Put your curiosity to the test.

After searching for yourself

Checking the middle number lets you rule out the half without the answer in one step.

Binary search can be used when the data is sorted. This experience uses numbers that are already sorted, so the cost of sorting is excluded from the comparison. If the target is at the very beginning, linear search may find it faster.

Read more about the principle · NIST: Binary Search

After breaking down the wait

File transfer speed and the time spent waiting for a response are separate factors.

Total time = one round-trip latency + file size × 8 ÷ connection speed. This simplified model assumes a constant transfer speed with no connection setup or congestion. Actual downloads vary with repeated communications, server processing, congestion, and transmission loss.

Read more about the principle · Cloudflare: Network Latency

After comparing for yourself

The lower the pH, the higher the approximate hydrogen ion concentration.

Here, we calculate [H₃O⁺] ≈ 10^−pH mol/L. Strictly, pH is defined using activity rather than concentration. A neutral pH of 7 applies at 25°C, and pH alone cannot determine safety or the strength of the acid itself. 0–14 is the exploration range for this experience; not all solutions have a pH within it.

Read more about the principle · OpenStax: pH and pOH

After pairing for yourself

The molecules that can be paired in the required ratio determine how many water molecules form.

Each group requires 2 H₂ molecules and 1 O₂ molecule. The number of possible groups is the smaller of the integer part of 'hydrogen count ÷ 2' and the oxygen count. An odd hydrogen count leaves 1 hydrogen molecule unpaired, so both kinds may remain. This is not an experiment reproducing actual reaction rates, conditions, or yields.

Read more about the principle · OpenStax: Limiting Reactants and Reaction Yields

After changing the circuit values

At constant resistance, current is proportional to voltage, and power is proportional to voltage squared.

Calculations use Ohm's law, I = V ÷ R, and power P = V² ÷ R. This is a DC model of a single resistor with constant resistance. In real materials, light bulbs, and electronic devices, resistance may change with temperature and operating conditions. This on-screen experience explains the relationships and does not provide instructions for actual wiring.

Read more about the principle · OpenStax: Ohm's Law · Power and Electrical Energy

After changing the position

Kinetic energy increases by the amount that potential energy decreases.

This model uses gravitational acceleration of 9.8 m/s² with no friction or air resistance. If the initial height is h and the fraction fallen is p, potential energy is mgh(1 − p), kinetic energy is mghp, and speed is √(2ghp). Potential energy is measured relative to the ground, and 100% means just before impact. Transformations into deformation, heat, and sound after impact are excluded.

Read more about the principle · OpenStax: Gravitational Potential Energy · NASA: Conservation of Energy

After making changes yourself

When incomes are similar, the mean and median are close.

The mean is the sum of all values divided by the number of people. The median is the middle value when the values are ordered by size. For six people, it is the mean of the two middle values. The appropriate measure of central tendency depends on the distribution and the question.

After making changes yourself

Changing colors gives the same layout a different impression.

The background color changes the impression of large areas, while the accent color changes where the eye is drawn. The brightness difference between text and background affects readability. The displayed value is the contrast ratio calculated from the relative luminance of the two colors.

After shelving for yourself

The numbers group books by subject and indicate where to find them on the shelf.

KDC is a system for classifying books by subject. Here, you practice ordering books using classification numbers for broad subjects. In actual libraries, books with the same classification number are ordered using other parts of the call number, such as author marks, and library rules.

After making changes yourself

With compound interest, future interest is earned on both the principal and the interest already accumulated.

Simple-interest amount = principal × (1 + annual interest rate × period); annually compounded amount = principal × (1 + annual interest rate)^period. For calculations, divide the percentage interest rate by 100. This educational calculation excludes taxes, fees, and changes in prices, and does not predict returns on actual financial products.

Read more about the principle · Investor.gov: How Compound Interest Works

After making changes yourself

When spam is rare, even a low false-positive rate can apply to many legitimate messages.

Correct alerts = 10,000 × spam rate × detection rate; false alerts = 10,000 × legitimate-message rate × false-positive rate. The share of spam among alerts is correct alerts ÷ all alerts. Even with the same detection performance, results depend on how common spam is to begin with.

Read more about the principle · OpenStax: Probability and Conditional Probability

After making changes yourself

As the population approaches the environment's limit, there is less room for further growth.

The logistic model uses dN/dt = rN(1 − N/K). N is population size, K is the fixed environmental carrying capacity, and r is the growth coefficient. This experience displays the equation's continuous-time solution. This simplified model excludes migration, seasons, and changes in food supply, and does not predict actual ecosystems.

Read more about the principle · OpenStax: The Logistic Equation

After making changes yourself

As width and height become more equal, the same fencing encloses more space.

Width + height = perimeter ÷ 2, and area = width × height. Among rectangles with a fixed perimeter, a square with equal width and height has the greatest area. Compare using maximum area = (perimeter ÷ 4)².

Read more about the principle · OpenStax: Comparing Areas with the Same Perimeter

After making changes yourself

A larger scale denominator means the same map length represents a larger real-world space.

At a scale of 1 : S, actual distance (km) = map length (cm) × S ÷ 100,000. The actual area of a square is its actual side length (km) squared. Map projection distortion and terrain slopes are excluded. Zooming in on a map on-screen does not automatically change the numbers in its printed scale label.

Read more about the principle · USGS: Guide to Maps and Scale

After making changes yourself

Even at the same seasonal position, sunlight angles differ between the Northern and Southern Hemispheres.

Solar-noon altitude ≈ 90° − |latitude − solar declination|. For a 1 m stick standing perpendicular to a horizontal surface, shadow length is calculated as 1 ÷ tan(altitude). At an altitude of 90°, the shadow length is 0. Atmospheric refraction and terrain are excluded, and representative seasonal declinations are used.

Read more about the principle · NOAA: Solar Position Equations

After making changes yourself

Doubling both width and height quadruples the pixel count and data size.

Pixel data (bytes) = width in pixels × height in pixels × bits per pixel ÷ 8. Here, 1 MB is 1,000,000 bytes. File headers, palettes, end-of-row padding, and compression are excluded, so actual image file sizes differ. 8 bits is one way to represent data such as grayscale brightness; the extra 8 bits in 32 bits represent transparency in this example.

Read more about the principle · Microsoft: Bitmap Pixel Data and Storage Structure

After making changes yourself

Doubling the solution volume halves the concentration, but the amount of solute stays the same.

Initial concentration × initial volume = final concentration × final volume. Use the same units for both volumes. This model assumes that the solute neither reacts nor evaporates, mixes completely, and that the volumes of the solution and water can be added directly.

Read more about the principle · OpenStax: Solution Concentration and Dilution

After making changes yourself

Adding the intensities of two sounds at the same level raises the level by about 3 dB.

Combined level of N sounds = level of one sound + 10 × log₁₀(N). This assumes sounds with equal levels at the same observation point that are independent and maintain no fixed phase relationship. This model does not calculate the sum of phase-aligned waves, perceived loudness, or hearing recommendations.

