Teaching-mechanic library

gear_ratio_gate · math · tuner.formula

Concept: Ratios & proportion

Player does: Set gear tooth counts so the portcullis rises at the target speed

Breaks: Ratios scale by adding the same amount

A ratio scales by multiplying both parts by the same factor, never by adding to them.

percent_shop · math · tuner.formula

Concept: Percent change

Player does: Apply successive markups and discounts to hit an exact price

Breaks: −20% then +20% returns to the start

Successive percent changes multiply, so a 20% drop and a 20% rise do not cancel.

graph_terrain · math · tuner.curve

Concept: Slope & intercept

Player does: Tune m and b so the ramp meets the ledge

Breaks: Slope is run over rise

Slope is rise over run, the vertical change divided by the horizontal change, not the other way around.

projectile_architect · math · tuner.curve

Concept: Quadratics (standard form)

Player does: Tune a, b, c so the arc threads the rings

Breaks: c changes how wide the parabola is

In standard form the leading coefficient a controls the parabola's width, while c only shifts the y-intercept.

move_the_arc · math · tuner.curve

Concept: Vertex form

Player does: Drag h, k, a to move the parabola onto its target

Breaks: (x − 3)² shifts the graph left

In vertex form, (x − h)² shifts the graph right by h, so (x − 3)² shifts right, not left.

reverse_growth · math · tuner.formula

Concept: Logarithms

Player does: Choose how many doublings reach the target population

Breaks: A logarithm is division

A logarithm answers how many times to multiply by the base to reach a value; it is the inverse of exponentiation, not division.

mirror_angles · math · tuner.formula

Concept: Reflection angles

Player does: Set mirror angles so the beam hits the crystal

Breaks: Light bounces straight back off any mirror

Light reflects so the angle of incidence equals the angle of reflection, not straight back unless it hit head-on.

pythagoras_bridge · math · tuner.formula

Concept: Pythagorean theorem

Player does: Cut the plank that spans the diagonal gap

Breaks: The hypotenuse is a + b

The hypotenuse's length is the square root of a² + b², not the simple sum of the legs.

scale_shifter · math · tuner.formula

Concept: Similarity; square-cube law

Player does: Grow ray: double your size, 8× your weight, too big for the gap

Breaks: Doubling length doubles volume

Volume scales with the cube of linear size, so doubling length multiplies volume by eight, not two.

volume_pour · math · tuner.formula

Concept: Volume

Player does: Fill differently shaped tanks to the same volume

Breaks: The taller container holds more

Volume depends on the full shape, not just height, so a shorter but wider container can hold more.

phase_gate · math · tuner.oscillator

Concept: Period

Player does: Dial the period so the vault rings lock

Breaks: A bigger b makes the period longer

The period of sin(bt) is 2π/|b|, so a bigger b squeezes the wave and shortens the period.

pulse_matcher · math · tuner.oscillator

Concept: Frequency

Player does: Match pulses per second to the beacon

Breaks: Frequency and period are the same number

Frequency is the reciprocal of the period: cycles per unit time, not time per cycle.

oscillation_reach · math · tuner.oscillator

Concept: Amplitude

Player does: Set the swing so the platform just touches each checkpoint

Breaks: Amplitude is peak-to-trough

Amplitude is the distance from the midline to a peak, half of the full swing.

wave_alignment · math · tuner.oscillator

Concept: Phase shift

Player does: Slide one wave in time until the peaks align

Breaks: sin(x + π/2) shifts right

Adding inside the argument shifts the graph left, because each output now happens earlier.

arc_builder · math · tuner.formula

Concept: Arc length s = rθ

Player does: Lay rope along the rim; the angle is arc ÷ radius

Breaks: An angle's radian measure depends on the circle's size

Radian measure is arc length divided by radius, so it is the same for any circle.

peak_efficiency · math · tuner.optimize

Concept: Optimization

Player does: Tune the parameter to maximize the machine's output

Breaks: The max is always at an endpoint

An optimum can occur at a critical point in the interior, not just at an endpoint of the domain.

