Lightning distance
Count the seconds from flash to thunder, divide by 3 for kilometres (by 5 for miles).
Read the rule →
Air cools as you climb
Temperature drops about 6.5 °C for every 1,000 m of altitude.
Read the rule → Open tool →
Pressure halves every ~5.5 km
Atmospheric pressure falls by half roughly every 5,500 m you climb.
Read the rule → Open tool →
Ten metres of water = one atmosphere
Every 10 m of seawater adds about 1 atmosphere of pressure.
Read the rule → Open tool →
Boiling point falls with altitude
Water boils about 1 °C cooler for every 300 m of altitude.
Read the rule → Open tool →
Dew point is the comfort number
Dew point above 20 °C feels oppressive to most people; below 10 °C feels dry.
Read the rule → Open tool →
Fog watch
Clear night + calm air + temperature within ~2.5 °C of the dew point = expect fog.
Read the rule → Open tool →
Solar panels: 3–5 sun-hours a day
1 kW of panels yields roughly 3–5 kWh per day in temperate climates, averaged over the year.
Read the rule → Open tool →
The sun moves 15° per hour
The sun (and the stars) sweep 15 degrees of sky per hour.
Read the rule → Open tool →
"Feels like" switches formula, not just number
Below about 10°C, wind chill drives "feels like." Above about 27°C, humidity (heat index) does. In between, it’s just the thermometer.
Read the rule → Open tool →
Frostbite risk is about minutes, not degrees alone
At wind-chill near −28 °C, frostbite can start in under 30 minutes on exposed skin; near −40 °C, in under 10.
Read the rule → Open tool →
Most time zones are whole hours — not all
A handful of zones sit on the half-hour or even quarter-hour: India (+5:30), Nepal (+5:45), parts of Australia (+9:30, +8:45).
Read the rule → Open tool →
A 100W device left on all month costs about $10
100 W running continuously for a month is roughly 72 kWh — about $10 at a typical $0.15/kWh rate.
Read the rule → Open tool →
Where you charge beats what you drive
Home off-peak to public rapid is roughly a tenfold difference in price per kWh. No two mainstream EVs differ in efficiency by even twofold — so the socket sets your running cost, not the car.
Read the rule → Open tool →
Radiators decide it, not the cold
A radiator gives about a third of its rated output at 45 °C flow instead of 70 °C. Emitter size, not outdoor temperature, is what stops a heat pump working.
Read the rule → Open tool →
The bolt-torque shortcut
Tightening torque ≈ 0.2 × preload × bolt diameter (dry steel).
Read the rule → Open tool →
Torque mostly fights friction
Roughly 90% of tightening torque is lost to friction; only ~10% actually stretches the bolt.
Read the rule → Open tool →
Deflection limits: span/250
Keep beam deflection under span/250 generally, span/360 where people will notice a bouncy floor.
Read the rule → Open tool →
Depth beats width, cubed
Double a beam’s depth and its deflection drops 8-fold. Double its width and it only halves.
Read the rule → Open tool →
Keep water at 1–2.5 m/s
Size pipes so water flows between about 1 and 2.5 m/s.
Read the rule → Open tool →
One pipe size up fixes most pressure problems
Friction pressure drop scales with 1/d⁵ — a modest increase in diameter slashes losses.
Read the rule → Open tool →
Laminar below 2,300
Pipe flow is laminar below Re ≈ 2,300 and reliably turbulent above ~4,000.
Read the rule → Open tool →
Steel grows 1 mm per metre per 85 °C
A metre of steel expands about 1 mm when heated ~85 °C.
Read the rule → Open tool →
O-ring squeeze: 20–30% static
Compress an O-ring 20–30% of its cross-section for static seals, only 10–20% for dynamic.
Read the rule → Open tool →
Double the load, an eighth of the life
Ball-bearing fatigue life scales with 1/load³.
Read the rule → Open tool →
Safety factor 2 — or 4 with people involved
Use a safety factor around 2 for well-known static loads, 4 or more for dynamic loads, uncertainty, or human safety.
