Teaching-mechanic library

mimic_chest · general · truth_finder.mimic

Concept: Any concept with a listed misconception

Player does: Several claims, exactly one false; expose the lie

Breaks: Any listed misconception of the concept

Testing each claim against what the source actually says is how you catch a plausible-sounding falsehood.

chrono_bridge · general · sequencer.linear

Concept: Any process, timeline, or procedure

Player does: Place the steps in the right order; decoys don't belong

Breaks: The order of steps doesn't matter

A procedure works only because each step prepares the next; reordering breaks the chain.

type_matched_weapon · general · sorter.type_match

Concept: Any set of categories

Player does: Weapon modes are categories; only the right one hurts each labeled enemy

Breaks: Items that look alike belong to the same category

Category membership follows a defining feature, not surface resemblance.

grapple_anchors · general · linker.pairs

Concept: Any set of matched pairs (term–definition, part–function)

Player does: Chain lightning jumps only between matching pairs

Breaks: Similar-sounding terms mean the same thing

Each term has one partner; matching by meaning, not by sound, is what makes the link hold.

cycle_wheel · general · sequencer.cycle

Concept: Any cyclic process

Player does: A loop of rooms rotates one stage per clear; trigger the stages in order

Breaks: A cycle has a fixed start and end

A cycle has no first stage; what matters is which stage feeds which.

number_line_leap · math · mapper.number_line

Concept: Magnitude, fractions, negatives

Player does: Land on 3/8 or −2.5 with only coarse landmarks labeled

Breaks: A bigger denominator means a bigger fraction

A fraction's size comes from how far along the line it lands, and a bigger denominator makes each step smaller.

equivalent_landing · math · mapper.number_line

Concept: Equivalent fractions

Player does: Two different-looking fractions must land on the same stone

Breaks: Different numerals mean different amounts

Equivalent fractions name the same point on the line, no matter how they are written.

log_scope · math · mapper.number_line

Concept: Scientific notation, orders of magnitude

Player does: Place 10⁻⁹ m, 10³ m, 10⁶ m on a zoomable log line

Breaks: 10⁻⁹ is a large negative number

A negative exponent means a tiny positive number; each step on a log line multiplies by ten.

integer_lift · math · mapper.number_line

Concept: Signed-number arithmetic

Player does: Ride the elevator through basement floors: −3 − (−5)

Breaks: Subtracting a negative makes a number smaller

Subtracting a negative moves you up the line, because it undoes a move down.

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.

unit_pipeline · math · transformer.composition

Concept: Dimensional analysis

Player does: Chain conversion-factor pipes; joints connect only when units cancel

Breaks: Units are labels you can drop

Units multiply and cancel like numbers, which is exactly how a chain of conversions works.

order_of_ops_forge · math · transformer.trace

Concept: Order of operations

Player does: Predict what the calculator machine outputs for a nested expression

Breaks: Operations always go left to right

Operations follow precedence and grouping, not reading order.

balance_chamber · math · balance.equation

Concept: Solving linear equations

Player does: Apply the same operation to both pans until x stands alone

Breaks: You can move a term across without changing its sign

Moving a term to the other side is really applying the same operation to both sides, which flips its sign.

intersection_hunt · math · mapper.plane

Concept: Systems of equations

Player does: Mark the one point both laser paths pass through

Breaks: A solution only has to satisfy one equation

A system's solution must satisfy every equation at once, which is exactly where the lines intersect.

allowed_region · math · mapper.plane

Concept: Inequalities

Player does: Only tiles where y < 2x + 1 are solid; mark the safe region

Breaks: Multiplying by a negative keeps the inequality direction

Multiplying or dividing an inequality by a negative number flips the inequality's direction.

distance_field · math · mapper.number_line

Concept: Absolute value

Player does: Beacons light where |x − 3| < 2; mark the whole lit stretch

Breaks: |x| just deletes a minus sign, so |x| = x

Absolute value is distance from zero, so |x| equals x only when x is nonnegative; otherwise it flips the sign.

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.

function_factory · math · transformer.function_machine

Concept: What a function is

Player does: Feed inputs, watch outputs, predict the output for a new input

Breaks: A function must be a formula

A function is any rule pairing each input to exactly one output, whether given by a formula, table, or graph.

reverse_factory · math · transformer.inverse

Concept: Inverse functions

Player does: Run the machine backward to recover the input

Breaks: f⁻¹(x) means 1/f(x)

The inverse function f⁻¹ undoes f by swapping inputs and outputs; it is not the reciprocal 1/f(x).

machine_pipeline · math · transformer.composition

Concept: Composition

Player does: Order two machines so the crate exits as the target

Breaks: f(g(x)) = g(f(x))

Function composition is generally not commutative, so f(g(x)) and g(f(x)) usually differ.

input_filter · math · transformer.domain_filter

Concept: Domain & range

Player does: Choose which crates the √ or 1/x machine accepts without jamming

Breaks: Every real number is a valid input

A function's domain excludes inputs that break it, like negatives under a square root or zero in a denominator.

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.

sea_level_roots · math · function_world.roots

Concept: Zeros / roots

Player does: Mark every point where the terrain meets sea level

Breaks: Every quadratic has two real roots

A quadratic can have two, one, or zero real roots, depending on whether its graph crosses, touches, or misses the x-axis.

factor_forge · math · builder.tiles

Concept: Factoring

Player does: Arrange algebra tiles into a rectangle whose sides are the factors

Breaks: x² + 5x + 6 = (x + 5)(x + 1)

Factoring requires two numbers that multiply to the constant and add to the middle coefficient, here 2 and 3, giving (x + 2)(x + 3).

replication_chamber · math · simulator.reach_state

Concept: Exponential growth

Player does: Set the growth factor so the colony hits the target on day n

Breaks: Exponential growth is fast linear growth

Exponential growth multiplies by a constant factor each step, which eventually outpaces any linear growth no matter how fast.

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.

pattern_path · math · transformer.function_machine

Concept: Arithmetic & geometric sequences

Player does: Jump to the stone holding the next term

Breaks: Every sequence adds a constant

An arithmetic sequence adds a constant difference, but a geometric sequence multiplies by a constant ratio instead.

inheritance_machine · math · simulator.reach_state

Concept: Recursive sequences

Player does: Set a₁ and the rule so generation 5 matches

Breaks: Recursive and explicit rules can't describe the same sequence

A recursive rule and an explicit formula can generate exactly the same sequence, just computed differently.

exponent_mimic · math · truth_finder.mimic

Concept: Exponent rules

Player does: Expose the chest claiming x²·x³ = x⁶

Breaks: Multiply exponents when multiplying powers

When multiplying powers with the same base, exponents add, so x²·x³ = x⁵, not x⁶.

transform_key · math · transformer.geometric

Concept: Rigid transformations & symmetry

Player does: Rotate, reflect, translate the key until it fits the slot

Breaks: A line reflection equals a 180° rotation

A reflection flips a shape's orientation while a 180° rotation preserves it, so the two are not the same transformation.

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.

area_tiler · math · builder.tiles

Concept: Area vs perimeter

Player does: Tile the room, then fence it; compare both counts

Breaks: More area always means more perimeter

Area and perimeter can change independently, so a shape can have more area with less perimeter, and vice versa.

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.

congruence_stamp · math · sorter.bins

Concept: Congruence criteria

Player does: Sort triangle pairs into SSS / SAS / ASA / not enough info

Breaks: SSA proves congruence

Side-Side-Angle does not guarantee congruent triangles; it can produce two different triangles from the same measurements.

proof_ladder · math · sequencer.linear

Concept: Two-column proofs

Player does: Order the rungs of a proof; one decoy reason doesn't follow

Breaks: If it looks true in the diagram, it's proven

A geometric proof needs a chain of justified statements, since a diagram can look true without the relationship being proven.