Read more about the principle · University of Manchester: Adding the Intensities of Different Sounds

After making changes yourself

Even if your money grows, its purchasing power falls if prices rise faster.

Nominal amount = initial amount × (1 + amount growth rate)^period, amount at initial prices = nominal amount ÷ (1 + inflation rate)^period. The inflation-adjusted annual growth rate is calculated as (1 + amount growth rate) ÷ (1 + inflation rate) − 1. For calculations, divide percentage rates by 100. This educational model uses constant hypothetical rates and excludes taxes, fees, and differences in individual spending.

Read more about the principle · European Central Bank: Inflation and Purchasing Power

After making changes yourself

Crossing Aa with Aa gives aa in one of the four combinations. The theoretical probability of wrinkled seeds is 25%.

This is a single-gene model of complete dominance in which each parent passes on either allele with equal probability. The four cells show equally likely combinations; they do not mean that every four actual offspring must have a 3 : 1 ratio. Each fertilization is a separate probability event. Not all traits in all organisms follow this rule, and dominant does not mean better or more common.

Read more about the principle · OpenStax: Mendel's Laws of Inheritance and Punnett Squares

After making changes yourself

Doubling the number of supporting rope segments halves the force and doubles the pulling distance. The work needed to lift the same object to the same height stays the same.

Pulling force = mass × gravitational acceleration ÷ number of supporting rope segments; pulling distance = number of supporting rope segments × height; work done = force × distance pulled. This ideal model uses gravitational acceleration of 9.8 m/s², ignores rope and pulley mass and friction, and lifts the object at constant speed. The illustration explains how force is shared and does not provide instructions for rigging actual ropes.

Read more about the principle · OpenStax: Pulleys and Mechanical Advantage

After making changes yourself

Half the Moon is always illuminated by sunlight, but its apparent shape changes as the direction from which Earth views that lit half changes.

Lunar phases are not caused by Earth's shadow. Earth's shadow falling on the Moon causes a lunar eclipse. This model assumes uniform motion in a circular orbit and an average synodic month of 29.53 days. The illuminated percentage is calculated as 50 × (1 − cos position angle), and elapsed days as 29.53 × position angle ÷ 360. This is not a calendar for predicting the Moon's actual position, lunar calendar dates, lunar eclipses, or observation times.

Read more about the principle · NASA: Moon Phases and the Illuminated Side

After making changes yourself

At the same proportion, quadrupling the sample size halves the margin of error. The margin does not shrink in direct proportion to the sample size increase.

Margin of error (%p) = 1.96 × √(p × (1 − p) ÷ sample size) × 100, where p is the percentage in favor divided by 100. This is a normal approximation for independent random sampling from a very large population. It is less accurate for small samples or proportions close to 0% or 100%. If surveys were repeated using the same method to construct intervals, about 95% of those intervals would contain the true proportion. It does not mean that this particular interval has a 95% probability of containing the true proportion. Increasing the sample size does not resolve response bias or the exclusion of groups from the survey.

Read more about the principle · NIST: Confidence Intervals for Proportions and the Normal Approximation

After making changes yourself

With the same camera position and sensor, a longer focal length captures a narrower view. A smaller sensor width also narrows the angle of view.

Horizontal angle of view = 2 × arctan(sensor width ÷ (2 × focal length)); captured width = subject distance × sensor width ÷ focal length. Enter sensor width and focal length in the same units; the angle of view is converted to degrees for display. This ideal rectilinear pinhole model excludes distortion and changes in angle of view caused by focusing. Subject distance is measured from the model's center of projection, not the front of an actual lens barrel. Changing lenses from the same position changes the angle of view; perspective changes that alter the relative sizes of near and far objects occur when the camera position moves.

Read more about the principle · Edmund Optics: Focal Length and Angle of View

After trying the combinations

Even with the same conditions, the result depends on whether you use AND, OR, XOR, or NOT.

This logic model uses only two values, 0 and 1. AND returns 1 when both A and B have that value; OR returns 1 when at least one has that value; XOR returns 1 when they differ. NOT returns 1 when A is 0, and 0 when A is 1. It shows the structure of conditional decisions without reproducing actual circuit voltages or timing.

Read more about the principle · OpenStax: Conditions and Logical Operations · Analog Devices: XOR Logic

After moving through time

At equilibrium, the amounts of A and B need not be equal. Their conversion rates are equal in both directions, so each amount stops changing.

This is a closed hypothetical model in which A ↔ B converts in a 1 : 1 ratio, and each conversion rate equals the amount of the substance × its rate coefficient. The coefficients are constant, and the total remains 100. At equilibrium, amount of B = 100 × forward coefficient ÷ (forward coefficient + reverse coefficient). Conversion continues in both directions, but the net change approaches 0. Complex rates involving actual substances, temperatures, and reaction steps are not included.

Read more about the principle · OpenStax: Chemical Equilibrium and Dynamic Equilibrium

After moving up the levels

If 10% is transferred at each level, 1% of the initial energy reaches two levels up.

Energy at the selected level = producer energy × (transfer percentage ÷ 100)^{number of levels}. Energy not transferred is converted to heat through life processes or moves along other pathways through uneaten parts, waste, and dead organisms. It has not disappeared. Nutrients cycle through ecosystems, but energy enters, moves through them, and ultimately disperses into the surroundings as heat.

Read more about the principle · OpenStax: Energy Flow in Ecosystems

After dividing the work

A makes both products faster, but gives up less Y when making X.

To make 1 unit of X, A gives up 0.5 units of Y, while B gives up 1 unit. The person with this lower opportunity cost has a comparative advantage in X. If A spends 75% of the time on X and B spends 0%, output is X: 24 and Y: 12, producing 4 more units of X than initially. This hypothetical example assumes output is proportional to working time and excludes quality, travel, and coordination costs.

Read more about the principle · OpenStax: Absolute and Comparative Advantage

After moving through generations

The type that contributes more to the next generation makes up a larger proportion of the population over successive generations.

With the initial proportion of A as p₀ (0–1), contribution coefficients wA·wB, and generation count g, the proportion of A is p₀ × wA^g ÷ [p₀ × wA^g + (1 − p₀) × wB^g]. This educational model assumes that both types reproduce asexually, pass on their traits unchanged, and have a constant environment and coefficients. It differs from a model of genotype crosses and excludes mutation, migration, and random genetic drift. It compares proportions without predicting actual population counts. Evolution has no predetermined goal or endpoint.

Read more about the principle · UC Berkeley: A Proportion Model of Natural Selection (PDF) · OpenStax: Environment and Relative Fitness

After trying a discount

A price cut reduces revenue if sales quantity does not increase enough.

Arc elasticity is the percentage change in sales quantity divided by the absolute percentage change in price, with each percentage change based on the average of the values before and after the discount. Below 1, revenue falls after the discount; at 1, it stays the same; above 1, it rises. Choosing price 27.5 and Q = 100 − 2P gives equal revenue of 1,237.5 before and after the discount. This hypothetical linear demand model excludes costs, competition, seasons, and differences among consumers. It compares the interval before and after a discount without predicting demand or returns for actual products.

