170 experiments found
Clear filtersFor a resistor at constant temperature, current is proportional to voltage: V = I × R
Measure the current through resistors and a bulb as you add cells, plot current against voltage, and find the resistance.
Electrical energy is measured in units: 1 unit = 1 kilowatt-hour (kWh) = power in kW × hours of use; the bill counts the units, charged at a higher rate for each slab, so efficient appliances and less use save money
Choose the bulbs, the TV time, the way you cook and whether you have a fridge, then read the meter and work out the month's units and the bill.
The potential energy of water high on a hill becomes kinetic energy in the pipe, then electrical energy in the turbine and generator; the power is about 9.8 × flow (m³/s) × fall (m) kW, times the efficiency, and the rest is lost as heat and sound, never destroyed
Run a village micro-hydro plant. Change how far the water falls, how much flows, and the turbine, and see how many homes you can light, and how much energy is lost on the way.
Momentum is mass × velocity; in a collision the total momentum before equals the total after, so trolleys that stick move at (m₁v₁ + m₂v₂) ÷ (m₁ + m₂), and some kinetic energy turns into sound and heat
Send trolley A into trolley B so they stick together. Change their masses and velocities, and check that the total momentum is the same before and after, while some kinetic energy is lost.
When one object pushes on another, the second pushes back with an equal and opposite force
Launch a balloon rocket along a string. Change the amount of air, the nozzle and the load, and compare the action and reaction forces.
Objects keep moving at the same velocity unless a force acts on them (Newton's first law)
Run a trolley carrying a block into a stopper and watch the block keep going. Change the speed, the block's mass and whether it is strapped down.
Distance is the length of the path travelled; displacement is the straight-line distance from the start to the finish, in a direction; after a whole lap the distance is 400 m but the displacement is zero
Run part of a lap, a whole lap and more round a 400 m track. Compare the distance you ran with your displacement, and your average speed with your average velocity.
Heat can travel as infrared radiation; dull black surfaces absorb and emit it well, shiny surfaces poorly
Warm a matt black can and a shiny silver can of water in the sun, then let hot water cool in each, and compare how well they absorb and give off heat.
Concave mirrors can form real or magnified virtual images; convex mirrors always form small, upright, virtual images with a wide view
Move an object in front of concave, convex and plane mirrors, trace the reflected rays, and describe each image.
A magnetic field runs from N to S around a magnet; a current in a wire makes circular field lines whose direction follows the right-hand grip rule
Map the field around a bar magnet with a plotting compass, then pass a current through a straight wire and see the circular field it makes.
The right instrument depends on the quantity, its size and the accuracy needed: a tape for metres, a vernier caliper for tenths of a millimetre, a screw gauge for hundredths
Decide which instrument to use for each measurement of length, mass, time and volume, thinking about the size of the quantity and the accuracy needed.
Density = mass ÷ volume, a property that helps identify a material
Weigh blocks and a stone on a balance, find their volume by measuring their sides or by displacement, and calculate their density.
The slope of a distance-time graph is velocity; the slope of a velocity-time graph is acceleration
Follow a car moving steadily, speeding up, slowing down or parked, record its position and speed each second, and plot motion graphs.
Pressure in a liquid increases with depth and with the liquid's density (p = ρgh)
Open holes at different depths in a tall container of water and oil, and compare how hard the liquid squirts out.
Scientists repeat a measurement, leave out an odd result and take the average; the ruler falls d metres in t = √(2d ÷ g) seconds, so the catch distance gives your reaction time
A friend drops a ruler and you catch it. Catch it five times, find your average reaction time, spot the odd catch, then try again while looking at a phone.
A windlass is a wheel and axle: its velocity ratio is the handle's radius ÷ the axle's radius; friction makes the mechanical advantage smaller than the VR, and efficiency = MA ÷ VR × 100%
Lift a bucket of water from a well by turning a crank. Change the length of the handle, the thickness of the axle, the bucket and whether the axle is oiled, and work out the VR, MA and efficiency.
Stars are huge balls of hot gas that make their own light; planets and moons shine by reflected sunlight; a constellation is a pattern of stars seen from the Earth, and a galaxy is a vast collection of billions of stars
Look at thirteen things in the sky, from the Sun and Dhruvatara to Venus, Saptarishi and the Milky Way. Find out whether each makes its own light and how far away it is, then sort them.
Gravitational potential energy (mgh) turns into kinetic energy (½mv²); some becomes heat when there is friction
Release a trolley from different heights on smooth and rough ramps and watch its potential energy turn into kinetic energy.
Work done = force × distance moved in the direction of the force; power = work ÷ time
Lift boxes of different masses onto shelves at different heights, calculate the work done, and compare the power of quick and slow lifts.
An acid and an alkali neutralise each other to form a salt and water; at the end point the pH is 7
Add dilute sodium hydroxide drop by drop to dilute hydrochloric acid with an indicator, watch the pH change, and find the exact point of neutralisation.
Air is a mixture: about 78% nitrogen, 21% oxygen, 1% argon and a little carbon dioxide
Trap air in a tube with damp iron wool over water, watch the water rise as the iron rusts, and compare fresh air with breathed-out air.