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.

coordinate_treasure · math · mapper.plane

Concept: Distance & midpoint

Player does: Mark the midpoint between two torches on the grid

Breaks: The midpoint is the difference of the coordinates

The midpoint is the average of the two points' coordinates, not their difference.

circular_navigator · math · mapper.plane

Concept: Unit circle

Player does: Walk the unit circle to where the x-shadow reads the given cos θ

Breaks: sin θ is the x-coordinate

On the unit circle cos θ is the x-coordinate and sin θ is the y-coordinate.

projection_machine · math · mapper.number_line

Concept: sin & cos as projections

Player does: Predict where the rotating point's shadow lands at θ

Breaks: sin and cos are unrelated functions

Sine and cosine are the two shadows of one rotating point, which is why they share a period and trade peaks.

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.

radian_rune_line · math · mapper.number_line

Concept: Radian measure

Player does: Step onto 5π/6 on the unrolled circle

Breaks: π radians is a full turn

A full turn is 2π radians because the unit circle's circumference is 2π; π is half a turn.

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.

identity_circuits · math · truth_finder.mimic

Concept: Trig identities

Player does: Several routes transform the signal; expose the non-equivalent one

Breaks: Identities only hold at special angles

An identity holds for every angle; one counterexample is enough to expose an impostor.

trig_solve_bridge · math · sequencer.linear

Concept: Solving sin x = k

Player does: Lay the planks: isolate, reference angle, quadrants, all solutions

Breaks: sin x = ½ has one solution

Solving a trig equation means isolating, finding the reference angle, then using every quadrant where the sign matches.

approach_target · math · function_world.limit

Concept: Limit as x → a

Player does: Walk toward x = a while the f(x) marker approaches its destination

Breaks: A limit is the value at the point

A limit describes where f(x) approaches as x nears a, which can differ from f(a) or exist even when f(a) does not.

two_path_convergence · math · function_world.limit

Concept: Two-sided limits

Player does: Approach by left and right routes; the gate opens only if both reach the same height

Breaks: One side is enough

A two-sided limit exists only when the left-hand and right-hand limits agree.

directional_scanner · math · function_world.limit

Concept: One-sided limits

Player does: Scan the machine from one side only; predict where the output heads

Breaks: x → a⁻ means x is negative

x → a⁻ means x approaches a from values less than a, not that x itself is negative.

broken_tile · math · function_world.limit

Concept: Limit exists at a hole

Player does: The tile at x = a is missing, but both paths converge on one spot

Breaks: No f(a), no limit

A limit can exist at a point even when the function itself is undefined there, as long as both sides converge to the same value.

decoy_destination · math · function_world.limit

Concept: Limit ≠ function value

Player does: Nearby paths converge on one platform, but the tile at a teleports elsewhere

Breaks: The limit always equals f(a)

A limit describes where nearby values head, which can differ from the function's actual defined value at that point.

split_gate · math · function_world.limit

Concept: Jump discontinuity (DNE)

Player does: Approaching from the left raises one gate half; from the right, a different height

Breaks: Every function has a limit at every point

When the left and right limits disagree, as at a jump, the two-sided limit does not exist.

runaway_elevator · math · function_world.limit

Concept: Infinite limits

Player does: Near the forbidden floor the elevator climbs faster and never settles

Breaks: An infinite limit is a very large number

An infinite limit means the function grows without bound, describing behavior, not a specific large numeric value.

long_horizon · math · function_world.asymptote

Concept: Limits at infinity

Player does: Travel far; the terrain levels toward a fixed altitude

Breaks: A curve can never cross its asymptote

A function can cross a horizontal asymptote near the middle of its domain; the asymptote only describes its far-away behavior.

closing_walls · math · function_world.squeeze

Concept: Squeeze theorem

Player does: Upper and lower walls close in; predict where the trapped orb ends

Breaks: The trapped function can escape between the bounds

If two bounding functions converge to the same limit, the squeezed function trapped between them must converge there too.

precision_zone · math · function_world.epsilon_delta

Concept: ε-δ definition

Player does: The target band shrinks; choose how close x must stay to a

Breaks: You pick δ first, then ε

The ε-δ definition works the other way: for any tolerance ε chosen first, a matching δ must be found.

unbroken_track · math · function_world.continuity

Concept: Continuity

Player does: A cart rides the track; mark where holes, jumps, and blow-ups stop it

Breaks: Continuous just means defined everywhere

Continuity requires the limit to exist and equal the function's value at every point, not merely being defined there.

repair_the_track · math · function_world.continuity

Concept: Removable discontinuity

Player does: Place the missing piece where the surrounding track says it belongs

Breaks: Any discontinuity can be patched with one point

Only a removable discontinuity, where the limit exists but the function value doesn't match, can be fixed by redefining one point.

slope_scanner · math · function_world.slope

Concept: Slope depends on position

Player does: Cross curved terrain with a tangent scanner; mark the steepest spot

Breaks: A curve has one slope

A curve's slope, given by its derivative, changes from point to point rather than staying fixed.

speed_snapshot · math · function_world.secant

Concept: Instantaneous rate

Player does: Freeze time over shorter windows to read the vehicle's speed

Breaks: Instantaneous rate = average rate over the trip

Instantaneous rate is the limit of the average rate as the time window shrinks to zero, not the average over the whole trip.

closing_gates · math · function_world.secant

Concept: Difference quotient → derivative

Player does: Two checkpoints slide together while Δy/Δx updates

Breaks: The secant slope is the derivative

The derivative is the limit of the secant slope as the two points merge, not the secant slope at any fixed separation.

momentum_direction · math · function_world.slope

Concept: Sign of f′

Player does: Mark where the terrain pushes you forward vs back

Breaks: f′ < 0 means f < 0

The sign of f′ tells whether f is increasing or decreasing, which is unrelated to whether f itself is positive or negative.

zero_force_zones · math · function_world.slope

Concept: Critical points

Player does: Find every spot where the horizontal push vanishes

Breaks: f′ = 0 always means a max or min

A critical point where f′ = 0 can be a local max, local min, or neither, like a saddle point.

curvature_gravity · math · function_world.slope

Concept: Concavity

Player does: Steering bends with how the slope itself changes

Breaks: Concave down means decreasing

Concavity describes how the slope itself is changing, which is independent of whether the function is increasing or decreasing.

gravity_flip · math · function_world.slope

Concept: Inflection points

Player does: Mark where the world's curvature flips

Breaks: An inflection point is where f′ = 0

An inflection point is where concavity changes, marked by f″ = 0, not necessarily where f′ = 0.

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.

derivative_machine · math · transformer.function_machine

Concept: Differentiation rules

Player does: The machine turns f into f′; predict its output for a new f

Breaks: d/dx x³ = 3x³

The power rule brings the exponent down as a coefficient and reduces it by one, so d/dx x³ = 3x², not 3x³.

tile_the_region · math · accumulator.riemann

Concept: Riemann sums

Player does: Place narrower and narrower blocks under the curved ceiling

Breaks: Left sums always underestimate

Whether a left or right Riemann sum over- or underestimates the true area depends on whether the function is increasing or decreasing.

fill_the_reservoir · math · accumulator.area

Concept: Definite integral as area

Player does: Stop the gate where the reservoir holds the target volume

Breaks: Area under a curve needs a geometric formula

The definite integral gives exact area under a curve even when no simple geometric shape formula applies.

energy_ledger · math · accumulator.signed

Concept: Signed area

Player does: Area above the axis charges the battery; below drains it

Breaks: Area below the axis counts as positive

The definite integral treats area below the x-axis as negative, subtracting from the total rather than adding.

resource_collector · math · accumulator.rate_total

Concept: Accumulation from a rate

Player does: Ride through changing production rates; predict inventory at t

Breaks: Total = rate × time even when the rate changes

When the rate changes, total accumulation is the integral of the rate over time, not a single rate times time.