Read more about the principle · OpenStax: The Midpoint Method and Price Elasticity · The Relationship Between Elasticity and Revenue

After changing the altitude

As altitude increases, less air weight remains above, so atmospheric pressure falls.

This is an ideal isothermal atmosphere model with pressure P = P₀ × exp[−Mgh ÷ (RT)]. Sea-level pressure P₀ is 1,013.25 hPa, air molar mass M is 0.0289644 kg/mol, gravitational acceleration g is 9.80665 m/s², and the gas constant R is approximately 8.31446 J/(mol·K). Here, h is altitude (m), and T is temperature in Celsius + 273.15 (K). Temperature, composition, and gravity are constant throughout the atmosphere; water vapor and weather changes are excluded. This differs from standard-atmosphere calculations in which temperature decreases with altitude, and cannot be used for actual pressure forecasts or altitude measurements.

Read more about the principle · NASA: Ideal Gases Under Gravity and the Isothermal Approximation

After changing the note

One octave contains 12 semitone steps. Moving up 12 steps doubles the frequency; moving down halves it.

In 12-tone equal temperament, frequency = reference frequency × 2^{(semitones ÷ 12)}. Each semitone step multiplies frequency by approximately 1.05946, so the difference in Hz for the same number of steps depends on the reference note. The two sine-wave tones play at equal amplitude only when the button is pressed. Actual instrument timbres, harmonics, and tuning differences are excluded. Doubling the frequency ratio does not mean doubling perceived loudness.

Read more about the principle · UNSW: Note Frequencies and 12-Tone Equal Temperament

After grouping repeated characters

Longer runs of identical characters let you store the original in fewer groups.

This method is run-length encoding (RLE). For example, AAAAAAAA can be stored as “8, A” and restored by writing A 8 times. If A and B alternate every time, each character uses 2 bytes, doubling the size. Counts are displayed as readable numbers, but the storage model counts each number as 1 byte. This educational model uses only A and B and excludes headers; it does not reproduce the actual BMP file format or a complete general-purpose compression program. A compression method's effectiveness depends on the data pattern.

Read more about the principle · Microsoft: RLE Storage Using Run Counts and Color Values

After floating the object

Increasing volume increases both weight and maximum buoyant force. The ratio of the two densities determines the floating fraction.

By Archimedes' principle, buoyant force equals the weight of the displaced liquid. When an object floats, buoyant force = weight, so submerged volume fraction = object density ÷ liquid density. If this ratio exceeds 1, the object cannot float at the surface, and the submerged fraction is shown as 100%. The object is assumed to have uniform density and be incompressible; the liquid is still, and surface tension, fluid drag, and bottom contact are excluded. The downward force is weight minus buoyant force after full submersion, not sinking speed. N is a unit of force, and gravitational acceleration is 9.80665 m/s².

Read more about the principle · OpenStax: Archimedes' Principle

After increasing the group size

Among 23 people, there are 253 pairs to compare. The chance of any shared birthday is much higher than the chance of a birthday on a specific date.

The probability of a shared birthday is 1 minus the probability that everyone's birthday is different. With n people, there are n(n − 1) ÷ 2 pairs to compare. Shared-birthday events for all pairs cannot be treated as mutually independent, and multiplying the number of pairs by 1/365 does not give the overall probability. The model assumes birthdays are uniformly distributed over 365 days and independent between people, excluding leap day and actual seasonal birth patterns. It means the probability becomes high even in a small group, not that a shared birthday is guaranteed.

Read more about the principle · UC Berkeley: The Birthday Problem and Complementary Events (PDF)

After cooling the air

Cooling lowers saturation vapor pressure, raising relative humidity. Below the dew point, some water vapor condenses into liquid water.

Relative humidity = current vapor pressure ÷ saturation vapor pressure at that temperature × 100%. Saturation vapor pressure over liquid water is calculated using the Magnus approximation: eₛ(T) = 6.112 × exp[17.62T ÷ (243.12 + T)] hPa. The dew point is the temperature at which saturation vapor pressure equals the initial air's vapor pressure. Below that target temperature, excess vapor condenses, reducing vapor pressure to saturation, and relative humidity is shown as 100%. “Excess above saturation” is the difference between vapor pressure before condensation and saturation vapor pressure at the target temperature, not droplet mass or condensation rate. Dew points below freezing are also shown relative to liquid water and differ from frost points. This equilibrium model excludes supersaturation, pressure changes, and heat transfer rates.

Read more about the principle · NPL: Dew Point and Relative Humidity

After adding the arrows yourself

Even with polar bonds, the sum can be 0 if they point symmetrically in opposite directions.

If two bond dipoles of equal magnitude m form an angle θ, the resultant vector has magnitude 2m × cos(θ/2). Understanding molecular polarity requires considering both bond angles and directions. This activity is a geometric model that adds only the vectors of two identical bonds; it does not calculate a real molecule's full electron distribution or dipole moment. The angle 104.5° is an example for comparing bent shapes, and the relative units shown are neither measured values for water nor Debye units. Actual bond angles are determined by factors such as electron-pair arrangements and cannot change freely like the slider.

Read more about the principle · OpenStax: Molecular Structure and Polarity

After adding substrate yourself

When most enzymes are processing substrate, adding more substrate increases the rate only slightly. Increasing the enzyme amount also increases the maximum rate.

This uses the Michaelis–Menten initial-rate model v = Vₘₐₓ × [S] ÷ (Kₘ + [S]). Substrate concentration [S] and Kₘ are in mM, with Kₘ fixed at 2mM. The maximum rate Vₘₐₓ is set to the relative enzyme amount × 10 in hypothetical rate units. When the substrate concentration equals Kₘ, the rate is half the maximum regardless of enzyme amount. The model assumes substrate is in sufficient excess over enzyme and follows simple saturation kinetics. It excludes inhibition, cooperativity, substrate depletion over time, and enzyme denaturation. It does not predict total product yield or the actual rate of a particular experiment.

Read more about the principle · MIT: Enzyme Catalysis and Reaction Rates (PDF)

After changing each person's choice yourself

Whatever the other person chooses, defection gives you a higher individual score. But the combined score when both defect is lower than when both cooperate.

This payoff structure is called the prisoner's dilemma. Both decide simultaneously just once, with no communication or binding agreements before choosing. If each considers only their own score, mutual defection is the Nash equilibrium. Equilibrium means neither can benefit by changing their choice alone; it does not mean the outcome is best for everyone. The scores are hypothetical rewards, not prison terms. Results may change when trust, retaliation, negotiation in repeated relationships, or concern for others is included.

Read more about the principle · MIT: The Prisoner's Dilemma and Nash Equilibrium (PDF)

After changing the queue conditions yourself

As the arrival rate approaches the service rate, there is less spare capacity to clear the backlog, so waiting time rises sharply.