Read the rule → Open tool →
A litre of water is a kilogram
1 L of water ≈ 1 kg ≈ 10 N of weight. A cubic metre is a tonne.
Read the rule → Open tool →
An inch is exactly 25.4 mm
Many “round” imperial–metric pairs are exact by definition (1 in = 25.4 mm, 1 lb = 0.453 592 37 kg). Others are measured.
Read the rule → Open tool →
Density and viscosity are independent
Mercury is 13× denser than water yet flows just as easily; honey is barely denser than water yet far more viscous.
Read the rule → Open tool →
Diesel compresses harder because it doesn’t pre-mix
Diesel engines run compression ratios of 16–22:1; petrol engines top out around 8–12:1.
Read the rule → Open tool →
Every gear mesh flips the direction
One meshed gear pair reverses rotation direction. An even number of stages ends up spinning the same way as the input; an odd number reverses it.
Read the rule → Open tool →
A cylinder is weaker coming back, and faster for it
Retract works on the annulus, not the bore: the rod\’s cross-section is missing from the piston face. A 100 mm bore with a 50 mm rod gives 75% of the force on the way back — and 133% of the speed for the same pump flow.
Read the rule → Open tool →
On retract, the return line carries more oil than the pump sends
Retracting, the small annulus end takes the pump flow while the big bore end empties. A 100 mm bore with a 50 mm rod expels about 53 L/min while the pump delivers 40 — so the return line is the one that has to be bigger.
Read the rule → Open tool →
Cycle a spring near its own resonance and it stops behaving like a spring
A compression spring has a surge frequency of its own, ½·√(k/m) with its full active mass. Work it above about a thirteenth of that and the coils stop moving together, pass a wave along the spring instead, and clash.
Read the rule → Open tool →
A spring usually runs out of travel before it runs out of strength
Solid height is the wire diameter times the total coil count. Once the coils touch, the spring is a solid bar — the load path goes rigid with no warning, and the stress check that said it was fine no longer applies.
Read the rule → Open tool →
One planetary set, three ratios — pick which member you hold
Hold the ring and a sun-in, carrier-out set reduces by 1 + ring/sun. Hold the sun instead and the same metal gives 1 + sun/ring. Hold the carrier and it reverses, at ring/sun. Same three gears, three different answers.
Read the rule → Open tool →
Ring minus sun must be even, and the planets must divide
The planet has (ring − sun) ÷ 2 teeth, so an odd difference asks for half a tooth and cannot be built. Separately, sun + ring must divide by the number of planets or they will not space equally.
Read the rule → Open tool →
Below 17 teeth, the cutter eats the tooth root
A standard full-depth gear needs about 2/sin²α teeth before the cutter starts undercutting the root — 17 at a 20° pressure angle, but 32 at 14.5° and only 12 at 25°.
Read the rule → Open tool →
A 4-stroke camshaft turns at half crank speed
Because four strokes take two full crankshaft revolutions, the camshaft that opens each valve exactly once per cycle must turn at exactly half the crankshaft\’s speed — never the same speed.
Read the rule → Open tool →
It is piston speed that limits an engine, not rpm
Mean piston speed is 2 × stroke × rpm. Production engines sit under about 20 m/s, competition engines run 20–25, and past roughly 25 m/s cast pistons stop surviving — so a redline means nothing until you know the stroke.
Read the rule → Open tool →
Spark advance is a clock, not an angle
Combustion takes roughly a fixed time — call it 2 ms — so the advance needed to put peak pressure just after TDC grows in direct proportion to rpm: about 12° at 1,000 rpm and 72° at 6,000.
Read the rule → Open tool →
Double the wire, sixteen times the spring rate
A helical spring’s stiffness scales with wire diameter to the fourth power — doubling it multiplies the rate by 16.
Read the rule → Open tool →
Double the piston diameter, quadruple the force
In a hydraulic cylinder, force scales with the piston diameter squared, not the diameter itself.
Read the rule → Open tool →
The groove, not the ring, usually fails first
A circlip’s rated thrust load is set by the groove wall in the shaft or bore yielding under the ring’s edge, not by the ring itself breaking.
Read the rule → Open tool →
Every circlip rating assumes a corner you cannot machine
Published circlip thrust ratings assume a perfectly sharp groove corner. Whatever radius your cutting tool leaves comes straight off the load-bearing face — subtract it from the groove depth before you believe the number.