rate_total_machine · math · accumulator.rate_total

Concept: Fundamental Theorem

Player does: Switch between the rate gauge and the storage tank; predict each

Breaks: Integrals and derivatives are unrelated

The Fundamental Theorem of Calculus links them directly: differentiation and integration are inverse operations.

integration_window · math · accumulator.area

Concept: Changing bounds

Player does: Drag the window's edges to hit the target total

Breaks: Swapping the bounds doesn't matter

Swapping the limits of integration flips the sign of the definite integral's value.

level_the_reservoir · math · accumulator.average_value

Concept: Average value

Player does: Flatten uneven water into a rectangle of equal area

Breaks: Average value is the average of the endpoints

The average value of a function over an interval is its integral divided by the interval length, not just the average of the endpoints.

rate_runner · math · accumulator.rate_total

Concept: Position from velocity

Player does: Sketch your ghost's velocity; position is the area under it

Breaks: The velocity graph shows the path's shape

A velocity-time graph shows speed and direction over time; its shape is not the shape of the object's actual path.

vector_winds · math · mapper.plane

Concept: Vector addition

Player does: Place fans; mark where their combined push carries you

Breaks: Vectors add like numbers (3 + 4 = 7)

Vectors add component-wise (or tip-to-tail), so their combined magnitude generally is not the simple sum of their lengths.

component_split · math · mapper.plane

Concept: Components

Player does: Split thrust into x and y engines that land you on target

Breaks: Components add up to the magnitude

A vector's magnitude comes from the Pythagorean combination of its components, not their simple sum.

transformation_grid · math · transformer.matrix

Concept: Matrices as transformations

Player does: Pick the 2×2 matrix that turns the rune into the door's shape

Breaks: Matrix multiplication is commutative

Matrix multiplication generally is not commutative, so applying two transformations in a different order gives a different result.

determinant_press · math · transformer.matrix

Concept: Determinant as area scale

Player does: Predict the stamped area after the matrix press

Breaks: The determinant is the sum of the entries

The determinant measures how a matrix scales area (or volume); it is computed from a product-and-difference formula, not a sum of entries.

repeat_experiment · math · simulator.sample

Concept: Variability

Player does: Rerun the identical experiment; predict the spread

Breaks: Same setup, same result

Random variation means identical setups can still produce different results from trial to trial.

prediction_stabilizer · math · simulator.sample

Concept: Law of large numbers

Player does: Choose how many trials until the estimate stays in the band

Breaks: After five heads, tails is due

Independent trials have no memory, so past outcomes don't change future odds; the law of large numbers only stabilizes averages over many trials.

long_run_casino · math · simulator.sample

Concept: Expected value

Player does: Pick the table with the best long-run payoff, then watch 1,000 plays

Breaks: The biggest prize is the best bet

Expected value weighs each payoff by its probability, so a rare huge prize can still be a worse bet than a smaller frequent one.

population_collector · math · simulator.sample

Concept: Sampling bias

Player does: Sample only reachable districts; explain why the estimate drifts

Breaks: A bigger biased sample fixes the bias

A biased sampling method stays biased no matter how large the sample gets; only fixing the method removes the bias.

capture_the_parameter · math · simulator.sample

Concept: Confidence intervals

Player does: Throw interval nets; predict how many catch the hidden true value

Breaks: There's a 95% chance this one interval holds the value

A 95% confidence level describes the method's long-run capture rate across many intervals, not the chance for one already-computed interval.

spread_field · math · sequencer.rank

Concept: Standard deviation

Player does: Rank datasets with the same mean by spread

Breaks: Same mean, same data

Two datasets can share the same mean while differing greatly in standard deviation, which measures the spread around it.

scatter_landscape · math · mapper.number_line

Concept: Correlation

Player does: Place each scatter plot's r on the −1…1 line

Breaks: r = 0 means no relationship of any kind

A correlation of r = 0 rules out a linear relationship but not a strong nonlinear one.

hidden_variable_mystery · math · investigator.elimination

Concept: Correlation vs causation

Player does: Find the hidden third system driving both meters

Breaks: Correlation proves causation

A correlation can be explained by a hidden confounding variable driving both, rather than one causing the other.

evidence_updater · math · mapper.number_line

Concept: Bayes' theorem

Player does: After the test result, set the belief meter

Breaks: A 99%-accurate test means a 99% chance you're sick

Bayes' theorem shows the chance of being sick given a positive test also depends on the disease's rarity, which can make it far below the test's accuracy rate.

restricted_world · math · mapper.number_line

Concept: Conditional probability

Player does: Irrelevant outcomes vanish once the condition is known; place P(A|B)

Breaks: P(A|B) = P(B|A)

Conditional probability is not symmetric in general: P(A|B) and P(B|A) can be very different values.

odds_forge · math · simulator.sample

Concept: Probability & counting

Player does: Choose the fork with the best odds of loot

Breaks: Two outcomes means 50/50

Probability depends on how many ways each outcome can occur, not just how many outcomes are listed.

mean_median_seesaw · math · balance.torque

Concept: Mean vs median

Player does: Add an outlier; move the fulcrum (mean) while the median marker stays put

Breaks: Mean and median always move together

The mean is pulled toward outliers while the median resists them, so the two can move very differently as data changes.

z_score_rapids · math · mapper.number_line

Concept: z-scores, normal distribution

Player does: Place the raft at the reading's z-score

Breaks: z = 2 means twice the average

A z-score of 2 means the value is two standard deviations above the mean, not double the mean itself.

motion_predictor · physics · mapper.number_line

Concept: Velocity

Player does: Predict where the cart will be after Δt, then watch

Breaks: Speed and velocity are the same thing

Velocity includes direction as well as speed, so two objects with the same speed can have different velocities.

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.

free_fall_drop · physics · truth_finder.predict_reveal

Concept: Free fall

Player does: In the vacuum chamber, predict whether the boulder or the feather lands first

Breaks: Heavier objects fall faster

Without air resistance, gravity accelerates every object at the same rate regardless of mass.

frictionless_arena · physics · truth_finder.predict_reveal

Concept: Newton's 1st law

Player does: Predict the puck's path after you stop pushing

Breaks: Moving things need a force to keep moving

An object in motion stays in motion at constant velocity unless a net force acts on it.

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.

recoil_movement · physics · simulator.intervene

Concept: Newton's 3rd law

Player does: Move only by throwing mass the opposite way

Breaks: The bigger object pushes harder

Newton's third law pairs are always equal and opposite in force, regardless of the objects' relative size.

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.

energy_converter · physics · balance.ledger

Concept: Energy conservation

Player does: Route energy among potential, kinetic, and spring meters

Breaks: Energy gets used up

Energy is conserved and converts between forms like potential and kinetic; it is never simply used up.

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.

lever_door · physics · balance.torque

Concept: Torque

Player does: Place the weight at the distance that opens the lever door

Breaks: Only the force matters, not where it's applied

Torque depends on both the force and its distance from the pivot, so where a force is applied matters as much as its size.

balance_builder · physics · balance.torque

Concept: Center of mass

Player does: Stack cargo so the center of mass stays over the base

Breaks: Center of mass is always the geometric center

Center of mass depends on how mass is distributed, so an unevenly loaded object's center of mass shifts off its geometric center.

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.

orbit_slingshot · physics · simulator.reach_state

Concept: Gravity & orbits

Player does: Set the launch speed that reaches a stable orbit

Breaks: There's no gravity in space

Orbiting objects are in continuous free fall under gravity; an orbit is a balance of gravity and sideways speed, not an absence of gravity.

charge_navigator · physics · mapper.plane

Concept: Electric fields

Player does: Mark where the test charge feels zero net force

Breaks: Field lines are paths charges must follow

Field lines show the direction of force at each point; a charge's actual path depends on its motion and other forces, not a fixed track.

voltage_height_map · physics · truth_finder.predict_reveal

Concept: Electric potential

Player does: Predict where charges roll on the voltage landscape

Breaks: Voltage flows through a circuit

Voltage is a potential difference between two points, like height on a landscape; current, not voltage, flows through a circuit.

flow_network · physics · balance.ledger

Concept: Current

Player does: Fill each branch's current so every junction balances

Breaks: Current is used up by bulbs

Current is conserved through a circuit; a bulb converts electrical energy to light and heat, but the current itself is not consumed.