One mole of a substance is 6.022 × 10²³ particles (Avogadro's number), and its mass in grams equals the molar mass; so moles = mass ÷ molar mass, and particles = moles × 6.022 × 10²³
Atoms and molecules are far too small to count one by one, so chemists count them in moles. Put water, carbon, salt, carbon dioxide or glucose on a balance, change the mass, and find the moles and the number of particles.
Pure carbon exists in different forms called allotropes: crystalline ones (diamond, graphite, fullerene, graphene) have their atoms in a regular pattern, amorphous ones (coal, charcoal, coke, lampblack) do not; carbon compounds such as methane and sugar contain other elements too
Test twelve substances, from diamond and graphite to charcoal, soot and sugar, for hardness, electricity and what they are made of. Sort them into crystalline carbon, amorphous carbon and carbon compounds.
In a chemical reaction atoms are only rearranged, never made or destroyed, so the total mass stays the same (the law of conservation of mass); mass only seems lost when a gas escapes from an open container
Weigh a flask of vinegar with baking soda before and after they react, once with the flask open and once closed with a balloon. Find out whether mass is really lost.
Atoms are not created or destroyed in a reaction, so a balanced equation has the same number of each kind of atom on both sides
Change the numbers in front of each substance until every kind of atom is the same on both sides of the equation, watching the molecules as you go.
Higher temperature and smaller pieces make reactions faster
Drop fizzy tablets into cold, room-temperature and warm water, and crush one. Time which reaction finishes first.
Physical properties such as solubility, miscibility, melting and conductivity can be used to identify and group substances
Test common laboratory substances with water, heat and a conductivity tester, and group them by how they behave.
A solution's particles are smaller than 1 nm and let light through unseen; a colloid's (1 to 1000 nm) scatter a beam of light (the Tyndall effect) but pass through filter paper; a suspension's are bigger, settle and are filtered out
Shine a torch through twelve mixtures, from salt water and vinegar to milk, fog and muddy water, filter them, and sort them by the size of their particles.
The concentration of a solution by mass is mass of solute ÷ mass of solution × 100, where the solution is the solute plus the solvent; more solute or less water makes it stronger
Dissolve sugar in water to make sharbat. Change the grams of sugar and water, find the mass per cent, and see that different amounts can make solutions of the same strength.
Metals can be arranged in order of reactivity; a more reactive metal displaces a less reactive one
Put magnesium, zinc, iron and copper in dilute acid and compare the fizzing, then see which metals can push copper and zinc out of their salt solutions.
Metals above hydrogen in the reactivity series react with dilute acids to give hydrogen: Zn + 2HCl → ZnCl₂ + H₂; the amount of gas is set by whichever reactant runs out first, and hydrogen burns with a squeaky pop
Drop zinc, magnesium, iron or copper into dilute hydrochloric acid and collect the gas over water. Change the amounts and find out which runs out first, then test the gas with a burning splint.
The number of protons decides the element; electrons fill shells 2, 8, 8; ions have unequal protons and electrons; isotopes have different numbers of neutrons
Add protons, neutrons and electrons to build atoms of the first 20 elements, arrange the electrons in shells, and make ions and isotopes.
Particles spread from a high concentration to a low concentration by diffusion, faster at higher temperatures
Put a drop of food colouring into still water at different temperatures and measure how far the colour spreads over time.
Water moves by osmosis through a partially permeable membrane from a dilute solution to a more concentrated one
Soak potato strips in sugar solutions of different strengths, weigh them, and find the strength at which they neither gain nor lose water.
Every species has a two-part (binomial) scientific name: the genus, written with a capital letter, then the species; closely related species share a genus, and genera are grouped into families, orders, classes, phyla and kingdoms
Read the scientific names of eleven animals and plants found in Nepal, from the tiger and the jackal to the peepal and lali gurans, look up their families, and sort them by genus.
Living things have changed slowly over millions of years (organic evolution); fossils, homologous organs, vestigial organs and connecting links are evidence that today's species came from common ancestors
Look at a shaligram from the Kali Gandaki, a bat's wing, the human appendix, Archaeopteryx and more. Find out what each tells us, and sort them by the kind of evidence for evolution they are.
When the diaphragm moves down, the chest gets bigger, the pressure inside falls and air flows into the lungs
Pull and push the rubber diaphragm of a bell-jar lung model, watch the balloon lungs fill and empty, and see what a hole in the chest does.
Humans have a double circulation: the right side of the heart pumps blood to the lungs, and the left side pumps oxygen-rich blood to the body
Follow a drop of blood from the body through the four chambers of the heart, the lungs and back out to the body, checking whether it carries oxygen at each stage.
Living cells respire, using oxygen and giving out carbon dioxide; respiration is controlled by enzymes and depends on temperature
Use a respirometer to measure the oxygen used by germinating, dry and boiled seeds at different temperatures, and test their gas with limewater.
Mushrooms are fungi: they cannot make food, so they absorb it from the straw; they grow best when it is warm (about 20 to 25 °C), the air is damp and the straw has been pasteurised to kill moulds
Grow oyster mushrooms on bags of straw. Change the room's temperature, the dampness of the air and how the straw was treated, and find the conditions that give the biggest harvest.