Read the rule → Open tool →
A circlip never holds anything still
The standard groove is deliberately wider than the ring in it — by 0.10 mm at most sizes and 0.15 mm at the largest. The retained part can always move that far, even with everything made exactly to nominal.
Read the rule → Open tool →
Double the speed, quadruple the braking distance
Braking distance goes as the square of speed. Twice as fast is four times the distance under braking, because it is four times the kinetic energy for the same tyres to turn into heat.
Read the rule → Open tool →
To go supersonic, the pipe has to get wider
Below the speed of sound, squeezing a duct speeds the gas up — the garden-hose intuition, and it is right. Above the speed of sound it is exactly backwards: squeezing slows the flow, and only a WIDENING duct keeps it accelerating.
Read the rule → Open tool →
Axial load is free until it is not
Below a threshold e (roughly 0.22–0.44 of the radial load) thrust costs a deep-groove bearing nothing. Above it, the equivalent load jumps to 0.56Fr + YFa and life falls with the cube.
Read the rule → Open tool →
A motor asks for six to eight times its running current before it turns
Locked-rotor current is typically 6–8× full-load current. A 7.5 kW 400 V motor runs at 14 A and starts at 85–113 A.
Read the rule → Open tool →
Current follows voltage, torque follows voltage squared
Reduce a motor’s terminal voltage to a fraction a of nominal and it draws a times the current but produces a² times the torque.
Read the rule → Open tool →
Double the voltage, quadruple the power
For a fixed resistance, power scales with voltage squared — doubling voltage gives four times the power.
Read the rule → Open tool →
The ampacity table is a best case, not a rating
Published ampacity assumes 30 °C ambient and no more than three current-carrying conductors together. Six conductors in a 45 °C roof space leaves about 65% of the printed figure — enough to turn a correct 12 AWG answer into an undersized one.
Read the rule → Open tool →
90 °C wire does not buy you 90 °C amps
The current you may draw is capped by the lowest-rated part of the path — almost always a 60 °C or 75 °C terminal, not the cable. But you still derate from the cable\’s own 90 °C column, which is what makes the better insulation worth buying.
Read the rule → Open tool →
Ampacity and voltage drop are two different checks
A wire can be perfectly safe from an overheating standpoint and still starve a device of voltage over a long run — checking only the breaker rating misses the second failure mode entirely.
Read the rule → Open tool →
A poor power factor means paying for current you don’t use productively
Power factor below 1.0 means some current flows just to sustain a magnetic field (in motors, transformers) without doing useful work — utilities often bill industrial customers extra for this "reactive" demand.
Read the rule → Open tool →
Transformers don’t multiply power, only trade voltage for current
Step up the voltage and the current drops by the same ratio — a transformer changes the balance between volts and amps, never the total power available (minus small real-world losses).
Read the rule → Open tool →
Resistor bands read digit-digit-multiplier
On a 4-band resistor: first two bands are significant digits, third is the power-of-ten multiplier, fourth is tolerance.
Read the rule → Open tool →
Preferred values are spaced by their own tolerance
The E-series steps by 10^(1/n) per decade, sized so that consecutive tolerance bands overlap. E24 steps by 1.10 and carries 5%, which spans 1.11 — so every value in the decade is already covered, and a 5k resistor would be indistinguishable from the 5.1k next to it.
Read the rule → Open tool →
A resistor’s power rating assumes free air, and your board is not free air
Work out V squared over R, then pick a part rated for at least twice it. 5 V across 220 ohms is 0.114 W — arithmetic says an eighth-watt part, practice says a quarter.
Read the rule → Open tool →
Half a lead-acid battery is not yours to use
Usable capacity is about 50% of a lead-acid nameplate and 80–90% of lithium — and the Peukert effect takes more again at high draw.
Read the rule → Open tool →
Guess the direction, let the algebra correct you
Assign any current direction you like. If the answer comes out negative, the current flows the other way — the magnitude is still right.
Read the rule → Open tool →
Delta needs a third of the capacitance
For the same kVAR, a delta-connected bank needs one third the capacitance of a star one — because reactive power scales with the square of the voltage each capacitor sees.