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.

power_grid_builder · physics · builder.circuit

Concept: Series vs parallel

Player does: Wire the town so every house gets full voltage

Breaks: Adding bulbs in parallel dims all of them

Parallel branches each get the full source voltage, so adding a parallel bulb does not dim the others.

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.

heat_flow_rooms · physics · simulator.predict

Concept: Thermal equilibrium

Player does: Predict final temperatures when rooms share a wall

Breaks: Cold flows into warm objects

Heat flows from the warmer object to the cooler one until both reach thermal equilibrium; cold itself does not flow.

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.

phase_diagram_pin · physics · mapper.plane

Concept: Phases vs pressure & temperature

Player does: Place the substance on the P–T diagram to make ice, water, or steam

Breaks: Water always boils at 100 °C

Boiling point shifts with pressure, so water boils below 100°C at low pressure and above it at high pressure.

atom_conservation · chemistry · balance.chem_equation

Concept: Balancing equations

Player does: Set coefficients until every atom token balances and the reactor starts

Breaks: Change subscripts to balance

Balancing an equation means adjusting coefficients, since changing a subscript would change the substance itself.

reaction_factory · chemistry · balance.ratio

Concept: Stoichiometry

Player does: Feed reactants in the required ratio; leftovers pile up

Breaks: Reactants combine 1:1 by mass

Reactants combine in the mole ratios set by the balanced equation, not in equal masses.

production_bottleneck · chemistry · balance.ratio

Concept: Limiting reagent

Player does: Predict which feed runs out first and how much product ships

Breaks: The reactant with less mass is limiting

The limiting reagent is decided by mole ratio needed, not by which reactant has the smaller mass.

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.

dynamic_reaction_arena · chemistry · simulator.predict

Concept: Dynamic equilibrium

Player does: Watch forward and reverse reactions; predict when counts stop changing

Breaks: At equilibrium the reaction stops

At equilibrium, the forward and reverse reactions still run, just at equal rates, so concentrations stop changing but the reaction never stops.

disturb_the_reactor · chemistry · truth_finder.predict_reveal

Concept: Le Châtelier

Player does: Predict the shift, then add reactant or change pressure

Breaks: Adding product speeds the forward reaction

Adding product shifts the equilibrium backward, favoring the reverse reaction, not the forward one.

energy_barrier · chemistry · simulator.reach_state

Concept: Activation energy

Player does: Heat until enough molecules clear the barrier

Breaks: Exothermic reactions need no energy to start

Even an exothermic reaction needs activation energy to get started, before it releases net energy overall.

alternate_route · chemistry · truth_finder.predict_reveal

Concept: Catalysts

Player does: Unlock the tunnel; predict what changes and what doesn't

Breaks: Catalysts change how much energy is released

A catalyst lowers the activation energy path but does not change the overall energy released by the reaction.

proton_transfer · chemistry · linker.pairs

Concept: Brønsted acids & bases

Player does: Link each acid to its conjugate base by passing proton tokens

Breaks: Strong acid means concentrated acid

Acid strength is about how completely it donates protons, which is independent of its concentration.

ph_meter · chemistry · mapper.number_line

Concept: pH scale

Player does: Place solutions on the pH line; 10× concentration is one step

Breaks: pH 4 is twice as acidic as pH 5

pH is logarithmic, so each whole step is a tenfold change in acidity, meaning pH 4 is ten times more acidic than pH 5.

orbital_filling · chemistry · builder.electron_config

Concept: Electron configuration

Player does: Fill orbital slots obeying Aufbau, Hund, and Pauli

Breaks: Orbitals fill in number order (3d before 4s)

The Aufbau principle fills orbitals by increasing energy, which puts 4s below 3d in energy despite the smaller number.

periodic_landscape · chemistry · sequencer.rank

Concept: Periodic trends

Player does: Rank elements by atomic radius or electronegativity

Breaks: Atoms get bigger left to right

Atomic radius shrinks left to right across a period as nuclear charge pulls electrons in tighter.

escape_energy · chemistry · sequencer.rank

Concept: Intermolecular forces

Player does: Rank substances by boiling point from their attractions

Breaks: Boiling breaks covalent bonds

Boiling overcomes the intermolecular forces between molecules; the covalent bonds within each molecule stay intact.

molecule_builder · chemistry · builder.molecule

Concept: Bonding & valence

Player does: Assemble the requested molecule within valence rules

Breaks: Atoms can form any number of bonds

Each element's valence sets a limit on how many bonds it can typically form.

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.

reaction_type_sorter · chemistry · sorter.bins

Concept: Reaction types

Player does: Sort reactions into synthesis, decomposition, single/double replacement, combustion

Breaks: Any reaction with O₂ is combustion

Combustion specifically requires a fuel reacting rapidly with oxygen to release heat and light, not just any reaction involving O2.

polarity_sorter · chemistry · sorter.bins

Concept: Molecular polarity

Player does: Sort molecules polar vs nonpolar from their shape

Breaks: Polar bonds always make a polar molecule

A molecule's overall polarity depends on its geometry too, since symmetric arrangements of polar bonds can cancel out.

redox_tracker · chemistry · linker.pairs

Concept: Oxidation & reduction

Player does: Link each species to whether it's oxidized or reduced

Breaks: Oxidation always involves oxygen

Oxidation is the loss of electrons, a definition that applies even in reactions with no oxygen present.

particle_spread · biology · simulator.predict

Concept: Diffusion

Player does: Predict how particles spread from the crowded chamber

Breaks: Particles move toward empty space on purpose

Particles move randomly and simply spread out net from high to low concentration; they have no destination in mind.

membrane_balance · biology · simulator.predict

Concept: Osmosis

Player does: Predict which way water crosses and whether the cell swells or shrinks

Breaks: Salt crosses the membrane instead of water

In osmosis, water crosses the semipermeable membrane toward the higher solute concentration; the solute itself often cannot cross.

atp_gate · biology · simulator.intervene

Concept: Active transport

Player does: Spend ATP to pump ions against the gradient before the budget runs out

Breaks: Transport always goes down the gradient

Active transport uses ATP to move substances against their concentration gradient, unlike passive diffusion.

membrane_router · biology · sorter.bins

Concept: Membrane selectivity

Player does: Route molecules to simple diffusion, channel, or pump

Breaks: Anything small passes freely

The membrane is selectively permeable, so a molecule's charge and polarity, not just its size, decide its crossing route.

shape_docking · biology · linker.pairs

Concept: Enzyme specificity

Player does: Dock each substrate into its matching active site

Breaks: Enzymes are used up in reactions

An enzyme's active site releases the product and is reused, so the enzyme itself is not consumed.

dock_blocker · biology · simulator.predict

Concept: Competitive inhibition

Player does: Decoys occupy sites; predict how much more substrate restores the rate

Breaks: Inhibitors destroy enzymes

A competitive inhibitor blocks the active site reversibly; adding more substrate can outcompete it and restore the reaction rate.

genome_repair · biology · transformer.encode

Concept: DNA base pairing

Player does: Rebuild the damaged strand by complementary pairing

Breaks: A pairs with G

DNA base pairing is fixed: A pairs with T, and G pairs with C.

rna_assembly · biology · transformer.encode

Concept: Transcription

Player does: Read the template strand and assemble the mRNA (U for T)

Breaks: mRNA is an exact copy of the template strand

mRNA is complementary to the DNA template strand, not an identical copy, and it swaps thymine for uracil.