A dicot stem has protective, storage, transport and growing tissues; xylem carries water up and phloem carries food, arranged in a ring of vascular bundles
Look at a stained cross-section of a sunflower stem and label the epidermis, cortex, phloem, cambium, xylem and pith.
A tissue is a group of similar cells doing one job; animal tissues are epithelial (cover and line), connective (join, support and carry), muscular (contract to move) or nervous (carry messages)
Look at twelve parts of the human body under a microscope and find out what each does, then sort them into epithelial, connective, muscular and nervous tissue.
Energy flows one way along a food chain; at each step only about 10% is passed on and the rest is used for living and lost as heat, so each level has less energy and food chains rarely have more than four or five links
Follow the energy in a food chain from grass to grasshoppers, frogs, snakes and eagles. Change how much energy the plants store and how much is passed on, and see why there are so few eagles.
Biodiversity can be compared by sampling with quadrats; more samples find more species until the count levels off
Place 1 m quadrats in a school field, at a forest edge and on a trampled path, count the plant species, and find how many quadrats are enough.
A sector with angle θ is θ/360 of the circle: area = θ/360 × πr², arc length = θ/360 × 2πr
Set how far a sprinkler turns and how far it sprays, and work out the area it waters and the length of its curved edge.
Two crossing paths w wide cover w × l + w × b − w², because the square where they meet would otherwise be counted twice; the lawn is the rest
A school garden has two paths crossing in the middle, one along it and one across. Change the garden's size, the paths' width and the paving price, and find the area of the paths, the lawn left and the cost.
The angle at the centre is twice the angle at the circumference standing on the same arc
Walk a photographer round the edge of a circular park and compare the angle at which the gate is seen with the angle at the centre.
A dividend is declared as a per cent of the face value of a share: a cash dividend is paid in rupees (with 5% tax taken off), and a bonus (stock) dividend is paid in new shares
Your family owns shares in a hydropower company. Change the number of shares and the cash and bonus dividends the company declares, and find the cash you receive after tax and the new shares you get.
The bank buys foreign money at its buying rate and sells it at its higher selling rate; the difference is its margin
Change money sent home from abroad into rupees, buy Indian rupees for a trip, and see why the bank's buying and selling rates differ.
For grouped data the mean is estimated as Σfx ÷ Σf using mid-values; cumulative frequency finds the median class
Group the heights of a class into 10 cm classes, draw the histogram, and estimate the mean, the modal class and the median class.
Units used = new reading − last reading; the tariff charges each tier of units at its own rate, plus a service charge, so a big user pays more per unit
Read the meter, find the units used this month, and work out the bill tier by tier. Change the units from 15 to 200 and see why using more electricity costs more for each unit.
log_b x is the power of b that gives x (b^y = x means log_b x = y); multiplying numbers adds their logarithms: log (xy) = log x + log y
A logarithm asks how many times you multiply the base to reach a number. Change the base and two numbers, and see on a slide-rule scale that multiplying numbers adds their logarithms.
The solution of two linear equations is the pair (x, y) that makes both true at once; on a graph it is the point where the two lines cross
Try values of x and y in two equations at once, see each one checked, and watch the guess move on a graph of both lines until it sits where they cross.
For discrete data in a frequency table, the mean is Σfx ÷ N, the median is the ((N + 1)/2)th value and the mode is the value with the biggest frequency; new data move the mean every time, but the median and the mode only when enough values change
A survey counts the members in each family of a tole. Add more families of five to the frequency table, and watch how the mean, the median and the mode each change, or stay put.
The area of a triangle with sides a, b and c is √(s(s − a)(s − b)(s − c)), where s = (a + b + c) ÷ 2 is the semi-perimeter; any two sides together must be longer than the third
A farmer can measure the three sides of a triangular field but not its height. Change the sides, find the semi-perimeter and the area with Heron's formula, and see when three lengths cannot make a triangle.
When p(x) is divided by (x − a), the remainder is p(a); if p(a) = 0, then (x − a) is a factor of p(x) and a is a zero, where the graph meets the x-axis
Divide a cubic polynomial p(x) by (x − a) without long division: the remainder is just p(a). Change a and the polynomial, and find the values of a that make (x − a) a factor.
After cards are removed, probability = favourable cards left ÷ cards left
Take all the kings, the clubs, the face cards or the aces out of the pack before you draw. See how the number of cards left and the favourable cards both change, and how that changes the probability.
P(A or B) = P(A) + P(B) − P(A and B); for mutually exclusive events the overlap is 0
Draw one card and win if it is a king or a queen, then a king or a heart. Count the cards that win and find out when you can just add the probabilities and when you must take away the overlap.
For independent events, multiply along the branches: P(A and B) = P(A) × P(B); P(at least one) = 1 − P(none)
Set how often each traffic light is green and use a tree diagram and an area model to find the chance of a trip with no stops.
Solving −x² + bx + c = h finds where a parabola meets a line; the discriminant tells whether there are two, one or no real solutions
Shape a fountain jet y = −x² + bx + c and find where it reaches a balcony, or prove with the discriminant that it never can.