linked_machines · math · tuner.formula

Concept: Related rates

Player does: Set how fast the radius grows so volume rises at the target rate

Breaks: Linked quantities change at the same rate

Related quantities change at rates connected by the chain rule, which usually differ rather than match.

thrust_vehicle · physics · tuner.formula

Concept: Acceleration

Player does: Set thrust so velocity hits the target at the checkpoint

Breaks: Zero velocity means zero acceleration

Acceleration is the rate velocity changes, so an object can have zero velocity while still accelerating, like at the top of a toss.

formula_engine · physics · tuner.formula

Concept: Newton's 2nd law (and any formula)

Player does: Same push on different masses; set the force that hits the target acceleration

Breaks: A constant force produces constant speed

A constant net force produces constant acceleration, not constant speed, since F = ma.

collision_arena · physics · tuner.formula

Concept: Momentum conservation

Player does: Set mass or speed so the post-collision motion hits the target

Breaks: Momentum is lost in collisions

Total momentum is conserved in a collision; what looks lost has transferred between the colliding objects.

force_time_shield · physics · tuner.formula

Concept: Impulse

Player does: Stop the boulder with a short hard shield or a long soft one

Breaks: Stopping force depends only on speed, not stopping time

Impulse equals force times time, so stretching the stopping time over a longer duration reduces the force needed.

height_bank · physics · tuner.formula

Concept: Gravitational PE

Player does: Choose the ledge height that banks exactly enough energy

Breaks: PE depends on the path taken

Gravitational potential energy depends only on height above the reference, not on the path taken to get there.

force_path · physics · tuner.formula

Concept: Work

Player does: Push at an angle; only the component along the path counts

Breaks: Any force on a moving object does work

Work is done only by the force component along the direction of motion; a perpendicular force does no work.

surface_strategy · physics · tuner.formula

Concept: Friction

Player does: Pick the surface coefficient that stops the sled on the pad

Breaks: Friction depends on contact area

Friction force depends on the normal force and the surfaces' coefficient, not on the contact area.

catapult_range · physics · tuner.formula

Concept: Projectile motion

Player does: Set the angle to clear the wall without landing in the lava

Breaks: Horizontal speed slows during flight

Without air resistance, a projectile's horizontal speed stays constant; only its vertical speed changes under gravity.

flow_restrictors · physics · tuner.formula

Concept: Resistance, Ohm's law

Player does: Choose the resistor that sets the bulb to target brightness

Breaks: More resistance means more current

By Ohm's law, more resistance at a fixed voltage means less current, not more.

wave_generator · physics · tuner.formula

Concept: v = fλ

Player does: Tune the frequency so the wavelength fits the chamber

Breaks: Raising frequency raises wave speed

Wave speed is set by the medium; raising frequency shortens the wavelength instead of increasing the speed.

resonance_breaker · physics · tuner.optimize

Concept: Resonance

Player does: Drive the oscillator at the frequency that shatters the crystal

Breaks: Pushing harder always makes bigger oscillations

Oscillation amplitude grows largest when the driving frequency matches the system's natural frequency, resonance, not just from pushing harder.

wave_bridge · physics · tuner.oscillator

Concept: Superposition, interference

Player does: Set the second wave's phase so the sum flattens into a path

Breaks: Two waves always make a bigger wave

Superposition can produce destructive interference, where waves cancel instead of adding to a bigger wave.

light_bender · physics · tuner.formula

Concept: Refraction (Snell's law)

Player does: Angle the beam so it bends through the water onto the sensor

Breaks: Light bends away from the normal entering water

Light slows and bends toward the normal when entering a denser medium like water, not away from it.

gas_balloon · physics · tuner.formula

Concept: Ideal gas law

Player does: Heat the balloon until lift reaches the ledge

Breaks: Gas pressure doesn't depend on temperature

By the ideal gas law, pressure rises with temperature at fixed volume, so temperature and pressure are directly linked.

half_life_vault · physics · tuner.formula

Concept: Radioactive decay

Player does: Set the wait so exactly the target fraction remains

Breaks: After two half-lives it's all gone

Each half-life halves the remaining amount, so after two half-lives a quarter remains, not zero.