protein_factory · biology · transformer.encode

Concept: Translation

Player does: Route codons to amino acids with the codon table to build the chain

Breaks: Each nucleotide codes for one amino acid

A group of three nucleotides, a codon, specifies one amino acid, not a single nucleotide.

chromosome_logistics · biology · sequencer.linear

Concept: Mitosis

Player does: Duplicate and separate chromosome sets in stage order

Breaks: Chromosomes duplicate during prophase

Chromosomes duplicate during interphase, before mitosis's stages of prophase, metaphase, anaphase, and telophase begin.

meiosis_shuffle · biology · simulator.sample

Concept: Meiosis & variation

Player does: Shuffle homolog pairs; predict how many gamete types appear

Breaks: Meiosis makes identical cells

Meiosis shuffles homologous chromosomes to produce genetically varied gametes, not identical cells.

punnett_forge · biology · builder.genetics

Concept: Mendelian genetics

Player does: Fill the Punnett square to breed a familiar with the needed trait

Breaks: Dominant traits are always the most common

A trait's frequency in a population depends on allele frequency, not simply on whether the allele is dominant.

survival_sim · biology · simulator.predict

Concept: Natural selection

Player does: Change the environment; predict which trait spreads

Breaks: Individuals evolve because they need to

Natural selection acts on existing variation across a population over generations; individuals themselves do not evolve by need.

drift_sim · biology · simulator.sample

Concept: Genetic drift

Player does: Run tiny populations and watch alleles vanish by chance

Breaks: Evolution always improves fitness

Genetic drift changes allele frequencies by random chance, especially in small populations, regardless of fitness.

ecosystem_network · biology · linker.network

Concept: Food webs

Player does: Remove a species; trace the cascade through the web

Breaks: Removing a predator helps every other species

Removing a predator can trigger a cascade that harms some species, such as prey overpopulating and depleting their own food.

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.

feedback_controller · biology · simulator.intervene

Concept: Homeostasis

Player does: Release insulin or glucagon to keep glucose in the safe band

Breaks: Hormones switch on and stay on

Hormonal control in homeostasis is a continuous feedback loop, releasing and easing off as the level shifts, not an on/off switch.

threshold_pulse · biology · simulator.intervene

Concept: Action potentials

Player does: Stack inputs until threshold fires an all-or-none spike

Breaks: Stronger stimuli make bigger action potentials

An action potential is all-or-none once it fires; a stronger stimulus raises firing frequency, not the spike's size.

organelle_city · biology · linker.pairs

Concept: Organelle functions

Player does: Assign each city service to its organelle

Breaks: Plant cells don't have mitochondria

Plant cells have mitochondria for cellular respiration in addition to chloroplasts for photosynthesis.

taxonomy_tower · biology · sorter.hierarchy

Concept: Classification hierarchy

Player does: Nest kingdom through species into a climbable tower

Breaks: Species that look alike are closest relatives

Taxonomic closeness is based on shared ancestry and genetics, and can differ from superficial resemblance.

levels_tower · biology · sequencer.linear

Concept: Levels of organization

Player does: Climb cell → tissue → organ → system → organism

Breaks: Organs are built directly from cells

Cells form tissues, tissues form organs, so organs are built from tissues, not directly from cells.

photosynthesis_recipe · biology · transformer.composition

Concept: Photosynthesis

Player does: Route light, water, CO₂ through the stations to make glucose

Breaks: Plants get their mass from the soil

Most of a plant's mass comes from carbon dioxide fixed by photosynthesis, not from the soil.

respiration_forge · biology · transformer.composition

Concept: Cellular respiration

Player does: Run glucose through glycolysis, Krebs, and the ETC to charge ATP

Breaks: Plants don't do cellular respiration

Plants perform cellular respiration to make ATP just as animals do, in addition to photosynthesis.

krebs_wheel · biology · sequencer.cycle

Concept: Krebs cycle

Player does: Rotate the wheel so each intermediate feeds the next

Breaks: Glucose enters the Krebs cycle directly

Glucose is broken down to pyruvate and then acetyl-CoA before entering the Krebs cycle, not directly.

heart_run · biology · sequencer.linear

Concept: Blood flow

Player does: Sprint through the chambers in blood-flow order; valves are one-way gates

Breaks: All arteries carry oxygen-rich blood

The pulmonary artery carries deoxygenated blood to the lungs, so arteries are not defined by carrying oxygen-rich blood.

boss_anatomy · biology · linker.pairs

Concept: Parts & functions

Player does: Strike the weak point whose function the announcer names

Breaks: The right side of the heart pumps to the body

The right side of the heart pumps to the lungs, while the left side pumps oxygenated blood to the body.

immune_defense · biology · sorter.type_match

Concept: Innate vs adaptive immunity

Player does: Deploy the right defender against each invader

Breaks: Antibiotics kill viruses

Antibiotics target bacterial processes and have no effect on viruses.

vaccine_memory · biology · simulator.predict

Concept: Immunological memory

Player does: Predict how fast the second infection is cleared

Breaks: Vaccines give you the disease

A vaccine trains memory cells to recognize a pathogen without causing the actual disease.

rock_cycle_wheel · earth_space · sequencer.cycle

Concept: Rock cycle

Player does: Rotate processes so each rock type becomes the next

Breaks: Rocks only change in one direction

The rock cycle has no fixed starting point; igneous, sedimentary, and metamorphic rock can each transform into any other type.

strata_xray · earth_space · sequencer.rank

Concept: Relative dating

Player does: X-ray the cliff; order layers oldest to youngest

Breaks: Layer order says nothing about age

The law of superposition means undisturbed rock layers get younger from bottom to top.

plate_boundary_sorter · earth_space · sorter.bins

Concept: Plate boundaries

Player does: Sort features to convergent, divergent, or transform boundaries

Breaks: Earthquakes only happen at convergent boundaries

Earthquakes occur at all three boundary types, including the sliding motion of transform boundaries.

earthquake_triangulation · earth_space · mapper.plane

Concept: Locating epicenters

Player does: Draw three distance circles; mark where they meet

Breaks: One station can locate the epicenter

A single seismic station gives only a distance, so at least three stations are needed to triangulate the epicenter.

water_cycle_wheel · earth_space · sequencer.cycle

Concept: Water cycle

Player does: Restart the cycle by triggering stages in order

Breaks: Clouds are water vapor

Clouds are condensed liquid droplets or ice, while water vapor itself is invisible gas.

moon_phase_orbit · earth_space · mapper.plane

Concept: Moon phases

Player does: Place the Moon in its orbit to show the requested phase

Breaks: Phases are Earth's shadow

Moon phases come from the changing angle at which we see the Moon's sunlit half, not from Earth's shadow.

seasons_tilt · earth_space · truth_finder.mimic

Concept: Seasons

Player does: Expose the claim that summer is when Earth is closest to the Sun

Breaks: Seasons come from distance to the Sun

Seasons come from Earth's axial tilt changing sunlight angle and day length, not from distance to the Sun.

greenhouse_dial · earth_space · simulator.intervene

Concept: Greenhouse effect

Player does: Adjust emissions to keep temperature in the band

Breaks: The greenhouse effect is the ozone hole

The greenhouse effect traps heat via gases like CO2, which is a separate process from ozone depletion.

star_life · earth_space · sequencer.linear

Concept: Stellar evolution

Player does: Route the star through its life stages by mass

Breaks: All stars end as black holes

A star's mass decides its fate: only very massive stars can collapse into a black hole; smaller stars end as white dwarfs or neutron stars.

weather_fronts · earth_space · truth_finder.predict_reveal

Concept: Air masses & fronts

Player does: Predict the weather as the cold front arrives

Breaks: Warm air slides under cold air at a front

Warm air is less dense, so at a front it rises up and over the advancing cold air, not the other way around.

state_containers · cs · transformer.trace

Concept: Variables & assignment

Player does: Predict each container's value after the instructions run

Breaks: x = x + 1 is an impossible equation

In code, = is assignment, not algebraic equality, so x = x + 1 replaces x's stored value with one more than itself.