In any triangle ABC, the line joining the mid-points D of AB and E of AC is parallel to BC and DE = ½ BC, whatever the shape of the triangle
Mark the mid-points D and E of two sides of a triangle and join them. Change the base, move the top corner and make it taller or lower: DE always stays parallel to BC and exactly half as long.
In an arithmetic sequence the same number d is added each time, so the nth term is a + (n − 1)d, and the sum of n terms is n/2 × (2a + (n − 1)d)
The school hall has rows of seats: a few at the front and more in each row behind. Change the seats in the first row, the extra seats in each row and the number of rows, and find the seats in the last row and in the whole hall.
n(A ∪ B) = n(A) + n(B) − n(A ∩ B), because the students in both are counted twice; A − B is A without B, and the complement (A ∪ B)′ is everyone in U outside both sets
Sixty students say whether they like football, cricket, both or neither. Change the numbers, watch the Venn diagram fill in, and find n(A ∪ B), n(A − B) and the students who like neither.
Triangles with the same angles are similar: their sides are in the same ratio, the scale factor
Use a stick and two shadows to find the height of a tree, a pole or a temple, whatever the height of the sun.
A prism has the same cross-section all along it, so its volume is the area of its end × its length; its surface is the two ends plus the rectangles round it
A trekking group is choosing a tent. Change the width and height of its triangular end and its length, and find how much air it holds (volume) and how much canvas it needs (surface area).
Income tax is charged in slabs: each part of the income is taxed at its own rate, so the total tax is the sum of the slabs, and the average rate is less than the rate on the last rupee
Change a person's yearly income from 4 lakh to 25 lakh rupees. See the income cut into slabs, the tax on each slab, the total tax, and why the tax as a per cent of income is always less than the top rate.
Two triangles are congruent if three sides match (SSS), two sides and the angle between them match (SAS), two angles and a side match (ASA or AAS), or a right angle, the hypotenuse and a side match (RHS); equal angles alone, or two sides with an angle not between them, do not fix one triangle
Twelve pairs of triangles, from a bridge truss to two ladders against a wall. Check what is given and which parts match, and sort each pair by the test that proves it congruent, or by “not enough”.
For an angle θ in a right-angled triangle, sin θ = opp ÷ hyp, cos θ = adj ÷ hyp and tan θ = opp ÷ adj; they depend only on the angle
Lean a ladder against a wall at different angles and see how sin, cos and tan compare its sides; find the safe angle.
In a right-angled triangle, tan θ = opposite ÷ adjacent, so height = distance × tan(angle of elevation) + eye height
Stand back from a temple tower, aim a clinometer at its top and use tan of the angle of elevation to find how tall it is.
A cone holds a third of the cylinder with the same base and height; a sphere holds two thirds of the cylinder that just fits round it
Fill a cone or a ball with water and pour it into a drum of the same size to discover the volume formulas.
Dynamic programming solves a big problem from the answers to smaller ones and remembers them: the ways to climb n stairs are the ways for n − 1 plus the ways for n − 2
You climb a staircase one or two steps at a time. In how many different ways can you reach the top? Build the answer up stair by stair from the answers before it, and see why remembering answers beats working them out again and again.
Halving the range each time finds any number from 1 to 100 in at most 7 guesses, because 7 halvings are enough to shrink 100 numbers to 1 (2⁷ = 128): the idea behind binary search
Think of a number from 1 to 100 and let the computer guess. It always guesses the middle of what is left, and each “too high” or “too low” throws half the numbers away. Count its guesses and find the number that makes it work hardest.
The work a program does grows with its input: one loop over n items takes about n steps, a loop inside a loop takes n × n, and halving n each time takes only about log₂ n; Big O names this growth
Count the steps three small programs take for the same n: one loop, a loop inside a loop, and a loop that halves n. Double n and see one count double, one grow four times and one hardly move: a first look at Big O.
Big O describes how an algorithm's steps grow with the size of its input n: O(1) stays the same, O(log n) grows by one when n doubles, O(n) doubles, O(n log n) a little more than doubles, and O(n²) quadruples
Race a hash lookup, binary search, linear search, merge sort and bubble sort as the number of items grows from 8 to 1024. See some lines barely rise while bubble sort shoots up past half a million steps, and learn the Big O names for how each one grows.
Breadth-first search (BFS) visits cells in order of distance using a queue, so its path is always a shortest one; depth-first search (DFS) follows one way as far as it can using a stack, so its path can be longer
Send a search through a maze, one cell at a time. Breadth-first search spreads out in rings with a queue and finds the shortest way; depth-first search dives down one corridor with a stack. Compare the paths they find and how many cells each one visits.
Bubble sort compares each pair of neighbours and swaps them when they are in the wrong order; each pass carries the largest number left to the end, and a pass with no swaps means the list is sorted
Watch bubble sort put a list of numbers in order, one comparison at a time. Count the comparisons and swaps, see the largest number bubble to the end after every pass, and compare a jumbled, a sorted and a reversed list.
A network of friends is a graph; when every link counts as 1, the shortest path is the fewest links, found by settling people in order of how many links away they are, nearest first
In a small network of friends, find how many links separate two people: a friend is 1 link away, a friend of a friend 2. A shortest-path search spreads out from one person, nearest friends first, and finds the fewest links, or shows that someone cannot be reached at all.