This is a steady-state M/M/1 model with one counter serving people in arrival order. It assumes an independent Poisson arrival process, independent exponentially distributed service times, and sufficient waiting space. With average arrival rate λ and average service rate μ in people/min, waiting time = 60λ ÷ [μ(μ − λ)] seconds, time in the system = 60 ÷ (μ − λ) seconds, and number waiting = λ² ÷ [μ(μ − λ)] people. Finite steady-state averages exist only when λ < μ, and λ/μ is then the counter's utilization. When λ ≥ μ, λ/μ is the ratio of requested load to service capacity, not actual utilization. This means stable averages cannot be shown, not that every customer necessarily waits forever. These values do not predict instantaneous queue lengths or crowding in real stores with reservations, multiple counters, or departures during service.

Read more about the principle · MIT: The M/M/1 Queueing Model (PDF)

After flipping bits yourself

Even parity adds a check bit so the number of 1s is even. An error is detected if the received number of 1s is odd.

The four data bits represent values of 8, 4, 2, and 1 from left to right. The rightmost check bit is not included in the data value. An odd number of ultimately changed bits among the five transmitted bits is detected, but an even number can go undetected. 'Not detected' does not guarantee correct data. Changing only the check bit also triggers an error, while the data value can remain unchanged. This method alone cannot identify the incorrect bit or restore the original data. This activity compares at most two bit flips.

Read more about the principle · MIT: Digital Information and Parity Checks

After changing both sides of the membrane yourself

With equal hydrostatic pressure on both sides, water moves overall from lower to higher solute concentration. Equal concentrations produce no net movement due to this difference.

The osmotic pressure of a dilute ideal nonelectrolyte solution is π = CRT. C is in mol/m³, R is approximately 8.31446J/(mol·K), and T is the Celsius temperature + 273.15 in K. Numerically, 1mM = 1mol/m³, so pressure in kPa is the displayed concentration × R × T ÷ 1,000. The inside − outside osmotic pressure difference is the inside − outside hydrostatic pressure difference needed to prevent net water movement; it does not mean this hydrostatic pressure difference is actually present initially. Only initial concentrations are compared; changes in volume or concentration over time and movement rates are not calculated. This model does not predict complex solutes and membrane transport in real cells or responses of particular organisms.

Read more about the principle · OpenStax: Osmotic Pressure of Solutions

After changing the problem mix yourself

The overall success rate is an average of group success rates weighted by problem counts. With different proportions of easy problems, A can lead in each group yet trail overall.

This reversal is called Simpson's paradox. If a and b are the proportions of easy problems, ranging from 0 to 1, A overall = 90a + 40(1 − a), and B overall = 80b + 30(1 − b). With equal proportions, A always leads by 10%p, but at A 10% and B 90%, overall success rates become A 45% and B 75%. Differences within difficulty levels and the overall difference answer different questions. These are hypothetical data assigning different mixes to the two methods; they alone cannot establish which method actually causes success. Real comparisons also require examining how groups were defined and why their compositions differ.

Read more about the principle · Williams College: Weighted Averages and Simpson's Paradox (PDF)

After following the heat transfer yourself

Delivered heat includes both the input energy and heat taken from the colder side. Delivering more heat than the electrical energy supplied does not create new energy.

The ideal limit for a reversible Carnot heat pump is heating COP = T_hot ÷ (T_hot − T_cold), cooling COP = T_cold ÷ (T_hot − T_cold). Temperatures must be calculated as absolute temperatures in K by adding 273.15 to Celsius temperatures. Heat delivered to the hotter side = heat taken from the colder side + work input, and the two COPs always differ by 1. Both sides are heat reservoirs whose temperatures remain constant despite heat exchange, and all electrical energy is assumed to become work. These are theoretical upper limits excluding friction, compression losses, and temperature differences needed for heat exchange in real equipment. The displayed values cannot predict the performance or running costs of real air conditioners or heaters.

Read more about the principle · OpenStax: Key Equations for Thermodynamics and Heat Pumps

After choosing electrodes yourself

Metal releases electrons on the side with the lower standard reduction potential, while metal ions gain electrons on the higher-potential side. Electrons in the external circuit move from anode to cathode.

The standard voltage of an ideal galvanic cell is the cathode's standard reduction potential − the anode's standard reduction potential. The voltage magnitude is shown for the spontaneous reaction direction. Identical metals at identical ion activities have no potential difference. Comparisons assume pure solid metals and an activity of 1 for each M²⁺ ion at 25°C, referenced to the standard hydrogen electrode. Table potentials are rounded to three decimal places. The magnitude of maximum electrical work is nFE, displayed in kJ/mol using n = 2mol electrons/mol reaction and F = 96,485.33212C/mol. Concentration changes, internal resistance, overpotential, and side reactions in real cells are excluded. Voltage differs from capacity, the amount of stored charge; this value alone cannot determine actual power output or operating time.

Read more about the principle · OpenStax: Electrode Potentials and Cell Voltage · Table of Standard Half-Cell Potentials

After changing the light's direction yourself

Light bends toward the normal when entering a material with a higher refractive index. It bends away from the normal when entering one with a lower index, and undergoes total internal reflection above the critical angle.

The refraction angle is calculated using Snell's law, n₁sinθ₁ = n₂sinθ₂. A critical angle θc = asin(n₂/n₁) exists only when n₁ is greater than n₂. At the critical angle, the refraction angle is 90°, the limiting direction along the boundary; total internal reflection occurs only above the critical angle. 'Conditions not met' means the two refractive indices do not allow total internal reflection. The model assumes a flat boundary between two transparent, uniform materials with fixed refractive indices, excluding real wavelength and temperature dependence. This activity compares light directions and does not calculate the intensity of partially reflected light, polarization, or absorption. With equal refractive indices, light does not change direction optically, and no boundary reflection is shown.

Read more about the principle · OpenStax: Refraction and Snell's Law

After crossing income brackets yourself

The average tax rate is the share of total income paid in tax; the marginal tax rate applies to the next additional income. These figures answer different questions.

Bracket-based progressive taxation adds each bracket's income portion × its tax rate. At an income of 400 units, tax on the first 200 is 0 and tax on the next 200 is 20, totaling 20 units. Further income is taxed at 20%, so the marginal rate at a boundary is shown for the 'next income increment.' All bracket rates in this model are below 100%, so after-tax income increases with income and does not drop when crossing a boundary. Only hypothetical taxable income and tax rates are used; deductions, tax credits, welfare benefits, and other taxes are excluded. This activity compares the principle of bracket taxation rather than calculating actual taxes or take-home income.

Read more about the principle · IRS: How Tax Applies to Income Within Each Bracket

After comparing the two waves yourself

The arrival-time gap between the faster P wave and slower S wave grows with distance. Finding the distance requires applying both speeds to this gap.

