workshop
Latest
Coned
112×132 pixel wheelset with five loops: the weave it traces, the two contacts it rides on, and what it slides. A train has no differential — two wheels welded to one axle — so the only steering it owns is the 1:20 cone on the tread. Three things fall out. It weaves, at a wavelength of 14.48 m that is a fact about distance and not about speed, so the same shape runs past at 0.96 Hz on a subway and 2.76 Hz on a mainline. It is unstable: a free wheelset with creep forces grows from a 0.3 mm nudge to flange contact in 150 m, while the same wheelset sprung, let go hard against a rail, comes back down a clean exponential at 2.6 to a swing — what makes a train stable is not the cone but the springs. And it runs out almost at once — 7 mm of play against a cone that needs 35.39 mm to steer a 150 m curve, so the tightest radius it manages unaided is 758 m. Past that the steering is the flange: a 150 m curve asks the contact for 0.995% creepage, the cone can give 0.197%, and the 0.798% left over is 1.88 m of tread slid through a quarter circle by a force 4.7 times what friction can hold. Grind the cone steeper and the curves come back — 1:3.3 steers 126 m — and the model's critical speed falls from 330 km/h to 131.
High Water
A growable array over a real allocator, so the growth factor can be charged for both of the things it costs. Doubling copies each element 1.3 times and never — at any n, for an arithmetic reason — takes back one element of the 131,071 it has freed. ×1.25 takes them back 23 times and copies each element 4.8. The threshold between the two is the golden ratio, close enough that rounding the new capacity up instead of down is what decides which side you are on.
Dip
Draw a ramp between two pegs, release the sled, and find out that the fastest track to a point can go under it. The hinge is exact: at a run of π/2 times the drop the fastest curve bottoms out precisely at the finish and the sled arrives travelling horizontally, and past that ratio the curve dives below the flag and climbs back. Four numbers fall out. At a run of four times the drop the dive is 0.504 drops deep — 2.01 m under a flag that is only 4 m down — and the straight line, which never dives, takes 53.7% longer; at eight times it takes 97.6% longer, nearly double. On a cliff, a run a quarter of the drop, the straight line is 0.76% off and the whole trick is worth nothing. Galileo's answer, an arc of a circle, is never worse than 4.84% anywhere from 1:4 to 8:1 — which is how a wrong answer survived 58 years — even though at 4:1 it overshoots to 1.125 drops, more than twice as deep as it should. And the winning curve has a second property you can play against: on the π/2 arc a sled released anywhere reaches the bottom in 1.7376 s no matter what height it started from, so three sleds dropped at three heights land together, with 61 microseconds between them. None of the clocks are simulated — a polyline is a chain of constant-acceleration ramps, so a segment takes 2·ds/(v₁+v₂) exactly, which is why the time updates while your finger is still moving.
Too Wide
87×120 pixel tenpin deck in plan with four loops. Two pins are 12 in apart and 4.766 in thick, so the clear air between them is 7.234 in and the ball is 8.500 — too wide by 1.266. Flood the ball's configuration space and five of the ten pins turn out to have no position it can ever touch them from. It is not an energy problem: toppling a pin costs 0.0280 ft·lb and the ball arrives with 144.9, which is 518 times the whole rack. At the book pocket the ball touches three pins and the 1-3 door is 11.26 in wide by the time it gets there; take the entry angle out and the door is still 6.57, the ball has to shove, and it leaves the 8.
Evict
Cuckoo hashing drawn as the two claims it actually makes — every lookup reads exactly two slots at any load factor, and the entire cost of that moved into insert, where a chain of displacements either reaches a free slot or goes round forever. Read the table as a graph, one vertex per slot and one edge per key, and the jam stops being bad luck in the insertion order: it is a component holding more keys than it has slots, which no order places.
Air Time
Juggling, played as a booking problem. A throw of value h made on beat b lands on beat b+h, so a pattern is possible exactly when no two throws land on the same beat — and the mean of the values is the ball count, exactly, always. Four numbers fall out. At 2.5 throws a second a 3 is airborne 0.98 s and peaks 1.18 m above the hands, a 5 is 3.30× that height and a 7 is 6.93×, because the apex goes as the square of the air time. Keeping a 5 at the height of a 3 means 4.54 throws a second instead of 2.50 — 1.8× the hand rate — and keeping a 7 there would need 6.58/s, which no pair of hands has. Taking one hand out of the pattern doubles every value and so costs 4.95× the height, not 2×. And the arithmetic is not negotiable: throw one ball higher than its share and the demo will show you the exact later beat where the bill comes due, because the mean cannot move.