Read the rule → Open tool →
A crash is good news right up until you stop saving
Run the same run of returns over your saving years and your retirement and it gives opposite answers. While contributing, the bad years arriving first leave you better off. While withdrawing, they are the thing that ends the portfolio.
Read the rule → Open tool →
At 3% a year, prices double in a working generation
Divide 72 by the inflation rate to get the doubling time. At 3% — close to the long-run UK and US average — prices double in 24 years, so cash left alone that long buys half as much.
Read the rule → Open tool →
The Rule of 72
Money doubles in about 72 ÷ (interest rate %) years.
Read the rule → Open tool →
Double the tax, round up
In the US, doubling the sales tax on the receipt gives a quick tip estimate.
Read the rule → Open tool →
114 triples, 144 quadruples
Divide 114 by the rate for tripling time, 144 for quadrupling.
Read the rule → Open tool →
Ten years earlier ≈ twice the pot
At typical market returns, starting to invest 10 years earlier roughly doubles your final balance.
Read the rule → Open tool →
The 4% rule
A retirement pot can sustain withdrawals of about 4% a year — so you need roughly 25× your annual spending.
Read the rule → Open tool →
28/36 for housing debt
Keep housing costs under 28% of gross income, and all debt payments under 36%.
Read the rule → Open tool →
Each 1% of rate ≈ 10% of buying power
A one-point rise in mortgage rates cuts what you can borrow for the same payment by roughly 10%.
Read the rule → Open tool →
At 6%, interest ≈ the house itself
On a 30-year mortgage above ~5.5%, total interest roughly equals — then exceeds — the amount borrowed.
Read the rule → Open tool →
20 / 4 / 10 for car buying
Put 20% down, finance for no more than 4 years, and keep total car costs under about 10% of gross income.
Read the rule → Open tool →
PCP balloon is deferred cost, not a discount
A low UK PCP monthly usually means a large optional final payment (GFV) is waiting at the end — not that the car is cheaper.
Read the rule → Open tool →
50/30/20 as a first budget
Aim 50% of take-home pay at needs, 30% at wants, 20% at saving and debt paydown.
Read the rule → Open tool →
"Rent is throwing money away" ignores what the down payment could be doing
A renter who actually invests the cash a buyer would tie up in a down payment isn’t losing that money — they’re compounding it somewhere else instead of in home equity.
Read the rule → Open tool →
Avalanche saves more money, snowball keeps you motivated
Paying off the highest-interest debt first (avalanche) minimises total interest paid — paying off the smallest balance first (snowball) clears accounts faster and keeps momentum, even though it usually costs more overall.
Read the rule → Open tool →
Mid-market is not what you get at the bank
ECB reference rates are mid-market snapshots — a high-street FX quote usually costs you more both ways.
Read the rule → Open tool →
A UK student loan is a tax, not a debt
In the UK you repay 9% of income above a threshold for a fixed number of years and the rest is cancelled. If you will not clear it before the write-off, overpaying buys you nothing.
Read the rule → Open tool →
Every tuning rule is an opinion about one ratio
Ziegler–Nichols, Cohen–Coon and lambda tuning all read the same three numbers off a process — gain, time constant and dead time — and differ only in how close to the stability edge they are willing to sit. On a τ=2 s, θ=0.5 s loop they give Kp of 5.6, 4.8 and 1.0.
Read the rule → Open tool →
Without dead time there is barely a tuning problem
Dead time is what creates a gain ceiling. Remove it and every standard tuning formula divides by zero, because there is no longer a phase limit to respect — the difficulty was never the lag, it was the delay.
Read the rule → Open tool →
Sample at least twice the fastest signal
To capture a signal faithfully, sample above 2× its highest frequency — in practice 5–10×.
Read the rule → Open tool →
Aliases fold around half the sample rate
A tone above fs/2 reappears mirrored below it — 900 Hz sampled at 1 kHz reads as 100 Hz.
Read the rule → Open tool →
Find the edge, then back off
Tuning by hand: raise proportional gain until the loop just oscillates, then halve it.
Read the rule → Open tool →
Read the oscillation
Slow rolling oscillation = too much integral. Fast jitter = too much derivative amplifying noise.