For the same distance d, the arrival-time difference is Δt = d/vS − d/vP, so d = Δt ÷ (1/vS − 1/vP). This activity assumes both waves travel along the same straight path at constant speeds through a uniform solid. Hypocentral distance is the distance between the station and the underground earthquake source, not the surface distance to the epicenter on a map. P- and S-wave times are model travel times with the earthquake's start set to 0. 'Distance for a time gap of 1 second' is a distance conversion factor, not the speed of either wave. The model excludes speed changes between Earth's layers, refraction, and differing paths. Determining the hypocenter's direction and location requires records from multiple stations and a more detailed model. This activity does not predict earthquakes.

Read more about the principle · USGS: Locating Earthquakes Using Seismic Waves and Arrival Times

After changing the storage policy yourself

Cache effectiveness depends on replacement policy and request order as well as slot count. Follow how the same requests are stored step by step.

FIFO preserves insertion order, even on cache hits. LRU moves a hit item to the most recently used position. When an item is missing and all slots are full, each policy replaces its oldest item. For this sequence, FIFO misses increase from 9 with 3 slots to 10 with 4 slots, a phenomenon called Belady's anomaly. LRU with the same 4 slots has 8 misses. This result alone does not mean FIFO is always bad or LRU is better for every sequence. Hit rate is hits ÷ processed requests and is undefined for 0 requests. This replacement-policy model treats each number as one equal-sized data item. It does not calculate real browser cache expiration, communication time, capacity, or write costs.

Read more about the principle · MIT: FIFO, LRU, and Memory Replacement

After changing model complexity yourself

A model passing through every observation can still deviate more at other reference points. Examine errors at both the fitted points and the separate comparison points.

Observed x values are −1, −0.6, −0.2, 0.2, 0.6, and 1. To y = 3 + 2x, we add the perturbation multiplier × [0.2, −0.2, 0.3, −0.3, 0.2, −0.2]. The polynomial is fitted by minimizing the sum of squared errors at these six points. Reference x values are twenty points from −0.95 to 0.95 at intervals of 0.1; reference y is the unperturbed 3 + 2x. This activity compares reference errors for hypothetical data with a known generating process; the reference points are not independently collected observations. A degree-5 polynomial can pass through all six perturbed observations but may curve sharply between them, increasing reference error. This does not mean complex models are always bad or simple models always good. With perturbation set to 0, all three models fit the same straight line. These values do not guarantee errors for future outcomes that are unknown in practice.

Read more about the principle · scikit-learn: Underfitting and Overfitting

After changing species proportions yourself

Even with the same species count, diversity decreases when a few species dominate the distribution. Species count and evenness of proportions are different characteristics.

For species proportions p, Shannon diversity is H = −Σp ln(p), using natural logarithms. exp(H) is the effective number of species in an even distribution with the same H; it does not mean a fractional number of species actually exists. Evenness is displayed as H ÷ ln(species count) × 100 and equals 100% when all species have equal proportions. With 4 species, a first-species proportion of 25% makes all proportions 25%; at 70%, the others are each 10%. The slider moves in 5% increments, so the exactly even proportion ⅓ for 3 species cannot be selected. This compares ideal distributions with all proportions positive, without addressing sample size, observation error, or relationships among species. A higher index does not always mean a healthy ecosystem; species identity, rarity, and conservation value require separate consideration.

Read more about the principle · Hokkaido University: Diversity Indices

After changing the task durations yourself

The longest task path that determines the completion time is called the critical path. Shortening only tasks on a shorter path may leave the overall completion time unchanged.

The total completion time is 2 + max(B + D, C + E) + 1. The “sum of task durations” adds the durations of all six tasks and differs from actual elapsed time because the two paths run simultaneously. Each path’s slack is the longer path’s duration minus its own duration; it can be delayed by that amount without delaying overall completion. With the defaults, the lower path takes 8 minutes and the upper path takes 7 minutes, leaving 1 minute of slack on the upper path. When the paths have equal durations, both are critical paths; shortening just one still leaves you waiting for the other. Shortening tasks can also change the critical path itself. This dependency model assumes separate teams share no resources and each task duration is fixed. It gives the earliest completion time, excluding staffing conflicts, interruptions, and uncertainty in durations.

Read more about the principle · MIT: Critical Path Method (PDF)

After changing the dissolution limit yourself

There is a limit to how much can dissolve under given conditions. If you add more, the excess remains undissolved. If cooling lowers the limit, some dissolved material comes out as solid again.

Maximum soluble amount = solubility × water mass ÷ 100; dissolved amount = min(amount added, maximum soluble amount); remaining solid = amount added − dissolved amount. The displayed ratio is dissolved-solid mass ÷ water mass × 100, which differs from mass percent based on the total solution mass. We assume enough time has passed to reach dissolution equilibrium under each condition. A solution is saturated when the maximum soluble amount is dissolved; undissolved solid does not have to be present. Water evaporation, supersaturation, dissolution rate, and heat of dissolution are excluded. This hypothetical substance is defined to become more soluble as temperature rises; not all solids follow this relationship.

Read more about the principle · OpenStax: Solubility and Saturated Solutions

After changing the rate of change yourself

The same magnetic field change produces a larger average induced voltage magnitude when it happens over a shorter time. Reversing the direction of change also reverses the voltage’s sign.

Magnetic flux measures the magnetic field passing through the coil plane, accounting for area and direction. In a uniform magnetic field, Φ = BAcosθ. Here, θ is the angle between the normal perpendicular to the coil plane and the magnetic field. Faraday’s law gives the average induced electromotive force as ε = −NΔΦ ÷ Δt, where N is the number of turns. 1 mWb equals 0.001 Wb. The sign is shown relative to the chosen normal and winding direction; a negative value does not mean a smaller voltage magnitude. The minus sign in Lenz’s law indicates a direction that opposes the change in magnetic flux. We assume every coil turn has the same area and magnetic field, excluding detailed time waveforms, coil resistance, and self-induction. This compares average voltage over a set time interval without calculating actual current or power.

Read more about the principle · OpenStax: Magnetic Flux · Faraday’s and Lenz’s Laws

After adding the omitted costs yourself

When harm creates costs that the parties to a transaction do not pay, market output can differ from the output that maximizes society’s total net benefit.

External costs are costs borne by 3rd parties who do not directly participate in a transaction. Social marginal cost = private marginal cost + marginal external cost. In a competitive market, output reflecting only private costs is q = 120 − private cost; output maximizing total net benefit including external costs is q = 120 − private cost − external cost. Between these output levels, social cost exceeds additional benefit. The lost net benefit is the triangular area: ½ × output difference × marginal external cost. This loss differs from the total external harm from all production. This continuous quantity model includes only negative externalities from production, excluding estimates of actual pollution damage, market power, policy enforcement costs, and positive externalities. It compares hypothetical products and cost units and does not predict the effects of actual regulations or taxes.

Read more about the principle · OpenStax: The Economics of Pollution and External Costs

After changing reflectance yourself

Reflecting more reduces absorbed energy and lowers the equilibrium temperature at which the same amount is radiated away. Celsius temperature is not proportional to absorption.