Backing
A drag is a friction clutch, so what it holds constant is a torque — and the line leaves from a radius the fish spends the whole fight making smaller. On this reel that is 2.43× between the lip and the arbor, so a 6 lb knob is 6 lb at the rod tip and 14.4 lb two hundred metres later, with nothing on the reel to say so. Against a 48 lb fish in open water that leaves exactly one fixed setting, 7 lb, that lands all twelve — and it wins by being spooled to within 5 m of the knot. Eleven pounds, a drag that is lighter than the winning one for the first 113 m of every run, parts the line 12 times out of 12 at an average of 133 m. Wind the knob down as the line leaves so the tension at the fish never moves, and 14.5 lb — never once lighter than the setting it beats — lands 40 out of 40, takes 37% off the fight and keeps 45 m more on the spool. The other half of the lesson is that the softest spring in the system is the line you have out: 20 lb braid at 150 m needs the fish to move 4.04 m to part it, and the same braid at 15 m needs 0.48.
Bind
A lock has no partial credit — five pins at ten depths is 79,666 legal keys and 79,665 of them do exactly as well as a stick. Then the factory drills the chambers a thousandth of an inch out of line, one pin takes the whole load, and 39,833 median guesses collapse into about fifty moves. Tighten the shear line to stop that and the key fails first, because the key has to land all five at once and the pick lands them one at a time.
Egress
You are the voice on the PA and the only dial you have says how hard everyone is trying — which moves their walking speed and how well they still follow a route, both at once. Underneath is a Helbing social-force crowd on an Eikonal floor field, and it disagrees with the poster. Wrecking the crowd's navigation costs nothing measurable (112 s against 115); over-urging costs 9.6%; and leaving the dial at a stroll costs 74%, so the instinct worth distrusting is politeness, not panic. The door decides everything else: the same hundred people clear 1.50 m in 25 s and 0.80 m in 123 s, because throughput goes as width^2.24 rather than linearly — a 2.4 m door carries 12.6x what a 0.8 m one does on 3x the width, which is also why splitting one opening into two doors of half the width is 1.8x to 2.8x slower than leaving it whole. And one bollard beats a bare doorway by 16.7% when it sits 0.6 m upstream and a full metre off the centreline, while 11 of the 36 places you could put it seal the room and nobody gets out at all.
Oche
A dartboard is not a target, it is an arrangement, and the 20 is flanked by the 1 and the 5 — the meanest pair of neighbours on the board. Below 16.88 mm of scatter the treble twenty is the best place to aim and above it nothing is, so the piece measures how straight you actually throw and then plays your leg of 501 from there, one aim point at a time.
Sink
Ten metres down your lungs are half the size they were and you stop floating, so the rest of the way to the plate is free. The oxygen gauge in your chest then reads its best number of the whole dive at the bottom — because it is a fraction multiplied by a pressure you are about to give back — and 59% of the blackouts land in the last ten metres of a dive that went to fifty.
Flare
The engine is gone and the rotor is a flywheel with about a megajoule in it. Everything you do from here spends it, you only get to spend it once, and the landing is the last two seconds — so the whole game is refusing to buy anything comfortable on the way down.
Send
A nine-metre kite on twenty-four-metre lines, and one fact holding the whole thing up — a kite that is sitting still pulls with the wind, and a kite that is moving pulls with its own apparent wind, which is several times bigger and enters squared. Two hundred and twenty metres of beach, one hand on the bar, and nowhere to put the kite down.
Gather
Hot glass on the end of a pipe, and one rule underneath all of it — viscosity. Blowing wants it hot, holding a shape wants it cool, and the only thing standing between a piece and the floor is that you never stopped turning. Three shapes, a minute or two each, and the physics is soda-lime's own.
Pebble
A curling end where the ice is the mechanic. Curling ice is not smooth — it is sprayed with droplets and frozen — and a stone rides the tops of them, which is the only reason it slides twenty-eight metres instead of five and the only reason it curls at all. Set the broom a metre off the target and find out why.
High Water
A growable array over a real allocator, so the growth factor can be charged for both of the things it costs. Doubling copies each element 1.3 times and never — at any n, for an arithmetic reason — takes back one element of the 131,071 it has freed. ×1.25 takes them back 23 times and copies each element 4.8. The threshold between the two is the golden ratio, close enough that rounding the new capacity up instead of down is what decides which side you are on.