Linear search compares the target with each item in order; it needs 1 comparison at best and as many as there are items at worst, or when the target is missing
Search a list of test marks or shop prices for a number, checking one item at a time. Count the comparisons when the number is first, in the middle, last, or not in the list at all.
A greedy algorithm makes the best-looking choice at each step: for change, it gives the largest note that still fits, again and again, until nothing is left
A shopkeeper's program gives change in Nepali notes and coins. Run it for different amounts and watch the greedy rule at work: always give the largest note that still fits. Count the notes, and find out when being greedy is not the best plan.
Insertion sort keeps a sorted part on the left and inserts the next number into it by sliding it past every bigger number; on a list that is nearly sorted it does very little work
Watch insertion sort take each number in turn and slide it left into its place among the numbers already sorted, the way you sort cards in your hand. Count its comparisons and swaps, and compare it with bubble sort and selection sort on a jumbled, a sorted and a reversed list.
A number is prime when only 1 and itself divide it; a program tests divisors with %, stops at the first one that divides, and only needs to try divisors up to the square root, because any bigger factor comes with a smaller partner
Choose a number and watch a program try divisors 2, 3, 4 … with the remainder. It stops at the first divisor it finds, or as soon as the divisor times itself is bigger than the number. Count the tries and find out why it never needs to go further.
Binary search compares the target with the middle of a sorted list and throws away the half it cannot be in, so 16 items need at most 5 comparisons instead of 16; it only works on sorted input
Search 16 numbers two ways: one by one, and by halving the list at each step. Compare the comparisons each needs, then run binary search on a list that is not sorted and see it miss a number that is there.
Merge sort divides the list into halves until each part has one number, then merges sorted parts by always taking the smaller front number; its comparisons grow like n log n, far slower than the n² of bubble sort
Split a list of eight numbers in half again and again until every part holds one number, then merge the parts back, always taking the smaller front number. Count the comparisons and compare them with bubble sort and insertion sort on jumbled, sorted and reversed lists.
Selection sort looks through the unsorted part for the smallest number and swaps it into the next place at the front; it always makes the same number of comparisons, but at most one swap per pass
Watch selection sort put a list in order by finding the smallest number left and swapping it to the front, pass after pass. Count its comparisons and swaps, and compare it with bubble sort on a jumbled, a sorted and a reversed list.
Dijkstra's algorithm settles towns in order of their distance from the start; each settled town's distance is final, and trying the roads out of it can only make the other towns' distances shorter
Find the shortest way by road from Kathmandu or Pokhara to another town. Watch Dijkstra's algorithm settle the nearest town first, try the roads out of it, and keep the shortest distance it has found to every town.
An algorithm is a list of exact steps done in order; changing the order or one step changes the result, and debugging means finding the step where the plan breaks
Put move and turn blocks in order so a robot crosses a 5 × 5 grid to its battery. Run a broken plan to find the block where it bumps into a wall, fix it, and then find the shortest plan.
A Caesar cipher replaces each letter with the letter a fixed number of places (the shift) further along the alphabet, going round from Z back to A; shifting back by the same number decodes it
Hide a message by moving every letter the same number of places along the alphabet, then read it back by moving the other way. Watch the program code a word letter by letter, see what happens after Z, and crack a coded word.
A state machine is always in one of a few states and moves to the next state by a rule; a traffic light keeps its state in a variable and goes round red, green, yellow with (state + 1) % 3
Run a traffic light for a number of ticks. A loop moves it from red to green to yellow and back to red, using one variable for the state and % to go round again. Count each colour's turns and predict the light at any tick.
A big problem is easier when it is broken into small parts (decomposition); each part becomes a few steps, and a mistake in one part, such as one move too many, breaks the whole plan
Guide a robot through a 7 × 7 maze to its battery. Break the long route into short parts, find the block where a plan with a counting mistake breaks, and build the shortest plan.
A linked list stores each item in a node with a link to the next node; adding at the front is instant, but reaching node i means following i links from the head
Build a train where each carriage is hooked to the next one. Add a carriage at the front or after another, take one out, and find one, and count how many links you have to follow each time, because a linked list has no index to jump to.
A stack keeps items in a pile: push puts an item on the top, pop takes the top item off and peek only looks at it, so the last item in is the first one out (LIFO)
Push plates onto a stack, pop them off and peek at the top. See why the last plate you put on is always the first one you take off, and what happens when you try to pop from an empty stack.
A queue keeps items in a line: enqueue adds an item at the rear, dequeue takes the item at the front and front only looks at it, so the first item in is the first one out (FIFO)
Serve customers at a ticket counter with a queue: new people join at the rear, the person at the front is served first, and front shows who is next. See why a queue is fair, and what happens when nobody is left to serve.
In a binary search tree every number on a node's left is smaller and every number on its right is bigger, so a search throws away half the tree at each step, but only while the tree stays bushy and short
Insert seven numbers into a binary search tree, smaller ones to the left and bigger ones to the right. Search it with very few comparisons, print it in order, and see how numbers that arrive already sorted turn the tree into a long chain.