The fraction of sunlight reflected across all wavelengths and the entire planet is called the Bond albedo a. Average absorption over the entire surface = S(1 − a) ÷ 4, and blackbody emission = σT⁴. The temperature at which they are equal is T = [S(1 − a) ÷ (4σ)]^¼. We use S = 1,361 W/m² and σ = 5.670374419 × 10⁻⁸ W/(m²·K⁴); Celsius temperature is T − 273.15. This simple model treats an entire spherical planet as having a single temperature, absorbing only solar radiation and having a thermal emissivity of 1. It excludes the atmosphere and greenhouse effect, temperature differences between day and night or across latitudes, ice and cloud feedback, and the time needed to reach equilibrium. The displayed temperature is not Earth’s actual surface temperature or a climate prediction for any location.

Read more about the principle · NASA: Earth’s Energy Budget · Solar Radiation and Earth Data

After choosing characters yourself

The Hangul character ‘가’ is one code point but takes 3 bytes in UTF-8. Counting characters and measuring storage size use different criteria.

A Unicode code point is a number that identifies a character, and UTF-8 is an encoding that stores each Unicode scalar value in 1–4 bytes. JavaScript string length counts UTF-16 code units; one smiling emoji 😊 uses 2 code units. Extended grapheme clusters use Unicode boundary rules to group multiple code points together, as with combining accents and joined emoji. Although é (U+00E9) and e + combining accent (U+0065 U+0301) look similar, their stored code points differ, and their UTF-8 sizes are 2 and 3 bytes respectively. This activity compares six fixed examples; it does not normalize input or segment arbitrary strings. The number of shapes drawn on screen does not always equal the number of Unicode clusters. File headers, encoding markers, and compressed sizes are excluded.

Read more about the principle · Python Documentation: Unicode and UTF-8 · Unicode: Grapheme Cluster Boundaries

After changing the connections yourself

At intermediate points, entering and leaving paths form pairs. To traverse every path exactly once, consider both connectivity and the number of odd-degree points.

The number of paths meeting at a point is its degree in the graph. When all points with paths are connected, 0 odd-degree points allow an Euler circuit that starts and ends at the same point; 2 odd-degree points allow an open Euler trail starting and ending at those two points. With more odd-degree points, traversing every path exactly once is impossible. If the points with paths are disconnected, a single complete traversal is impossible even when all degrees are even. Isolated points with no paths are excluded from the connectivity condition. This activity uses hypothetical graphs without directions or travel restrictions. It does not calculate drawn path lengths, distances, or optimal routes on actual roads.

Read more about the principle · MIT: Graphs and Euler Trails (PDF)

After changing the overlap yourself

If marked individuals have mixed sufficiently throughout the population, their proportion in the second observation is expected to resemble their proportion in the whole population.

The Lincoln–Petersen point estimate of total population is N̂ = M × C ÷ R. It follows from assuming that the population’s marked proportion M/N is similar to the second observation’s marked proportion R/C. The number of distinct individuals directly observed is M + C − R, which differs from the total population estimate. If R is 0, this formula cannot produce an estimate. When R is small, a difference of just a few overlapping individuals can greatly change the estimate. Assumptions include a closed population with no births, deaths, or migration between observations; sufficient mixing; retained marks and accurate identification; equal capture probability for every individual; and no behavioral changes after marking. Small-sample bias corrections and confidence intervals are not calculated. This activity explores relationships among hypothetical observation counts; it does not provide instructions for capturing or marking actual animals or estimating a real population.

Read more about the principle · University of Wyoming: Mark–Recapture and Population Estimation

After moving the camera yourself

The ratio of the objects’ distances from the camera determines their projected size ratio. Changing only focal length at the same position changes both widths together while preserving their ratio.

In ideal pinhole projection, image-plane width w′ = f × w ÷ z. Here, f is focal length, w is actual width, and z is distance from the camera to the object. For two objects of equal actual width, the projected width ratio is far-object distance ÷ near-object distance = (d + separation) ÷ d; focal length cancels out. As the camera moves back, this ratio approaches 1. If you change focal length and also move the camera to maintain the same framing, the position change alters perspective. The layout above shows the distance relationships between the camera and objects. The bars’ 36 mm scale provides a fixed reference for comparing projected widths. This geometric model assumes parallel object and image planes and does not calculate lens distortion, defocus blur, depth of field, or changes in facial shape.

Read more about the principle · MIT Course Text: Images and Pinhole Projection · Columbia: Geometry of Perspective Projection

After changing the light’s travel conditions yourself

Concentration multiplied by path length determines absorbance. Doubling absorbance does not halve the fraction of light transmitted.

The Beer–Lambert law gives absorbance A = ε × concentration c × optical path length l = −log₁₀(transmitted light ÷ incoming light). This activity calculates transmittance as 100 × 10⁻ᴬ % and transmitted light as incoming light × 10⁻ᴬ. Absorbance is logarithmic and differs from the percentage of light absorbed. When A increases from 0.5 to 1, transmittance falls from about 31.623% to 10%. At the same concentration and path length, halving only incoming light halves both transmitted and absorbed light, while transmittance and absorbance remain unchanged. At a concentration of 0, all light passes through in this model. This ideal linear absorption model assumes monochromatic light passing through a uniform solution without reflection, scattering, or light emission, and excludes absorption by the container and solvent. It is not a tool for measuring actual substance concentrations or predicting a specific liquid’s transparency.

Read more about the principle · IUPAC: Beer–Lambert Law

After changing collision conditions yourself

If the external impulse on the two objects is 0, total momentum is the same before and after collision. Whether kinetic energy is conserved must be examined separately.

Total momentum is p = m₁v₁ + m₂v₂, and kinetic energy is ½m₁v₁² + ½m₂v₂². When the objects approach each other, their post-collision velocities are v₁′ = [p − m₂e(v₁ − v₂)] ÷ (m₁ + m₂) and v₂′ = [p + m₁e(v₁ − v₂)] ÷ (m₁ + m₂). Here, e is relative separation speed after collision ÷ relative approach speed before collision. When e = 1, kinetic energy is conserved; when e = 0, both objects move at the same velocity p ÷ (m₁ + m₂). Moving together does not always mean stopping. The decrease in kinetic energy is the portion converted into other forms such as deformation, heat, and sound; it does not mean total energy disappears. This is an ideal collision along a straight line with no external impulse. Friction, rotation, collision duration, and the deformation process are not calculated. No collision event is applied when the gap stays constant or increases.

Read more about the principle · University of Florida: Collisions Between Two Objects (PDF)

After changing how votes are counted yourself

Plurality considers only first choices, while Borda awards points for every rank. Even with the same preferences, the winner can change depending on what is added up.

Plurality gives one vote to each person’s rank-1 candidate. Borda awards 2 points for rank 1, 1 point for rank 2, and 0 points for rank 3, giving A = 2a + c, B = a + 2b, and C = b + 2c. The top-two runoff first compares rank-1 votes. A candidate with a majority—more than half the total—wins immediately. If nobody has a majority and the top two can be identified, each person votes for the one they prefer among those remaining candidates. If a tie for second place prevents selecting two candidates, “Additional rule needed” is displayed rather than choosing arbitrarily. Equal highest totals in plurality, Borda, and the final runoff are shown as ties. With the default preferences, the winners differ: A with 40 votes under plurality, B with 110 points under Borda, and A with 65 votes in the runoff. All preferences are hypothetical and complete; actual election rules, strategic voting, abstentions, and incomplete rankings are excluded. This does not mean one method is best in every situation.