Read the rule → Open tool →
One τ is 63%; five is settled
A first-order system covers 63% of a step in one time constant and is effectively settled after 4–5.
Read the rule → Open tool →
Dead time beats PID
When transport delay exceeds the process time constant, plain PID will disappoint.
Read the rule → Open tool →
No natural damping means P alone can’t save you
A system with zero built-in damping — like a ball rolling on a frictionless beam — can’t be stabilised by proportional control alone, no matter the gain.
Read the rule → Open tool →
Each pole costs 20 dB a decade and 90° of phase
Every additional real pole in a transfer function adds another −20 dB/decade of rolloff and, eventually, another −90° of phase lag.
Read the rule → Open tool →
A loop that only overshoots on big steps is winding up, not mistuned
Change the size of the step. If a small change behaves perfectly and a large one overshoots badly, the problem is actuator saturation, not tuning.
Read the rule → Open tool →
Stop integrating while the actuator is at its stop
Freeze the integral whenever the output is saturated and the error would drive it further in. One condition, no new parameter, and it cannot destabilise the loop.
Read the rule → Open tool →
Cycling is worst at half load, not on the coldest day
On-off cycling peaks at about 50% duty. On the coldest day the unit runs nearly continuously and barely cycles at all.
Read the rule → Open tool →
One step test gives you all three numbers
Step the input and wait. The delay before anything moves is L, the total change over the input change is K, and the time from the end of the delay to 63.2% of the final value is τ.
Read the rule → Open tool →
The ratio of dead time to time constant decides everything
Below L/τ ≈ 0.2 PID is easy and forgiving. Past about 1.0 it will disappoint no matter how you tune it.
Read the rule → Open tool →
2¹⁰ ≈ 10³
Every 10 bits multiplies by ~1,000 — kibi and kilo differ by just 2.4%.
Read the rule → Open tool →
Each refresh rate step buys less than the last
Frame time is the reciprocal of refresh rate, so equal steps in Hz are not equal steps in latency — 60 to 120 Hz saves 8.3 ms per frame, 240 to 480 Hz saves only 2.1 ms.
Read the rule → Open tool →
Each subnet bit halves the hosts
Every bit moved from host to network halves the addresses — a /24 has 254 usable.
Read the rule → Open tool →
RAID is not a backup
RAID protects against disk failure — deletion, ransomware and fat fingers replicate instantly.
Read the rule → Open tool →
3-2-1 backups
Keep 3 copies of anything that matters, on 2 different media, with 1 off-site.
Read the rule →
The ×100 latency ladder
RAM ~100 ns, SSD ~100 µs, spinning disk ~10 ms — each rung roughly 100× slower.
Read the rule →
Every nine costs 10×
Each extra "nine" of uptime — 99% → 99.9% → 99.99% — costs roughly an order of magnitude more effort.
Read the rule →
Big disks want dual parity
Beyond ~4 large drives, prefer RAID 6 over RAID 5 — rebuilds are when second failures strike.
Read the rule → Open tool →
A word is five characters
Every typing speed ever quoted is characters divided by five, divided by minutes. Not average words — a fixed five characters, spaces included.
Read the rule → Open tool →
Your ISP plan is a ceiling, not a promise
A "500 Mbps" plan means up to 500 Mbps under ideal conditions — not a guaranteed floor. Peak-hour congestion, Wi-Fi, and distance from the local exchange all pull real speeds below the advertised number.
Read the rule → Open tool →
Bits to bytes costs a factor of eight
Divide advertised Mbps by 8 for ideal MB/s — a 100 Mbps link is about 12.5 megabytes per second at full throttle.
Read the rule → Open tool →
Latency limits bandwidth, not just response time
TCP will not send more than one window before it hears back, so throughput is capped at window divided by round-trip time no matter what the line is rated at. A 64 KB window over a 10 ms hop is 52 Mbps; over a 150 ms ocean crossing it is 3.5 Mbps.
Read the rule → Open tool →
Never underestimate the bandwidth of a van
A 1 TB drive posted next-day moves data at an effective 93 Mbps, so it beats any connection slower than that. At 20 TB the courier is doing 1.9 Gbps.