To check brackets, push each opening bracket on a stack; a closing bracket must match the bracket on top, which is popped; at the end the stack must be empty
Every ( needs a ), every [ a ] and every { a }, closed in the right order. Watch a program check an expression with a stack: push each opening bracket, pop it when the matching one closes, and stop at the first bracket that does not fit.
A hash table turns each key into a bucket number with a hash function; adding and finding a key look in that one bucket, so they stay fast however many keys there are, as long as collisions are few
Add names to a table of buckets, where a hash function works out each name's bucket from its letters. See two names land in the same bucket (a collision), find a name by looking in one bucket only, and change the number of buckets to spread the names out.
A list (array) keeps items in order; each item has an index, counted from 0, so a list of n items has indexes 0 to n − 1
Keep a shopping list in a program. Read an item by its index, change one, add and remove items, and find out what happens when you ask for an index past the end of the list.
A two-dimensional array is a list of rows, each a list of values; grid[row][col] picks one value, a loop along a row or down a column adds them up, and two nested loops visit every cell
Store a class's marks in a table with a row for each student and a column for each subject. Read one mark with grid[row][col], add up a student's row or a subject's column, and find the highest mark with a loop inside a loop.
A traversal visits every node of a tree once; pre-order visits a node before its children, in-order between its left and right children, post-order after them, and level order row by row with a queue
Build a binary search tree of seven numbers and walk it four ways: in-order, pre-order, post-order and level by level. See four different orders of the same numbers, why in-order comes out sorted, and how a queue walks the tree row by row.
An if / else statement checks a condition: when it is true the program runs the then part, otherwise the else part, never both
Change the chance of rain and run a small program that decides whether to take an umbrella. Watch the condition turn true or false, see which branch runs, and fix a condition that gives the wrong answer.
NOT turns true into false; AND is true only when both parts are true; OR is true when at least one part is true, so a long condition can be worked out one part at a time
Tell the program whether it is raining, whether your homework is done and whether it is a holiday, and watch it work out each part of the rule with NOT, AND and OR before it decides whether you can play outside.
Conditions can be joined with AND (all must be true) and OR (one is enough), and an if / else can sit inside another, so the program checks one question after another until one is true
Enter marks in maths, science and English and run a result program: it works out the total and the percentage, checks with AND that every subject is passed, and then uses nested if / else to give the division. Find the marks that reach each branch.
A year is a leap year when it divides by 4 and not by 100, or when it divides by 400; the remainder % tests divisibility and AND / OR join the rules into one condition
Choose a year and run a program that decides whether February has 29 days. See how the remainder operator % tests divisibility, and how AND and OR join three rules into one condition that gets 1900 and 2000 right.
A loop condition joined with AND stops as soon as either part is false: here when the PIN is right or when three tries are used up
An ATM gives you three tries to type your PIN. Choose the PINs typed on each try and run the program: a while loop keeps asking while the card is locked AND fewer than three tries are used, then either welcomes you or blocks the card.
A function keeps a rule in one place: to_fahrenheit(c) returns c × 9 ÷ 5 + 32, and every call gives it a new argument for its parameter c
Write to_fahrenheit(c) once and use it for Kathmandu, Jumla and Biratnagar. Change each town's temperature, watch every call get its own c, and see one rule give three answers.
Euclid's rule gcd(a, b) = gcd(b, a % b) makes the numbers smaller at every call, so a recursive function reaches its base case, b = 0, where the answer is a
Find the greatest common divisor (HCF) of two numbers with a function that calls itself: gcd(a, b) calls gcd(b, a % b), and the numbers shrink until the remainder is 0. Watch the call stack, count the calls, and see which pairs share no divisor but 1.
fib(n) = fib(n − 1) + fib(n − 2) with fib(0) = 0 and fib(1) = 1; because each call makes two more, the number of calls grows very fast, and the same answers are worked out many times
Work out Fibonacci numbers with a function that calls itself twice. Count the calls as n grows, watch the same small calls happen over and over on the call stack, and see why remembering answers is so much faster.
A function is a named set of steps written once and used many times: its parameters receive the arguments of each call, and return sends its answer back to where it was called
Define a function area(length, breadth) once and call it for a kitchen, a hall and a bedroom. Watch each call open a new frame with its own parameters, see the value it returns, and change the bedroom's size to reach a target area.
A recursive function solves a problem by calling itself on a smaller version of it; a base case answers the smallest version directly, so the calls stop and the answers come back up the call stack
Work out factorials with a function that calls itself. Watch the call stack grow one frame for every call down to the base case, then shrink as each call returns its answer, and see why a recursive function needs a base case.
To move n disks, move the top n − 1 out of the way, move the biggest, then move the n − 1 back on top; the same plan solves the smaller towers, so the moves double and add one each time: 2ⁿ − 1
Move a tower of disks from peg A to peg C, one disk at a time, never a bigger disk on a smaller one. A recursive plan solves it for any number of disks: watch the calls and the moves, and find the pattern 2ⁿ − 1.
In nested loops the inner loop runs all the way through for every turn of the outer loop, so a table of r rows and c columns has r × c products
Print a multiplication table with two loops, one inside the other: the outer loop chooses the row, the inner loop goes along it. Change the size of the table, count the products, and see how the inner loop starts again on every row.
n % 10 is the last digit of n and floor(n / 10) is n without it, so a while loop that repeats both until n is 0 visits every digit once
Choose a number and watch a while loop take it apart: % 10 gives the last digit, dividing by 10 drops it, and the loop adds each digit to a total until the number reaches 0. Count the digits on the way.