Read more about the principle · OpenStax: Voting Methods

After changing water-flow conditions yourself

With the same head difference, a more conductive layer carries more water, while a longer flow distance reduces the difference per unit distance.

Darcy’s law gives hydraulic gradient i = Δh ÷ L, velocity based on the full cross section q = K × i, and discharge Q = q × cross-sectional area. The value q uses the entire cross section, including solids, rather than only the pores where water actually flows. With effective porosity nₑ = 0.3, average pore-water velocity is q ÷ nₑ, about 3.333 times the Darcy velocity here. When the head difference is 0, flow in this model is 0. Doubling only K also doubles discharge. Holding the head difference fixed and doubling distance halves both hydraulic gradient and discharge. This is steady flow through a uniform, fully water-saturated layer, with water properties, cross-sectional area, and porosity fixed. K uses hypothetical values chosen for comparison, not measurements of specific soils or rocks. The model does not predict rainfall infiltration into unsaturated soil, actual groundwater arrival times, contaminant transport, or ground stability.

Read more about the principle · USGS: Groundwater Hydraulics (PDF) · USGS: Pores and Groundwater Flow

After changing the execution order yourself

Even when the code adds once for each participant, one addition can be lost depending on the order of reads and writes to the shared value.

If A and B both read 10, each prepares 11. Even if they write 11 in turn, the final value is 11, so one update is lost. This is called a lost update. Protecting the entire read–calculate–write sequence with one lock lets the first person finish before the next reads the new value, making the final value 12. Locking only the write does not fix the earlier reading of an outdated value. The target 12 is the value after both additions finish, not the correct value at an intermediate step. When both have only finished reading, the shared value has not changed and no update has been lost yet. This activity models the order of operations without running actual browser threads. It does not cover synchronization across multiple computers or deadlocks.

Read more about the principle · OSTEP: Threads and Shared Values (PDF) · OSTEP: Locks (PDF)

After changing the number of tests yourself

Even with a small error threshold for each test, the probability of at least one false positive can grow when many tests are performed.

If p is the false-positive probability per test and m is the number of tests, the probability of at least one false positive among independent tests with all null hypotheses true is 1 − (1 − p)ᵐ. The average false-positive count is mp. Applying an unadjusted threshold of α = 5% to 20 tests gives an overall probability of about 64.15%. Bonferroni adjustment lowers each threshold to α ÷ m to keep the overall error rate at or below α. This upper-bound guarantee does not require independent tests, but the exact probability shown here assumes independence. The overall probability and the average count are different quantities. This activity does not generate test results or actual p-values. A p-value is not the probability that the null hypothesis is true or that an effect exists. Stricter thresholds can make small differences harder to detect, and actual research must also consider how hypotheses are grouped and how tests are related.

Read more about the principle · MIT: Error Probabilities in Multiple Testing (PDF) · NIST: The Bonferroni Principle

After changing one RNA letter yourself

Multiple codons can specify the same amino acid, so whether a single-letter change alters the amino acid depends on the codon.

A codon consists of three consecutive RNA bases. In the standard genetic code (table 1), GAA and GAG both specify Glu, so this is a synonymous change that leaves the amino acid sequence unchanged. GUA specifies Val (valine), making it a missense change, while a change to UAA is a nonsense change that brings the stop signal forward. If the second codon is a stop, translation ends after producing one Met, and the following UUC is not translated. Entering the original letter is classified as no change. The starting position and reading frame of three-letter groups are fixed; insertions and deletions are not covered. Genetic codes have exceptions across organisms and organelles, and synonymous changes are not always without effects in actual organisms. This model illustrates translation principles using a short synthetic RNA, not the effects of actual genes or diseases.

Read more about the principle · NCBI: The Standard Genetic Code · OpenStax: Codons and the Genetic Code

After changing the photo settings yourself

Aperture and shutter time change the incoming light, while ISO changes the model's brightness for that light. Motion trails can differ even at the same brightness.

Under otherwise identical conditions, light quantity is proportional to shutter time and inversely proportional to the square of the f-number. Opening from f/4 to f/2 quadruples the light, while shortening the time from 1/200 s to 1/800 s reduces it to ¼, so changing both gives the same quantity as the reference. The shorter time makes the motion trail ¼ as long. Using f/8 with ISO 400 gives ¼ of the received light but the same model brightness as the reference. A brightness difference of +1 stop means twice as bright, and −1 stop means half as bright. Model brightness uses linear amplification proportional to ISO. It does not reproduce highlight clipping, actual photo tone processing, metering, or noise. The motion trail is the relative length left by a point moving at constant speed, not a measure of camera shake or overall sharpness. Differences in lens light transmission, depth of field, and automatic exposure are also excluded.

Read more about the principle · Nikon: Aperture, Shutter, and Exposure Basics · Nikon: ISO Sensitivity

After changing the acid amount yourself

When the conjugate base accepts incoming H+ and becomes weak acid, it can reduce the pH change. But the amount available to accept it is limited.

First, calculate the stoichiometric reaction A− + H+ → HA. A− decreases and HA increases by the amount accepted by the conjugate base. The amounts in the diagram are calculated from neutralization, not exact ion amounts accounting for slight acid dissociation and water equilibrium. The Henderson–Hasselbalch approximation is pH ≈ pKa + log₁₀(A− ÷ HA). Two solutions with the same ratio have the same approximate initial pH, but the one containing more weak acid and conjugate base can accept the same amount of strong acid with a smaller pH change. This activity displays the approximation only when at least 10 mmol of A− remains after calculation and A− ÷ HA is within 0.1–10. Outside this range, it shows “Outside approximation range,” which does not mean the actual pH is nonexistent or 0. Once all the prepared conjugate base is used up, the remaining strong acid is shown, but this equation is no longer used to calculate pH. This is an ideal-solution approximation applied after the stoichiometric reaction of the weak acid and conjugate base. Exact acid dissociation equilibrium, ion activities, and volume changes are not calculated.

Read more about the principle · OpenStax: Buffers and Buffer Capacity

After changing where you apply the force

Torque depends on the force magnitude, its distance from the pivot, and its direction.

Torque magnitude is τ = rF sin θ. The lever arm r sin θ is the perpendicular distance from the rotation axis to the force's line of action. Applying the same force perpendicularly at twice the distance doubles the torque. At an angle of 30°, the torque is halved for the same distance and force. At 0° or 180°, parallel to the bar, torque is 0, so the right-side force alone cannot cancel the left-side torque. The “balancing force” is the calculated force required and may exceed the adjustable range of 10–40 N. A + net torque initially tends to turn the bar counterclockwise; − tends to turn it clockwise. 0 indicates rotational balance in this model, not that all forces on a free object are balanced. The pivot is fixed, and its reaction force constrains the bar's translation. The model assumes a massless rigid bar initially at rest with no rotational friction; motion over time is not calculated. Torque in N·m is not displayed as J, the unit of energy.