branch_doors · cs · transformer.trace

Concept: Conditionals

Player does: Predict which door the branching world opens for the current state

Breaks: Both branches of an if run

An if statement runs exactly one of its branches, chosen by whether the condition is true or false.

automation_track · cs · builder.program

Concept: Loops

Player does: Configure a repeating action instead of stepping manually

Breaks: A loop checks its condition only once

A loop re-checks its condition every iteration, which is what lets it keep running or stop.

reusable_machines · cs · builder.program

Concept: Functions

Player does: Define a machine once and call it at several stations

Breaks: Functions run when they're defined

A function's body runs only when it is called, not at the moment it is defined.

nested_rooms · cs · transformer.trace

Concept: Recursion

Player does: Each room holds a smaller copy; predict the result as base cases return

Breaks: Recursion never stops

Recursion terminates once it reaches a base case, which stops the chain of smaller calls from growing further.

indexed_inventory · cs · transformer.trace

Concept: Arrays & indexing

Player does: Fetch items by slot from a zero-indexed inventory

Breaks: Indexes count from 1 in every language

Many languages index arrays starting at 0, so the first element sits at index 0, not 1.

pointer_chain · cs · linker.chain

Concept: Linked lists

Player does: Relink next-pointers to insert a node

Breaks: You can jump straight to the 5th node

A linked list has no direct index access; reaching the 5th node means following the chain of pointers from the head.

tower_access · cs · sequencer.linear

Concept: Stacks

Player does: Predict the pop order after pushes and pops

Breaks: First in, first out

A stack is last in, first out: the most recently pushed item is the one that pops first.

processing_line · cs · sequencer.linear

Concept: Queues

Player does: Predict the service order at the processing line

Breaks: The newest arrival is served first

A queue is first in, first out, so the earliest arrival is served before newer ones.

branch_explorer · cs · sequencer.linear

Concept: Tree traversal

Player does: Order the rooms visited in pre-, in-, or post-order

Breaks: In-order means top to bottom

In-order traversal visits left subtree, node, then right subtree, which is not the same as visiting top to bottom.

half_split_hunt · cs · mapper.search

Concept: Binary search

Player does: Find the hidden number within ⌈log₂ n⌉ probes

Breaks: Binary search works on unsorted data

Binary search relies on the data being sorted so each probe can eliminate half the remaining range.

ordering_conveyor · cs · transformer.trace

Concept: Sorting algorithms

Player does: Predict the conveyor after pass k of bubble or insertion sort

Breaks: Every sort does the same work

Different sorting algorithms compare and move elements in different patterns, so their intermediate states differ.

network_explorer · cs · sequencer.linear

Concept: BFS vs DFS

Player does: Predict the visit order as BFS spreads and DFS dives

Breaks: BFS and DFS visit in the same order

BFS explores level by level with a queue while DFS dives deep with a stack, so they visit nodes in different orders.

weighted_path_planner · cs · linker.path

Concept: Shortest paths (Dijkstra)

Player does: Find the cheapest route across weighted bridges

Breaks: Fewest edges means cheapest

The shortest path minimizes total edge weight, which can require more edges than the fewest-edge route.

bucket_router · cs · transformer.function_machine

Concept: Hashing

Player does: Predict each key's bucket from h(k) = k mod m; spot collisions

Breaks: Hashing sorts the keys

A hash function maps keys to buckets for fast lookup; it does not order the keys.

memory_warehouse · cs · transformer.trace

Concept: Caching & locality

Player does: Predict hits and misses as requests arrive

Breaks: Every memory access costs the same

A cache hit is far cheaper than a miss, so locality of access strongly affects real performance.

race_switches · cs · truth_finder.predict_reveal

Concept: Race conditions

Player does: Predict the final counter when two agents interleave

Breaks: Two increments always add 2

Without synchronization, interleaved increments can overwrite each other, so two increments may add less than 2.

deadlock_locks · cs · linker.network

Concept: Deadlock

Player does: Find the cycle in the wait-for graph and break it

Breaks: Deadlock needs a bug in a single thread

Deadlock arises from a cycle of threads each waiting on a resource the other holds, not from a bug in one thread alone.

state_world · cs · linker.network

Concept: Finite state machines

Player does: Draw the transitions so the world accepts the input

Breaks: A state machine remembers its whole history

A finite state machine's next move depends only on its current state and input, not its full history.

big_o_race · cs · sequencer.rank

Concept: Complexity

Player does: Rank runners by growth rate as n explodes

Breaks: An O(n²) algorithm is always slower

Big-O describes growth as input size gets large; an O(n²) algorithm can still be faster than a worse-growth one on small inputs.

code_golem · cs · builder.program

Concept: Algorithms

Player does: Program the golem to build the staircase

Breaks: Computers do what you meant

A computer executes exactly what an algorithm specifies, not what its author intended.

cipher_door · cs · transformer.encode

Concept: Binary, hex, encodings

Player does: Translate the carving from binary or hex to unlock the door

Breaks: Hex digits only go 0–9

Hexadecimal digits run 0 through F, using A–F to represent values ten through fifteen.

bug_hunter · cs · truth_finder.error_hunt

Concept: Debugging

Player does: Strike the scroll line that causes the bug

Breaks: The bug is on the line where the error shows

A bug's root cause often sits earlier in the code than the line where the error finally surfaces.

osi_elevator · cs · sequencer.linear

Concept: Network layers

Player does: Ride a packet down and up the protocol stack

Breaks: Layers can be skipped

Each OSI layer depends on the one below it, so data must pass through every layer in order, never skipping one.

current_junction · engineering · balance.ledger

Concept: KCL

Player does: Balance every junction: current in = current out

Breaks: Current gets used up at junctions

Kirchhoff's current law says current in equals current out at every junction; none of it is consumed there.

voltage_loop · engineering · balance.ledger

Concept: KVL

Player does: Make rises and drops around the loop sum to zero

Breaks: Voltage is the same everywhere in series

Kirchhoff's voltage law says the rises and drops around a loop sum to zero, so voltage drops across each series element.

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.

logic_factory · engineering · builder.circuit

Concept: Logic gates

Player does: Wire gates so the output matches the target truth table

Breaks: You need AND, OR, and NOT to build any circuit

NAND alone is functionally complete, so every logic circuit can be built from NAND gates without AND, OR, or NOT.

state_explorer · engineering · builder.circuit

Concept: Truth tables

Player does: Test every input combination and fill the table

Breaks: Checking a few inputs proves the circuit

A circuit is verified only by testing every input combination in its truth table, not a sample of them.

grouping_territory · engineering · builder.tiles

Concept: Karnaugh maps

Player does: Cover the 1-cells with the fewest power-of-two rectangles

Breaks: Groups can be any size

Karnaugh map groups must be a power-of-two size (1, 2, 4, 8…) to correctly simplify the expression.

signal_railway · engineering · transformer.trace

Concept: Multiplexers

Player does: Set select lines so the right input track reaches the output

Breaks: Select lines carry the data

Select lines only choose which data input is routed through; they never carry the data themselves.

one_bit_room · engineering · transformer.trace

Concept: Flip-flops

Player does: Predict the stored bit after each clock edge

Breaks: Outputs change whenever inputs change

A flip-flop's stored output changes only at a clock edge, not whenever its inputs change.

safe_sampling_window · engineering · mapper.number_line

Concept: Setup & hold time

Player does: Place the input transition outside the window around the clock edge

Breaks: Any change before the edge is safe

A signal must be stable through both the setup time before and the hold time after the clock edge to be sampled safely.

instruction_factory · engineering · builder.tiles

Concept: Pipelining

Player does: Schedule instructions into stages; count total cycles

Breaks: Pipelining makes each instruction faster

Pipelining raises instruction throughput by overlapping stages; it does not shorten any single instruction's latency.

datapath_relay · engineering · sequencer.linear

Concept: Instruction execution

Player does: Route an instruction through fetch, decode, execute, memory, write-back

Breaks: The PC changes only on jumps and branches

The program counter advances after every instruction, not only on a jump or branch.

memory_distance · engineering · sequencer.rank

Concept: Memory hierarchy

Player does: Rank storage by latency and capacity

Breaks: Bigger memory is faster

Memory hierarchy trades capacity for speed: the fastest tiers, like cache, are the smallest.

path_prediction · engineering · simulator.sample

Concept: Branch prediction

Player does: Guess each branch; watch the pipeline flush on a miss

Breaks: Mispredictions cost nothing

A branch misprediction forces the pipeline to flush and restart, costing real cycles.