Evict
Cuckoo hashing drawn as the two claims it actually makes — every lookup reads exactly two slots at any load factor, and the entire cost of that moved into insert, where a chain of displacements either reaches a free slot or goes round forever. Read the table as a graph, one vertex per slot and one edge per key, and the jam stops being bad luck in the insertion order: it is a component holding more keys than it has slots, which no order places.
Canonical
The LC 71 Simplify Path walk drawn as the stack it is — a Unix path consumed left to right, a column of folder plates rising beside it, every component doing exactly one of three things (push, nothing, pop), and ".." at the root popping nothing without a special branch. The answer is never edited; it is whatever the stack holds when the input runs out. Two switchable bugs — an unguarded pop that throws, and a rule that reads "..." as "up".
Stride
The radix-2 FFT drawn with the part nobody draws — the bit-reversal permutation. Reverse the input's index bits and every stage pairs rows 1, 2, 4, 8 apart, so the whole transform runs in place with no scratch array; flip the ordering and the same 32 butterflies run in mirror image with the shuffle moved to the output. 32 complex multiplies against the direct DFT's 256, agreeing to 2e-15.
Lowest Common
The LC 236 lowest-common-ancestor walk answered bottom-up rather than searched top-down — every node lights with only what came back from its children (nothing, left, right, both), the first to light with both is the answer, and dragging one marker onto an ancestor of the other shows the same rule absorbing the trap case with no special code.
In Phase
A retry storm, simulated rather than diagrammed, so the claim can be counted instead of asserted. Sixty clients, a three-second outage, and a dependency that could clear the whole backlog in 900 ms of work — replayed under five retry policies over identical demand. Exponential backoff with no jitter takes 25.6 s and spends 95.8% of that recovery with nothing in flight at all; full jitter takes 6.0 s and spends 61 calls, one per client. Jitter is not a refinement of backoff. Backoff without it is the thing making the outage long.
Through
Reed–Solomon drawn as the one claim it makes — k bytes are the values of a degree k−1 curve, so any k of the n points you sampled put it back. Kill shards until exactly k are left and every byte returns; kill one more and the curve becomes a fan of curves, all still fitting, with 10^21 payloads consistent with what survived.
Ambiguous
One regex, one NFA, two walks over the same grid of (node, position) cells — and the only difference is the order. On ^(a+)+$ with 20 characters the backtracker spends 5,242,877 steps and the one-pass walk spends 82, both standing in the same 82 cells out of 132; one of them is entered 1,048,576 times. A 10-million-step budget buys 20 characters of backtracking or 2,499,999 of the other.
Unblocked
The LC 210 Course Schedule II topological sort drawn as the arithmetic it is — courses as discs each wearing the count of prerequisites it still waits on, a queue below holding every disc at zero, and each pop firing its edges so downstream counts tick down and new zeros drop into the queue — with the pick rule, the pair direction and the final count check each switchable, so the queue running dry with discs still on the board is the cycle, found without anyone going looking for it.
Coned
112×132 pixel wheelset with five loops: the weave it traces, the two contacts it rides on, and what it slides. A train has no differential — two wheels welded to one axle — so the only steering it owns is the 1:20 cone on the tread. Three things fall out. It weaves, at a wavelength of 14.48 m that is a fact about distance and not about speed, so the same shape runs past at 0.96 Hz on a subway and 2.76 Hz on a mainline. It is unstable: a free wheelset with creep forces grows from a 0.3 mm nudge to flange contact in 150 m, while the same wheelset sprung, let go hard against a rail, comes back down a clean exponential at 2.6 to a swing — what makes a train stable is not the cone but the springs. And it runs out almost at once — 7 mm of play against a cone that needs 35.39 mm to steer a 150 m curve, so the tightest radius it manages unaided is 758 m. Past that the steering is the flange: a 150 m curve asks the contact for 0.995% creepage, the cone can give 0.197%, and the 0.798% left over is 1.88 m of tread slid through a quarter circle by a force 4.7 times what friction can hold. Grind the cone steeper and the curves come back — 1:3.3 steers 126 m — and the model's critical speed falls from 330 km/h to 131.