Read more about the principle · OpenStax: Torque and Lever Arms · OpenStax: Force and Torque Balance

After changing the payment rule yourself

Even when the same person wins, the payment depends on whether they pay their own bid or the next-highest bid.

If there is a unique highest bid, that bidder wins under both rules. In a sealed-bid auction with price rank 1, the winner pays their own bid; with price rank 2, they pay the highest bid among the other participants. Nonwinners pay 0 points. A's net benefit is 80 − payment if A wins, and 0 otherwise. If A bids 90 points and wins at price rank 1, the net benefit can be −10 points. If the highest bids tie, the model does not arbitrarily select a winner; it displays payment and net benefit as “Undetermined.” The model uses one item, one round of bidding, and hypothetical valuations and points, with no reserve price, entry fees, or commissions. It does not calculate participants' strategic choices, information differences, or the best strategy across all auctions.

Read more about the principle · MIT: Auction Payment Rules and Strategies (PDF)

After changing the orbital conditions yourself

For the same central mass, more distant circular orbits have lower speeds and take longer to complete one revolution.

When gravity provides the centripetal force for a circular orbit, the period is T ∝ √(r³ ÷ M) and the speed is v ∝ √(M ÷ r). Here, radius r is in AU and M is in solar masses, normalized so that T = √(r³ ÷ M) model years and speed relative to the reference = √(M ÷ r)×. At the same mass, doubling the radius multiplies the period by √8 and the speed by 1 ÷ √2. At the same radius, doubling the central mass multiplies the period by 1 ÷ √2 and the speed by √2. Cumulative revolutions equal time ÷ period, and the current angle represents the position within one revolution. The model assumes the central star is much more massive than the orbiting object and the orbit is circular. Each setting compares a different possible orbit, rather than reproducing an instantaneous orbital change by a spacecraft. The star's drawn size is fixed and does not imply that its actual radius is calculated from its mass. Elliptical orbits, interactions among multiple bodies, and actual planetary positions are not calculated.

Read more about the principle · OpenStax: Gravity and Orbital Periods

After changing the table matches yourself

A join forms every pair of rows with the same key. Duplicate right-hand keys can turn one left-hand row into multiple result rows.

INNER JOIN includes every pair of rows with matching keys. Unmatched rows from either table are omitted. LEFT JOIN also includes every matching pair, but adds one row with NULL right-hand values for each left-hand row with no match at all. LEFT JOIN therefore preserves the left-hand data, but the result does not necessarily have the same 3 rows as the left table. If the right table has two A rows, the left A matches twice. The different keys D and E do not match A, B, or C on the left. This screen compares small fixed tables without connecting to an actual database. For illustration, rows follow a fixed left-table and right-table order; actual SQL requires ORDER BY to guarantee ordering.

Read more about the principle · PostgreSQL: Joining Tables · PostgreSQL: SELECT and Joined Tables

After dividing the area yourself

The area under a curve can be approximated by adding areas over small intervals. Even with the same number of pieces, the choice of heights makes a difference.

With n pieces, the width is Δx = 2 ÷ n. Rectangles sum x² × Δx at the selected positions; trapezoids sum the average of the endpoint heights × Δx. The exact area under this curve is ∫₀² x² dx = 8 ÷ 3. Because x² is increasing and convex upward here, left-endpoint and midpoint sums underestimate, while right-endpoint and trapezoidal sums overestimate. These relationships do not apply unchanged to every function. Displayed values are rounded to three decimal places; calculations use unrounded values. No actual physical unit is assigned to y, and the area under x² on [0, 2] is not interpreted as a probability. A probability density function requires separate checks, including that its total area is 1.

Read more about the principle · OpenStax: Approximating Area with Small Sums · OpenStax: Midpoint and Trapezoidal Integration

After changing the size and division yourself

As a sphere grows, its volume increases with the cube of its radius, while its surface area increases with the square. Dividing the same total volume into smaller spheres creates more surface.

A sphere's surface area is 4πr², its volume is 4πr³ ÷ 3, and its surface-area-to-volume ratio is 3 ÷ r. Dividing the volume of a sphere of radius R into N equal spheres gives a smaller radius of R ÷ ∛N and a total surface area ∛N times the original. With 8 spheres, the surface area is 2 times as large; with 64, it is 4 times as large. Division preserves only total volume, creates new surfaces, and assumes each sphere's surface is independently exposed. The diagram does not show smaller spheres packed without gaps inside the original, nor does it reproduce actual cell division. This model explains the relationship between surface area available for exchange and internal demand proportional to volume. Actual cells differ in shape, membrane folds, transport systems, and metabolic rates, so this ratio alone cannot predict exchange rates or viable sizes. Displayed decimals are rounded; calculations use unrounded values.

Read more about the principle · OpenStax: Cell Surface Area and Volume

After changing the line-breaking rules yourself

Filling each earlier line as much as possible may not minimize the total cost. You also need to consider the whitespace on later lines.

The greedy layout fills each line while the next word still fits, then starts a new line. The minimum-cost layout compares possible line breaks to find the smallest sum of squared or cubed unused widths. In this activity, the last line's cost is set to 0, and words on a line are separated by a space of 1 unit. Word order is preserved; omission, duplication, and splitting are not allowed. At width 6, starting with “one of” leaves 4 units of whitespace on the next “us” line, giving a squared cost of 16. Breaking into “one” and “of us” reduces the cost to 9 + 1 = 10. If minimum costs tie, the layout with fewer lines is preferred; if line counts also tie, the layout with more words on earlier lines is displayed. This may show one of several optimal layouts. This hypothetical cost model uses short English words with equal character widths and does not judge actual writing's beauty or readability. Proportional fonts, Korean punctuation, language-specific line breaking, and hyphenation are not calculated.

Read more about the principle · MIT: Line Breaking by Comparing Total Costs

After changing temperature and space yourself

For a fixed amount of ideal gas, pressure rises with higher absolute temperature or smaller volume. What you hold constant matters.

The ideal gas equation is PV = nRT. Here, n = 0.1 mol and R = 8.314462618 J/(mol·K). With volume in L and pressure in kPa, kPa × L = J. Pressure is calculated as nRT ÷ V, always using absolute temperature in K. Celsius is displayed as K − 273.15 but is not used in ratio calculations. Pressure × volume remains constant only at constant temperature; raising temperature increases this product too. Both gas diagrams use the same volume scale and the same number of representative dots. The dots do not reproduce actual particle counts or collisions. This ideal gas model ignores intermolecular forces and particle volume, so it does not predict real gases at high pressure, liquefaction, or phase changes. Displayed values are rounded; calculations use unrounded values.

Read more about the principle · OpenStax: Ideal Gas Pressure, Volume, and Temperature · OpenStax: The Kinetic Molecular Model of Gases