Read the rule → Open tool →
Sixty pixels per degree is where pixels vanish
20/20 vision resolves one arcminute, and a degree holds 60 of them — so about 60 pixels per degree is the point where a pixel grid stops being visible. Past it, more resolution changes nothing you can see.
Read the rule → Open tool →
A word is worth two characters
One more random character buys log₂(95) = 6.6 bits. One more Diceware word buys log₂(7776) = 12.9 — nearly twice as much, and you can remember it.
Read the rule → Open tool →
Crack time means nothing without naming the attacker
The same password takes minutes or millennia depending on who is guessing — ten tries a second at a login form against 10¹² a second on a leaked fast hash.
Read the rule → Open tool →
Hohmann is the cheapest transfer — until the ratio passes about 12
Two burns on opposite sides of an ellipse is the minimum-energy route between circular orbits, and LEO to geostationary costs about 3.9 km/s over five and a quarter hours. Past a radius ratio of roughly 11.94, a three-burn bi-elliptic route out to a distant apoapsis is genuinely cheaper.
Read the rule → Open tool →
A narrower gap spreads light more, not less
Diffraction angle goes as wavelength over aperture, so halving the opening doubles the spread — squeezing a beam tighter makes it fan out wider, not narrower.
Read the rule → Open tool →
Geostationary runs on the sidereal day, not the 24-hour one
The altitude falls straight out of the period: match Earth\’s rotation and the orbit radius is fixed at about 42,164 km, or 35,786 km up. But the period to match is the sidereal day, 86,164 s — using 86,400 puts the answer roughly 77 km wrong.
Read the rule → Open tool →
ζ = 1 is critical damping
When the damping ratio hits 1, the system returns to rest as fast as possible without overshoot — less damps and it rings; more and it crawls.
Read the rule → Open tool →
A metre swings in two seconds
A 1 m pendulum takes almost exactly 2 s per full swing.
Read the rule → Open tool →
Pendulum period: √length, not mass
Quadruple a pendulum’s length and it swings half as fast. Mass doesn’t matter at all.
Read the rule → Open tool →
Falling: 5 metres in the first second
A dropped object falls ~5 m in the first second and gains ~10 m/s of speed every second.
Read the rule → Open tool →
Braking distance goes as speed squared
Braking distance scales with the square of speed — 20% faster needs 44% more room, and twice the speed needs four times.
Read the rule → Open tool →
45° for range — in a vacuum
Without air drag, 45° maximizes projectile range; with drag, the real optimum is nearer 30–40°.
Read the rule → Open tool →
The slip angle is the friction coefficient
A block starts sliding when tan(θ) exceeds µ — so a 45° ramp needs µ ≥ 1 to hold.
Read the rule → Open tool →
A marble always beats a hoop, and mass has nothing to do with it
A rolling body accelerates at g sin(theta) divided by (1 + k), where k says how far its mass sits from the axis. A hoop gets exactly half the acceleration of a frictionless slide, a solid sphere keeps five sevenths — and mass and radius cancel out entirely.
Read the rule → Open tool →
A ramp saves force, never energy
A ramp divides the force by up to 1 over sin(theta), and multiplies the distance by exactly the same factor. The work is unchanged — and once friction is involved a ramp actively costs energy, the more so the gentler it is.
Read the rule → Open tool →
Cooling closes the gap in equal fractions
Cooling has a half-life: each equal time-slice closes the same fraction of the temperature gap.
Read the rule → Open tool →
Ice water beats freezer air
An ice-water bath chills a can 5–10× faster than freezer air.
Read the rule → Open tool →
Smash factor tops out near 1.5
Ball speed rarely exceeds about 1.5× clubhead speed, even on a perfect driver strike.
Read the rule → Open tool →
Square the year, cube the distance
For anything going round the Sun, the year squared equals the distance cubed — in Earth years and astronomical units, with no constant to look up. A planet four times further out takes eight times as long to get round.
Read the rule → Open tool →
Everybody measures light at the same speed
Measure a light beam and you get 299,792,458 m/s — standing still, running towards it, or chasing it at nine tenths of that yourself. Every other strange thing in relativity is forced by keeping that one number fixed.
Read the rule → Open tool →
Orbits are ellipses, not circles, and speed isn’t constant
A satellite moves fastest at its closest approach (perigee) and slowest at its farthest point (apogee) — a circular orbit is just the special case where those two points coincide.