Too Wide
87×120 pixel tenpin deck in plan with four loops. Two pins are 12 in apart and 4.766 in thick, so the clear air between them is 7.234 in and the ball is 8.500 — too wide by 1.266. Flood the ball's configuration space and five of the ten pins turn out to have no position it can ever touch them from. It is not an energy problem: toppling a pin costs 0.0280 ft·lb and the ball arrives with 144.9, which is 518 times the whole rack. At the book pocket the ball touches three pins and the 1-3 door is 11.26 in wide by the time it gets there; take the entry angle out and the door is still 6.57, the ball has to shove, and it leaves the 8.
Detached
238×120 pixel Swiss lever escapement in plan with four loops. The balance is connected to the rest of the watch for 7.5 ms of every 125 ms beat — 6% of the time — and the other 94% is what the whole mechanism is for.
Low D
200×142 pixel trumpet valve block in section with five loops. Each of the three loops is cut exactly right, and together they are 63.40 mm of tube short — 53.56 cents, a quarter tone and then some, on written low C#. The error is not an approximation of anything; it is precisely the four cross terms you throw away when you expand a product as a sum. And it cannot be engineered out: the flattest three tubes that exist are still 16.32 cents wrong, which is why the instrument has a ring on it instead.
Standout
216×152 pixel tape measure with five loops. The blade is a strip of steel bent cold into a 20 mm arc, and that arc is worth 911 times the same strip flat. It stands out to 2.00 m, folds with no warning at 19% of yield, and then will not stand up again until you have wound it back to 0.72 m — a hysteresis of 2.77 that everyone who owns one has met and nobody is told the reason for.
Wrap
184×134 pixel drum brake seen through the drum, with four loops. Two identical shoes, one cylinder, one pressure, and one of them does 2.14 times the work of the other — which is geometry, not lining. The equation has a zero in it where the brake grabs on its own, and the drum is standing in the only place you could put the hinge to reach it.
Pitch
184×126 pixel vending shelf in elevation with four loops. The coil turns exactly once, every time, in every case — including the ones where nothing came out. What decides it is an angle nobody prints on anything, 180 × depth / pitch, and the margin it leaves is one millimetre.
X-Sync
164×108 pixel focal-plane shutter with four loops, drawn from where the film sits. The dial does not open anything — it sets a delay between two curtains, and the shutter's only real speed is the 4 ms one nobody prints on it.
Let Off
150×88 pixel grand piano action in section with four loops. The key never touches the hammer at the moment that matters — a screw throws it away 2 mm short of the string, and that one number sets the quietest note the instrument can play.
Dip
Draw a ramp between two pegs, release the sled, and find out that the fastest track to a point can go under it. The hinge is exact: at a run of π/2 times the drop the fastest curve bottoms out precisely at the finish and the sled arrives travelling horizontally, and past that ratio the curve dives below the flag and climbs back. Four numbers fall out. At a run of four times the drop the dive is 0.504 drops deep — 2.01 m under a flag that is only 4 m down — and the straight line, which never dives, takes 53.7% longer; at eight times it takes 97.6% longer, nearly double. On a cliff, a run a quarter of the drop, the straight line is 0.76% off and the whole trick is worth nothing. Galileo's answer, an arc of a circle, is never worse than 4.84% anywhere from 1:4 to 8:1 — which is how a wrong answer survived 58 years — even though at 4:1 it overshoots to 1.125 drops, more than twice as deep as it should. And the winning curve has a second property you can play against: on the π/2 arc a sled released anywhere reaches the bottom in 1.7376 s no matter what height it started from, so three sleds dropped at three heights land together, with 61 microseconds between them. None of the clocks are simulated — a polyline is a chain of constant-acceleration ramps, so a segment takes 2·ds/(v₁+v₂) exactly, which is why the time updates while your finger is still moving.
Air Time
Juggling, played as a booking problem. A throw of value h made on beat b lands on beat b+h, so a pattern is possible exactly when no two throws land on the same beat — and the mean of the values is the ball count, exactly, always. Four numbers fall out. At 2.5 throws a second a 3 is airborne 0.98 s and peaks 1.18 m above the hands, a 5 is 3.30× that height and a 7 is 6.93×, because the apex goes as the square of the air time. Keeping a 5 at the height of a 3 means 4.54 throws a second instead of 2.50 — 1.8× the hand rate — and keeping a 7 there would need 6.58/s, which no pair of hands has. Taking one hand out of the pattern doubles every value and so costs 4.95× the height, not 2×. And the arithmetic is not negotiable: throw one ball higher than its share and the demo will show you the exact later beat where the bill comes due, because the mean cannot move.