A loop can make a decision on every repeat: i % 3 == 0 is true when 3 divides i exactly, and in a chain of ELSE IFs the first true check wins, so the most special case must come first
Count from 1 to n, but say Fizz for multiples of 3, Buzz for multiples of 5 and FizzBuzz for multiples of both. Watch the remainder decide each number, and see why the order of the checks matters.
A loop repeats the same lines: a for loop runs once for each value of its counter, an accumulator adds something on every repeat, and a while loop repeats until its condition becomes false
Choose how many times a loop repeats, then run or step through it: watch the counter go up, each repeat light up and the running total grow, and learn when a while loop stops, and when it never does.
A while loop repeats its body as long as its condition is true, so the body must change something that makes the condition false in the end, or the loop never stops
Save the same pocket money every week until you have enough for something you want. A while loop keeps adding while the savings are less than the price: count the weeks, see the loop stop, and find out what makes a loop run forever.
In nested loops the inner loop runs completely for every turn of the outer loop; when the inner loop goes from 1 to row, row r prints r stars and n rows print n(n + 1)/2 stars
Run two loops, one inside the other, to print a triangle of stars. Change the number of rows, count how many times the inner loop runs, and work out how to turn the triangle into a square.
A variable is a named box that holds one value; an assignment works out the right-hand side and puts the answer in the box, replacing what was there, and the program uses the latest value
Buy notebooks at a shop with a small program: put in the price, how many and the money you pay, then step through it and watch each variable's box fill and change until the total and the change are printed.
Repeatedly dividing a number by 2 gives its binary digits as the remainders, from the last bit to the first; each remainder is placed one place further left
Choose a number and run the algorithm that turns it into binary: divide by 2, keep the remainder as the next bit, and repeat until nothing is left. Watch the bits build up from the right and count how many a number needs.
A string is a list of characters with indexes from 0; a loop can visit each one in turn, and two pointers moving in from both ends test a palindrome with only half as many comparisons
Go through a word one letter at a time: count its letters and its vowels, write it backwards, and test whether it is a palindrome by comparing letters from both ends at once.
Every atom is made of protons and neutrons in a tiny nucleus with electrons around it; the number of protons decides which element it is
Add protons, neutrons and electrons one by one and see the atom you make: its name, its mass and how its electrons sit in shells. Then turn carbon into nitrogen with a single proton.
NaHCO₃ + CH₃COOH → CH₃COONa + H₂O + CO₂: one mole (84 g) of baking soda gives one mole (44 g, about 24 L) of carbon dioxide and needs one mole (60 g) of acetic acid
Drop 2 to 20 g of baking soda into vinegar and blow up a balloon with the carbon dioxide. Use the equation and moles to find the mass and volume of the gas and how much vinegar the soda needs.
Solubility is the most of a substance that dissolves in 100 g of water at a given temperature; sugar's solubility rises steeply with temperature, but salt's hardly changes
Stir sugar or salt into water at different temperatures until no more dissolves. See how hot water dissolves far more sugar, while salt hardly changes, and why syrup for sweets is made hot.
Food spoils when microbes grow in it; microbes need water, warmth and time, so food keeps longer when we take the water out, keep it cold, heat it to kill the germs, or pickle it in acid or oil
Look at twelve kitchen foods, from sun-dried fish to mango achar, find out how long each keeps and what stops the microbes, and sort them by the way they are preserved: drying, cold, heat or pickling.
Dissolving salt makes water denser; an object floats when the liquid is denser than the object, so an egg floats once the salty water is denser than the egg
Drop an egg into a glass of water and stir in salt, a spoonful at a time, until the egg floats. Then try an old egg, which floats in plain water, and find out why cooks use this to test eggs.
Water boils when its vapour pushes as hard as the air above it; higher up the air pressure is lower, so water boils at a lower temperature (about 1 °C lower for every 300 m) and food cooks more slowly
Boil a pot of water at the seaside, in Kathmandu, Shimla, Leh and at Everest base camp. See how the boiling point falls as you climb, and why dal takes so much longer to cook up high.
A battery's capacity in mAh is the charge it holds: hours to charge = capacity ÷ charger current (plus about 20% losses), and energy in Wh = capacity (Ah) × voltage
Pick a phone battery of 3000, 4000 or 5000 mAh and a 1 A, 2 A or 3 A charger. Work out how long it takes to charge from empty, how much energy it stores and how many hours the screen can stay on.
An LED takes a fixed forward voltage (about 2 V for red, 3.2 V for blue and white); a resistor in series takes the rest, so by Ohm's law R = (supply − LED voltage) ÷ current
Light a red, green, blue or white LED from a 5 V, 9 V or 12 V supply. Work out the resistor that lets just the right current flow, and how much power the resistor turns into heat.
Electricity and fuel cost money and resources, and making them often pollutes the air; small habits, such as switching off, using LED bulbs and covering pots, save energy, while leaving things on or open wastes it
Look at twelve everyday habits, from switching off lights and cooking with a lid to leaving the TV on standby. Find out why each one saves or wastes energy, and sort them.