particle_crates · chemistry · tuner.formula

Concept: The mole

Player does: Pack particles into standard crates; convert grams to crates

Breaks: A mole of anything weighs the same

A mole is a fixed count of particles, so its mass depends on the substance's molar mass, not a fixed weight.

titration_drip · chemistry · tuner.formula

Concept: Titration

Player does: Stop the burette at the equivalence point

Breaks: The equivalence point is always pH 7

The equivalence point's pH depends on the acid and base strengths involved, and is only 7 for a strong acid–strong base pair.

energy_pyramid · biology · tuner.formula

Concept: Energy flow (≈10% rule)

Player does: Size each trophic level from the energy passed up

Breaks: Energy is recycled like matter

Energy flows one way through an ecosystem and is mostly lost as heat at each level, unlike matter, which cycles.

network_compression · engineering · tuner.formula

Concept: Equivalent resistance

Player does: Collapse the resistor network; predict the single equivalent

Breaks: Parallel resistors add up

Parallel resistors combine by reciprocal sum, which always gives a smaller equivalent resistance than any branch, not a sum.

capacitor_reservoir · engineering · tuner.formula

Concept: RC charging

Player does: Choose the wait so the capacitor reaches 63% (one τ)

Breaks: Capacitors charge linearly

A capacitor charges exponentially, approaching its final voltage faster at first and more slowly over time.

dosage_calculator · health · tuner.formula

Concept: Dosage calculation

Player does: Recompute the pharmacist's dose from weight and concentration

Breaks: Every patient gets the same dose

Safe dosing scales the drug's amount to the patient's weight and the concentration on hand, not a flat number.

price_shock · economics · tuner.formula

Concept: Elasticity

Player does: Change the price to hit target revenue given elasticity

Breaks: Raising prices always raises revenue

Whether a price rise raises revenue depends on the good's elasticity, since demand may fall enough to offset the higher price.

opportunity_fork · economics · tuner.optimize

Concept: Opportunity cost

Player does: Split a fixed budget between two upgrades; name what you gave up

Breaks: Free things have no cost

Opportunity cost is the value of the best alternative given up, which exists even when no money changes hands.

two_nation_trade · economics · tuner.optimize

Concept: Comparative advantage

Player does: Allocate workers in two nations, then trade to beat going it alone

Breaks: A country better at everything gains nothing from trade

Comparative advantage means both countries can gain from trade by specializing where their relative opportunity cost is lowest, even if one is better at everything.

purchasing_power_world · economics · tuner.formula

Concept: Inflation

Player does: The same coins buy less each year; compute the real price

Breaks: Inflation raises every price by the same percent

Inflation is an average across goods; individual prices rise by different amounts, some more and some less.

one_more_unit · economics · tuner.optimize

Concept: Marginal thinking

Player does: Produce until marginal benefit meets marginal cost

Breaks: Decisions depend on totals, not margins

The best decision compares the marginal benefit and marginal cost of one more unit, not the totals.

crowded_factory · economics · tuner.optimize

Concept: Diminishing returns

Player does: Add workers until the next one adds less than she costs

Breaks: More workers always means proportionally more output

Diminishing returns mean each added worker eventually contributes less extra output than the one before.

compound_tower · finance · tuner.formula

Concept: Compound interest

Player does: Choose the rate or years so the tower reaches the target

Breaks: Interest grows by the same amount each year

Compound interest grows on the growing balance, so each year adds more than the last.

tempo_sync · music · tuner.oscillator

Concept: Tempo

Player does: Match the drum's beats per minute to the march

Breaks: Tempo means loudness

Tempo is the speed of the beat, measured in beats per minute, and is independent of how loud the music is.

color_mixer · art · tuner.formula

Concept: Subtractive color mixing

Player does: Mix paint ratios to hit the target swatch

Breaks: Mixing more paint colors makes brighter colors

Subtractive mixing removes reflected light, so combining more pigments makes a muddier, darker color, not a brighter one.

break_even · business · tuner.formula

Concept: Break-even analysis

Player does: Set the price so the stand breaks even at expected sales

Breaks: Covering variable costs means profit

Break-even requires covering fixed costs too, not just the variable cost of each unit sold.