Read the rule → Open tool →
Most of a rocket’s mass is fuel it will burn on the way up
The Tsiolkovsky rocket equation shows that reaching a given velocity needs a propellant mass fraction that grows exponentially with the target delta-v — doubling the required velocity change can multiply the fuel needed several times over, not just double it.
Read the rule → Open tool →
Elastic collisions conserve both momentum and energy — real ones rarely do
Two equal masses colliding elastically exchange velocities completely — the moving ball stops, the stationary one takes off at the exact same speed. Any energy lost to heat or sound (an inelastic collision) breaks this clean exchange.
Read the rule → Open tool →
The drop height decides the argument
Gravity accelerates everything equally. Whether you can SEE that depends entirely on how far you drop it — short drops hide air resistance, long drops exaggerate it.
Read the rule → Open tool →
Thickness, not weight
A steak’s cooking time is set by how thick it is, not how heavy it is — and it rises faster than the thickness does. Double the thickness and you more than double the time.
Read the rule → Open tool →
Paint coverage assumes one coat on a primed wall
A litre of paint claiming 12-14 m² coverage means one thin, even coat on primed drywall — dark-to-light changes, bare plaster, or textured surfaces routinely need a second coat, doubling real consumption.
Read the rule → Open tool →
Mixing paint colours by volume, not by eye
A target colour from two base paints is a ratio problem, not a guess — the same two paints in a different ratio produce a completely different, unrepeatable shade.
Read the rule → Open tool →
A colour looks different on four walls than on a tester patch
Light bounces. Every wall throws its own colour onto every other surface, so four walls of a colour read stronger and warmer than the A5 patch you judged it by.
Read the rule → Open tool →
Tap drill size is major diameter minus pitch
For a metric thread, the drill bit for tapping is simply the major diameter minus the thread pitch — M6×1.0 needs a 5.0mm hole, no lookup table required.
Read the rule → Open tool →
Coffee ratios are about weight, not scoops
A tablespoon of coffee varies by 4-8g depending on grind size and roast — the same "two scoops" can span nearly a 2x difference in actual coffee weight.
Read the rule → Open tool →
Wet concrete needs less volume than the dry ingredients that make it
Loose sand and aggregate have air gaps between the particles — mixing needs roughly 1.54 times the dry volume of materials to produce one unit volume of finished, settled concrete.
Read the rule → Open tool →
Water controls concrete strength more than the mix ratio does
Strength falls roughly exponentially with the water-cement ratio. Going from 0.50 to 0.60 is about two extra litres in a 25 kg bag — invisible in a mixer — and costs around a quarter of the final strength; 0.70 costs nearly half.
Read the rule → Open tool →
Cold does not weaken concrete, it slows it down
Concrete gains strength by chemical reaction, and the reaction follows temperature. At 5 degrees C a pour takes roughly twice as long to reach a given strength as it does at 20 degrees C — but it does get there.
Read the rule → Open tool →
Egg cooking time scales with mass to the two-thirds power, not linearly
A jumbo egg doesn’t need proportionally more time than a medium one to reach the same doneness — heat has to penetrate through the whole egg, and that penetration time scales with mass^(2/3), not mass itself.
Read the rule → Open tool →
A naive timer drifts, a correct one can’t
Adding up tick intervals compounds every small delay into permanent error — computing elapsed time as now minus a fixed start point means a single slow tick costs you nothing, because the next measurement is still anchored to the same origin.
Read the rule → Open tool →
Multiplication binds tighter than addition
2 + 3 × 4 is 14, not 20. Multiplication and division are resolved before addition and subtraction, whatever order you typed them in.
Read the rule → Open tool →
SOH-CAH-TOA is one triangle, memorised three ways
Sine, cosine and tangent are just three different ratios of the same right triangle’s sides — knowing any one angle and any one side determines every other side and angle.
Read the rule → Open tool →
Area scales with the square, volume with the cube
Double every dimension of a shape and its area quadruples while its volume grows eightfold — the exponent, not the shape, decides how fast size compounds.
