.625 and 243.04. Why an absolute number beats relative humidity for judging comfort, what temperature a window has to reach before it runs, and the one input the calculator does not check.
· 12 min read
The calculator runs the 2001 index, not the 1945 one, and refuses to answer above 50 °F or under 3 mph. Here is what its number is a model of — bare skin, in shade, walking — and the two things it says nothing about: a clothed body, and a parked car.
· 11 min read
Lumens are what a lamp emits; lux is what lands on a surface. You cannot convert one to the other without an area, or without a distance and a beam angle. Anything that offers you a bare conversion factor is selling you a fiction.
· 11 min read
Area grows with the square of the diameter, so an 18-inch pizza is not half as big again as a 12-inch — it is more than twice. Price per square inch settles the argument; the crust and the toppings complicate it.
· 15 min read
Series adds, parallel adds reciprocals — that part is easy. The parts that catch people: which member dominates the answer, why two 5 % resistors in parallel are still 5 %, and why the smallest resistor in a parallel pair takes the largest share of the power.
· 15 min read
Vout = Vin·R2/(R1+R2) is only true when nothing is connected to the output. Draw current and the voltage collapses, by an amount that follows one exact law. Here is that law, the ten-times rule checked against it, and why a lower-resistance divider costs you power forever.
· 8 min read
For sound, f' = f(v + v_o)/(v − v_s) — and getting the signs backwards is the classic error. Here is the convention spelled out, a 440 Hz source computed at four speeds, and why light needs a different equation entirely.
· 9 min read
Re = ρvL/μ compares inertia with viscosity, and the units really do cancel. See the number worked out for honey, a household pipe, an artery, a swimmer and a wing — and why the 2 300 threshold belongs to pipe flow alone.
· 8 min read
Hooke's law says force is proportional to stretch — but only below the elastic limit. Here is F = kx with worked numbers, what a 200 N/m spring actually feels like, and how springs combine.
· 6 min read
The upward force equals the weight of the fluid pushed aside. That one sentence decides whether something floats, and if it floats, exactly how much of it stays under.
· 7 min read
Five variables, four equations, and each equation is missing exactly one of them. Choose by looking at the variable the question never mentions.
· 6 min read
Three formulas cover the whole of projectile motion on level ground. The catch is level ground: the moment launch and landing heights differ, the 45° result stops being true.
· 7 min read
PV = nRT holds when the gas is dilute and far from condensing. The value of R depends entirely on the units you feed it, and the temperature is never in degrees Celsius.
· 3 min read
Split a bill and tip evenly across a group. Learn the tip percentage, total, and per-person formulas with a clear worked example so nobody over- or underpays.
· 4 min read
Life expectancy is an actuarial average by age and sex, not a personal prediction. Learn how life tables work, why the figure rises as you age, and how to read the numbers.
· 4 min read
Estimate your retirement age from how many years it takes your savings to reach a target. Learn the growth formula, a worked example, and why the target matters most.
· 9 min read
Solve for the missing term, then convert the answer into a bench instruction. The step almost everyone skips is that the solvent you add is V2 minus V1, not V2.
· 8 min read
One half-life leaves 50%, three leave 12.5%, ten leave less than a tenth of a percent. The formula is a single line — the part worth understanding is why the answer never reaches zero.
· 4 min read
Estimate reading time from word count and reading speed. Learn the simple formula, the average of about 250 words per minute, and a worked example for a full novel.