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.

reflex_arc · health · sequencer.linear

Concept: Reflex arc

Player does: Order receptor, sensory neuron, spinal cord, motor neuron, effector

Breaks: Reflexes go through the brain first

A reflex arc routes through the spinal cord, which is what makes it faster than a decision made by the brain.

antibiotic_resistance · health · simulator.predict

Concept: Resistance evolution

Player does: Stop treatment early; predict which bacteria survive

Breaks: People become resistant to antibiotics

Antibiotic resistance evolves in the bacteria population, as the drug-tolerant survivors of an incomplete course multiply.

timeline_reconstruction · history · sequencer.timeline

Concept: Chronology

Player does: Rebuild the events in order; dates reveal after placing

Breaks: Events happened in the order the textbook mentions them

A textbook's narrative order can differ from the true chronology of events, which is what dated evidence establishes.

domino_engine · history · linker.chain

Concept: Cause & effect chains

Player does: Place causes so the chain topples through to the consequence

Breaks: History is a list of unrelated events

Historical events form causal chains, where each link sets up the conditions for the next.

causal_weighting · history · investigator.weigh

Concept: Competing causes

Player does: Allocate importance among causes; the explanation shifts with the weights

Breaks: Every event has one cause

Major historical events usually have several contributing causes whose relative weight historians still debate.

evidence_lab · history · sorter.bins

Concept: Primary vs secondary sources

Player does: Inspect documents and classify each one

Breaks: Old means primary

A primary source comes directly from the time or people being studied; age alone does not decide the category.

source_seer · history · investigator.source_eval

Concept: Source reliability

Player does: Weigh testimony by author, timing, incentives, corroboration

Breaks: Eyewitnesses are always reliable

An eyewitness account's reliability depends on the author's position, timing, incentives, and corroboration, not on eyewitness status alone.

perspective_switch · history · investigator.perspective

Concept: Historical perspective

Player does: Replay the event as different actors; match accounts to motives

Breaks: People in the past thought like us

Historical actors reasoned from the beliefs and constraints of their own time, which often differ sharply from ours.

map_evolution · history · mapper.map

Concept: Borders & movement over time

Player does: Move borders, migrations, and battles to the right year

Breaks: Borders have always been where they are now

Borders have shifted repeatedly through conquest, treaty, and negotiation, so today's map is a recent snapshot.

evidence_board · history · investigator.elimination

Concept: Inference from evidence

Player does: Pin clues; eliminate hypotheses until one explanation stands

Breaks: Any evidence supports any claim

Evidence supports a specific hypothesis only when it is consistent with that hypothesis and inconsistent with the rivals.

counterevidence_attack · history · investigator.argument

Concept: Qualifying a claim

Player does: Opposing evidence weakens your claim until you qualify it

Breaks: A strong argument ignores counterevidence

A strong historical argument addresses counterevidence directly and qualifies its claim instead of ignoring it.

context_reconstruction · history · recall.cloze

Concept: Period vocabulary

Player does: Infer a term's meaning from its use in the source

Breaks: Terms meant then what they mean now

Words can shift meaning over time, so a period term's sense has to be inferred from its use in context, not today's usage.

oracle_of_consequence · history · truth_finder.predict_reveal

Concept: Decisions & consequences

Player does: Advise the ruler, then see the documented outcome

Breaks: The outcome was inevitable

Historical outcomes followed from real, contested decisions, not an inevitable script.

who_said_it · history · linker.pairs

Concept: Figures & ideas

Player does: Match quotes and positions to historical figures

Breaks: Reformers of an era all wanted the same thing

Reform movements within the same era often held competing, even conflicting, goals.

era_sorter · history · sorter.bins

Concept: Periodization

Player does: Sort artifacts and events into eras

Breaks: Eras begin and end on exact dates

Historical eras are analytical categories with fuzzy, debated boundaries, not events with exact start and end dates.

museum_palace · history · recall.memory_palace

Concept: Remembering lists

Player does: Tour the museum, then recall which exhibit stood in which room

Breaks: Rereading is the best way to memorize

Spatial memory techniques like a memory palace outperform passive rereading for recalling lists.

branch_router · civics · sorter.bins

Concept: Separation of powers

Player does: Route each power to the legislative, executive, or judicial branch

Breaks: The president makes laws

Congress makes laws; the president's executive branch enforces them.

bill_to_law · civics · sequencer.linear

Concept: Legislative process

Player does: Move the bill through every stage to become law

Breaks: A bill becomes law after one vote

A bill must pass committee, both chambers, and the president's signature or an override before it becomes law.

checks_network · civics · linker.network

Concept: Checks and balances

Player does: Draw which branch checks which, and how

Breaks: Checks only run in one direction

Checks and balances run between every pair of branches, not just from one branch down to another.

amendment_match · civics · linker.pairs

Concept: Rights & amendments

Player does: Match each scenario to the amendment it invokes

Breaks: The First Amendment binds private companies

The Bill of Rights restrains government action, not the policies of private companies.

federalism_venn · civics · sorter.venn

Concept: Federalism

Player does: Place powers in the federal, state, or shared regions

Breaks: States can't make laws of their own

Federalism divides power between national and state governments, and states retain law-making authority of their own.

precedent_court · civics · investigator.argument

Concept: Judicial precedent

Player does: Build the ruling from the facts and prior cases

Breaks: Courts decide on personal opinion

Courts ground rulings in the facts of the case and applicable precedent, not a judge's personal opinion.

electoral_math · civics · balance.ledger

Concept: Electoral systems

Player does: Allocate electoral votes to reach the threshold

Breaks: Winning the national popular vote guarantees the presidency

The presidency is decided by a majority of electoral votes, which can diverge from the national popular vote.

map_placement · geography · mapper.map

Concept: Locations & regions

Player does: Pin places onto the regions of the stylized map

Breaks: Maps show countries at their true size

Most flat maps distort area with the projection, so a country's map size is not its true relative size.

lat_long_navigator · geography · mapper.plane

Concept: Latitude & longitude

Player does: Sail to the coordinates on the grid

Breaks: Longitude lines are parallel

Longitude lines converge at the poles, while only latitude lines run parallel to each other.

climate_zone_sorter · geography · sorter.bins

Concept: Climate zones

Player does: Sort cities into climate zones from their data

Breaks: Latitude alone sets climate

Climate depends on elevation, ocean currents, and prevailing winds too, not on latitude alone.

migration_flows · geography · linker.network

Concept: Push & pull factors

Player does: Connect factors to the migrations they drove

Breaks: Migration is only about money

Migration is driven by a mix of push and pull factors, including safety, family, and opportunity, not money alone.

population_pyramid · geography · simulator.predict

Concept: Demographic transition

Player does: Predict the pyramid's shape a generation later

Breaks: Population always grows

As birth and death rates fall through the demographic transition, a population's growth can slow, plateau, or shrink.

living_marketplace · economics · simulator.predict

Concept: Supply & demand

Player does: Shift supply or demand; predict the new price and quantity

Breaks: Higher prices increase demand

Higher prices reduce quantity demanded along the demand curve; only a shift of the whole curve changes demand itself.