Read the rule → Open tool →
A column of the periodic table is a prediction
Elements in the same group share the number of electrons in their outermost shell, so they react alike — which is what lets you predict the behaviour of an element you have never handled from the one above it.
Read the rule → Open tool →
Atoms get smaller as you add electrons
Moving left to right along a period, atoms shrink even though electrons are being added — lithium is 167 picometres across and fluorine, six electrons later, is 42.
Read the rule → Open tool →
Advantage is worth most at even odds
Rolling twice and keeping the higher result is not worth a flat bonus — it is worth nothing when you were certain either way, and a full 25 percentage points when the outcome was a coin flip.
Read the rule → Open tool →
A long run of heads is not evidence of anything
In a run of n coin flips the longest streak of the same face lands near log₂n — about six or seven in a hundred flips — so a streak that looks rigged is usually the ordinary case.
Read the rule → Open tool →
The mean is not the middle when data is skewed
A handful of extreme values can drag the average well away from where most of the data actually sits — the median, not the mean, is the more honest "typical value" for skewed distributions like income or home prices.
Read the rule → Open tool →
A graph is just the equation, drawn
Every point on the curve of y=f(x) is a coordinate pair that makes the equation true — nothing more mysterious than that.
Read the rule → Open tool →
Percent of means multiply by hundredths
X% of Y is (X/100) times Y — and "Y plus X%" is Y times (1 + X/100).
Read the rule → Open tool →
Halving the effect quadruples the sample
Sample size scales with 1/effect². Detecting a 5% lift instead of a 10% one costs four times the traffic, not twice.
Read the rule → Open tool →
“No significant difference” usually means “too small to tell”
An underpowered test cannot distinguish a real effect from none, so a null result from one carries almost no information.
Read the rule → Open tool →
Week 1 is the week holding the first Thursday
ISO 8601 weeks run Monday to Sunday, and week 1 is the week containing the year’s first Thursday — the same as the week containing 4 January. Not simply the week containing 1 January.
Read the rule → Open tool →
The hour hand keeps moving
Angle = |30H − 5.5M|. At 3:30 that is 75°, not 90° — the hour hand has already drifted halfway to the 4.
Read the rule → Open tool →
Rest three minutes between heavy sets
Heavy strength work needs roughly three to five minutes between sets — hypertrophy work sixty to ninety seconds, and endurance work thirty to sixty. The rest is prescribed by the goal of the set, not by how recovered you feel.
Read the rule → Open tool →
Estimate a one-rep max from five reps or fewer
Rep-max formulas agree closely at low reps and fall apart at high ones — within about five kilos of each other at five reps on a hundred-kilo lift, but seventy-seven kilos apart at twenty.
Read the rule → Open tool →
Keep your waist under half your height
Waist circumference should be less than half your height. At 175 cm that is 87 cm; at 160 cm it is 80 cm.
Read the rule → Open tool →
BMI and waist disagree in both directions, and that is the point
A muscular 180 cm person at 95 kg is BMI-obese and waist-healthy. A sedentary 180 cm person at 78 kg is BMI-healthy with a 95 cm waist and a ratio of 0.53.
Read the rule → Open tool →
BMI cannot tell muscle from fat
A lean, heavily muscled athlete can score "overweight" or even "obese" on BMI despite having very low body fat — the formula only ever sees total mass and height, never composition.
Read the rule → Open tool →
Doubling your race distance roughly doubles your time, plus a bit more
A 25-minute 5K predicts almost exactly a 52-minute 10K, not a 50-minute one — endurance pace slows down as distance increases, not just because a race is longer but because fatigue compounds.
Read the rule → Open tool →
Activity level is an average, not your best day
Three hard gym sessions plus a sedentary desk job the other four days averages out to lightly active, not moderately active — overestimating activity is the single most common reason a calorie estimate ends up wrong.
Read the rule → Open tool →
Measure your own sweat rate; the guidelines fit almost nobody
Sweat rates run from 0.5 to 2.5 L/h between people doing the same work. Weigh yourself before and after one session and you know yours; no published figure can tell you.
Read the rule → Open tool →
220 minus age is an average, not your number
It is a population mean with a spread of roughly ±10 beats, so for about a third of people it is out by more than 10 — and every training zone moves with it.
Read the rule → Open tool →