No Torque
A spacewalker who has let go has nothing to push against and no angular momentum to spend, and can still turn — but only by drawing a shape that encloses area. Four numbers fall out. Sweeping both arms through their entire 250° of travel and back, elbows locked, returns you to the heading you started on: not nearly, exactly, and for as many strokes as you care to spend. Fold the elbows on one leg of that same stroke and the identical 500° of shoulder travel buys 10.59° of heading — a 2.1% gear ratio, 8.5 cycles and 26 seconds to get 90°. The fold is not a dial: folding halfway gives up 71% of the gain to save 16% of the clock, so the honest options are all the way or not at all. And the one move that is genuinely fast is the one you cannot undo — a 1.2 kg tool thrown at 4 m/s buys 2.98 kg·m²/s and about 14°/s, which you then have forever, at a rate that drifts between 10.4 and 14.1°/s as your own shape changes under it.
Run In
A mile of loaded freight handled from the one cab at the front, where the train is a chain with 5.21 m of lost motion in it and nothing in front of the engineer measures that. Four things fall out. The brake pipe is not a wire: a full service takes 5.50 s to reach the last car, and in that gap the tail is doing 2.03 mph more than the head, which closes 5.00 m of the 5.21 m available. The knuckle is a speed limit rather than a force limit — 3.27 mph of closing parts it, on a gauge that reads to 1 mph. Where you apply a brake matters more than how hard: 1200 kN all at the head runs the slack in at 1181 kN, the same 1200 kN spread over 80 cars does it at 435 kN, and the spread one stops the train faster. And the hidden state decides everything — the identical full service at 25 mph peaks at 861 kN with the slack stretched and 452 kN with it bunched. Meanwhile notch 8 up the 0.9% has already spent 72.5% of a knuckle before anyone touches anything, and because tractive effort is power-limited, the slower you get the less throttle you are allowed.
Get Out
A sheepdog with one lever: sheep inside its 17 m flight zone move off, sheep outside it cannot tell the dog exists. That makes distance a switch rather than a dial — a dog holding station 9.4 m behind the flock drives it at 0.98 m/s, and the same dog 15.3 m back drives it at 0.029 m/s, which is the grazing drift of a paddock with no dog in it. Direction is pure position: standing still, over 130° of bearing, the flock leaves along the dog→flock line with a mean lean of 2.7°. Pressing closer is faster per success and worse per attempt — standoff 0.30 pens in 31.7 s but opens the flock up half the time and fails 22%, while 0.55 takes 42.3 s, pens 91% and wins on expectation. Standing still in the right place is worth about half a metre, because the rear sheep walks out of the zone and the lever disconnects. And the sport is named after the outrun, which this model will not pay for: running wide instead of barging through is worth a metre on the lift and nothing at all on the outcome.
Loose
A counterweight trebuchet, where the only control worth having is the angle at which the sling's loop leaves the release pin — and it does not throw the stone, it points it. Across 130° of dial the launch direction follows the pin to within 0.69° of a straight line while the release speed moves 13.9%, so range against pin is a hump, and every range short of its 201.08 m peak has exactly two settings: 25.5° apart at 180 m, 61.5° apart at 100 m. The top of the hump is symmetric to 0.11% out to ±10° either way, so a short shot tells you nothing about which way to turn. 45° is wrong twice over — releasing 8.05 m up takes 1.12° off it and buying loft with speed takes another 3.98° — and the best shot leaves at 39.90°. Then the counterweight: the same half tonne bolted to the arm instead of hung from it releases 2.74× as much energy and throws 38.5% shorter, because 75.9% of that energy is still in the weight when the stone goes, and four times the bolted mass still only reaches 143.6 m against 201.1.
Half Again
A domino amplifier, where energy says each one could be 2.90× the last, the collision says 1.56×, and the same 1.5× finishes 14 builds out of 25 at one spacing and 25 out of 25 at another — the gap nobody thinks about doing all the work.
Two Pulses
A gantry crane whose load is a pendulum you can only steer by the pivot — so the swing you arrive with was decided by two acceleration edges and the gap between them, and the only cure for a swinging load is a second mistake, timed.
Paid Out
Lead climbing, where the rope has never once known how far you fell. A 2 m fall and a 60 m fall arrest at the identical 6.907 kN because the length cancels out of the algebra — and the two things that do decide the day are both invisible: a dead-straight pitch already throws away 31% of its rope at the top carabiner, and the rack's ordering is worth more than the rack, 0.0% of falls zippering against 16.6% on the same routes with the same gear in the other order.