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.

hidden_spillover · economics · simulator.predict

Concept: Externalities

Player does: Your factory's costs land on neighbors; predict the true social cost

Breaks: The market price captures every cost

An externality falls on people outside the transaction, so the market price can miss part of the true social cost.

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.

gdp_sorter · economics · sorter.bins

Concept: GDP components

Player does: Sort transactions into C, I, G, NX, or not counted

Breaks: Buying a used car counts toward GDP

GDP counts newly produced goods and services, so a resale like a used car is excluded.

policy_levers · economics · sorter.bins

Concept: Fiscal vs monetary policy

Player does: Assign each lever to the central bank or the legislature

Breaks: The president sets interest rates

Interest rates are set by the central bank's monetary policy, while the legislature controls fiscal policy like spending and taxes.

business_cycle_wheel · economics · sequencer.cycle

Concept: Business cycle

Player does: Turn the wheel through expansion, peak, contraction, trough

Breaks: Recessions come out of nowhere

Recessions follow the business cycle's contraction phase, which typically follows a peak of overextended growth.

money_multiplier · economics · simulator.reach_state

Concept: Money creation (textbook model)

Player does: Set the reserve ratio so deposits expand to the target

Breaks: Banks keep every deposit in the vault

Banks lend out most of a deposit and keep only a fraction in reserve, which is what expands the money supply.

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.

credit_card_trap · finance · truth_finder.predict_reveal

Concept: Minimum payments & APR

Player does: Predict the payoff time on minimum payments, then watch

Breaks: Paying the minimum clears debt quickly

Minimum payments are sized to barely cover interest, so they can take years to clear a balance.

budget_balance · finance · balance.ledger

Concept: Budgeting

Player does: Balance income against expenses and savings goals

Breaks: Budgets only need to track big purchases

A budget only balances when every regular expense, not just the big ones, is tracked against income.

diversification_sim · finance · simulator.sample

Concept: Risk & diversification

Player does: Build portfolios; watch thousands of market years

Breaks: Diversifying lowers returns without lowering risk

Diversification spreads risk across uncorrelated assets, which can lower volatility without a matching loss in expected return.

conditioning_lab · psychology · sorter.bins

Concept: Classical vs operant conditioning

Player does: Sort scenarios into classical or operant and label the parts

Breaks: Negative reinforcement is punishment

Negative reinforcement removes something aversive to increase a behavior, while punishment decreases a behavior.

reinforcement_schedule · psychology · simulator.predict

Concept: Reinforcement schedules

Player does: Predict response persistence under fixed vs variable schedules

Breaks: Rewarding every time builds the most persistent habit

Variable schedules of reinforcement produce more persistent behavior than rewarding every single response.

memory_stages · psychology · sequencer.linear

Concept: Memory model

Player does: Route information from sensory to short-term to long-term memory

Breaks: Memory records like a video camera

Memory is reconstructed through encoding, storage, and retrieval stages, not recorded and replayed like video.

bias_detector · psychology · linker.pairs

Concept: Cognitive biases

Player does: Match each scenario to the bias it shows

Breaks: Only other people are biased

Cognitive biases are automatic patterns in everyone's thinking, not a flaw limited to other people.

experiment_designer · psychology · sorter.bins

Concept: Experimental design

Player does: Label the IV, DV, controls, and confounds

Breaks: Correlational studies show causes

Only a controlled experiment that manipulates the independent variable can support a causal claim; correlation alone cannot.

brain_regions · psychology · linker.pairs

Concept: Brain structure & function

Player does: Match functions to brain regions

Breaks: We only use 10% of our brain

Brain imaging shows activity across nearly the whole brain over a day, not just 10% of it.

maslow_tower · psychology · sorter.hierarchy

Concept: Hierarchy of needs

Player does: Stack needs into the tower's levels

Breaks: Higher needs never matter until lower ones are fully met

Maslow's levels describe a general tendency, and people often pursue higher needs before lower ones are fully satisfied.

syllogism_gate · philosophy · truth_finder.counterexample

Concept: Validity

Player does: Find the counterexample world that breaks the invalid argument

Breaks: True premises and a true conclusion make an argument valid

Validity is about the argument's form, so it needs a conclusion that must follow, not just premises and conclusion that happen to be true.

fallacy_duel · philosophy · sorter.type_match

Concept: Fallacies

Player does: Parry each fallacy the Sophist throws with its name

Breaks: An argument with a fallacy has a false conclusion

A fallacy is a flaw in the reasoning, so a fallacious argument can still happen to reach a true conclusion.

argument_map · philosophy · investigator.argument

Concept: Argument structure

Player does: Connect premises to the conclusion; cut the irrelevant ones

Breaks: More premises make a stronger argument

An argument's strength comes from premises that actually support the conclusion, not from how many premises are listed.

ethics_lenses · philosophy · sorter.bins

Concept: Ethical frameworks

Player does: Sort judgments into utilitarian, deontological, or virtue reasoning

Breaks: Deontology judges actions by their consequences

Deontology judges actions by whether they follow a duty or rule, while consequences are what utilitarianism weighs.

truth_table_logic · philosophy · builder.circuit

Concept: Propositional logic

Player does: Fill the truth table for the compound statement

Breaks: "If P then Q" means "if Q then P"

A conditional and its converse can differ in truth value, so "if P then Q" does not imply "if Q then P".

plot_arc · literature · sequencer.linear

Concept: Plot structure

Player does: Place scenes along exposition → rising action → climax → resolution

Breaks: The climax is the ending

The climax is the turning point of highest tension, and the resolution afterward is what closes the story.

character_motive · literature · investigator.perspective

Concept: Characterization & motive

Player does: Match actions and quotes to each character's motive

Breaks: Characters always say what they mean

Characterization is built from actions and subtext as much as dialogue, so what a character says is not always what they mean.

device_hunter · literature · sorter.type_match

Concept: Literary devices

Player does: Tag each passage's device as it scrolls past

Breaks: Every comparison is a metaphor

A simile compares with "like" or "as" while a metaphor asserts one thing is another, so not every comparison is a metaphor.

theme_evidence · literature · investigator.argument

Concept: Theme

Player does: Support a theme claim with passages from the text

Breaks: The theme is the topic

A theme is the insight the work makes about its topic, not the topic itself.

unreliable_narrator · literature · investigator.source_eval

Concept: Narrative reliability

Player does: Flag where the narrator's account conflicts with the facts

Breaks: The narrator is the author

A narrator is a constructed voice the author creates, whose account can conflict with the author's own view or the facts of the story.

setting_map · literature · mapper.map

Concept: Setting

Player does: Place key events on the story's map

Breaks: Setting is just backdrop

Setting shapes plot and character choices, functioning as more than a passive backdrop.

thesis_forge · writing · truth_finder.mimic

Concept: Arguable thesis

Player does: Expose the "thesis" that's really a statement of fact

Breaks: A thesis announces the topic

A thesis makes an arguable claim about the topic, not just a statement of fact or a restatement of the topic.

appeal_sorter · writing · sorter.bins

Concept: Ethos, pathos, logos

Player does: Sort ad lines by rhetorical appeal

Breaks: Pathos means manipulating with emotion

Pathos is a legitimate appeal to emotion that can support an argument, not by definition a manipulation.

paragraph_builder · writing · sequencer.linear

Concept: Paragraph structure

Player does: Order claim, evidence, analysis, transition

Breaks: Evidence speaks for itself

Evidence needs analysis connecting it back to the claim; it rarely makes its point on its own.

evidence_fit · writing · investigator.argument

Concept: Supporting evidence

Player does: Choose evidence that actually supports the claim

Breaks: Any quote on the topic is evidence

Evidence must directly support the specific claim being made, not merely mention the same topic.

transition_bridge · writing · linker.pairs

Concept: Transitions

Player does: Match transitions to the relationship each signals

Breaks: "However" and "therefore" are interchangeable

Each transition word signals a specific logical relationship, so "however" and "therefore" cannot be swapped.