Energy = power × time: a 0.4 kW panel in 5 hours of sun gives about 0.4 × 5 × 0.8 = 1.6 kWh, after the losses of heat, dust and wiring
Put 2 to 10 solar panels on a roof and choose how many sunny hours a day it gets. See how many kilowatt-hours they make a day and a month, how many LED bulbs that lights for an evening, and how much CO₂ it saves against coal power.
Each fuel gives a fixed heat per gram, its calorific value, but a stove passes only part of it to the pot; grams burnt = heat needed ÷ (calorific value × efficiency), so a good fuel in a good stove burns far less
Boil 1, 2 or 5 litres of water on an LPG stove, a kerosene stove, a coal stove, an open wood fire or a dung-cake chulo. See the heat the water needs and how many grams of each fuel burn to give it.
On a spinner with equal sectors, P(colour) = sectors of that colour ÷ all sectors; the probabilities add up to 1, and the expected number of times = probability × number of spins
Colour the equal sectors of a spinner red, blue and green, and work out the chance of landing on each colour and how many reds to expect in 60 spins. Then make the game fair.
Batting average = runs ÷ times out (runs per dismissal); strike rate = runs ÷ balls faced × 100 (runs per 100 balls)
Change a batter's runs, balls faced and times out, and work out the batting average and the strike rate. Find out why one tells you how reliable a batter is and the other how fast they score.
Amounts in a recipe are in direct proportion to the number of people: each amount × (new number of people ÷ 4), so doubling the people doubles everything
A dal-bhat recipe feeds 4 people. Cook it for 2, 6, 8 or 10 people instead, and see how the rice, dal, water and salt all change by the same ratio.
In the Fibonacci sequence each number is the sum of the two before it (Fₙ₊₁ = Fₙ + Fₙ₋₁); the ratio of neighbours Fₙ₊₁ ÷ Fₙ gets closer and closer to the golden ratio φ = (1 + √5) ÷ 2 ≈ 1.618
Build the Fibonacci numbers 1, 1, 2, 3, 5, 8, … as a spiral of squares, and divide each by the one before. Watch the ratio swing above and below and settle on 1.618, the golden ratio found in sunflowers, pine cones and shells.
In a group of n people each person shakes n − 1 hands, but every handshake is shared by two people, so there are n(n − 1) ÷ 2 handshakes: the number grows much faster than the group
At a party everyone shakes hands with everyone else once. Change the number of people from 2 to 12 and count the handshakes, then find the rule that works for any group.
Each number in Pascal's triangle is the sum of the two above it; the k-th number of row n is C(n, k), the number of ways to choose k things from n, and row n adds up to 2ⁿ
Pick a row n and a place k in Pascal's triangle. Read the number C(n, k), see that it is the sum of the two above it, and find that every row adds up to a power of 2: the ways of choosing, of tossing coins and of walking a grid.
A loan is repaid in equal monthly instalments, EMI = P r (1 + r)ⁿ ÷ ((1 + r)ⁿ − 1), with r the monthly rate and n the months; a longer loan has a smaller EMI but much more interest
Borrow Rs 1 lakh to Rs 10 lakh at 8% to 12% a year and choose to pay it back in 1, 3 or 5 years. See the equal monthly instalment (EMI), the total you pay back and how much of it is interest.
A line of symmetry folds a shape into two halves that match exactly; some shapes have one, some have many, and some have none, even if they look balanced when turned
Look at twelve shapes from nature and daily life, from a butterfly and a snowflake to a shoe and a swirl. Find how many lines of symmetry each has, and sort them into one line, more than one, or none.
The minute hand turns 6° a minute and the hour hand 30° an hour plus 0.5° a minute, so the angle between them is |30h − 5.5m|, or 360° minus that if it is bigger than 180°
Set a clock to any time and work out the angle between its hands. Find the times when the hands make a right angle or a straight line, and see why the hour hand is never exactly on the number at half past.
With n people there are n(n − 1)/2 pairs, each with a 1 in 365 chance of sharing a birthday; the pairs grow so fast that the chance of a shared birthday passes 50% at 23 people: P ≈ 1 − e^(−n(n − 1)/730)
Fill a room with 5 to 70 people and find the chance that at least two of them share a birthday. See why a class of 23 already has an even chance, and why the number of pairs, not people, is the secret.
The nth triangular number is 1 + 2 + … + n = n(n + 1) ÷ 2, and a triangle with the one before it makes a square: Tₙ + Tₙ₋₁ = n²
Stack dots in rows of 1, 2, 3, 4 and more to build triangles. Count the dots in each triangle, find the rule n(n + 1) ÷ 2, and see how two neighbouring triangles fit together into a square.
Every measurement has a quantity (length, mass, volume or time) and a unit; we choose a unit that suits the size, so the number is neither huge nor tiny: kilometres for roads, millimetres for an ant
From the distance between two cities to a spoon of cough syrup, decide what each of twelve everyday things measures, length, mass, volume or time, and choose a unit that suits its size.
Body mass index compares mass with height: BMI = mass (kg) ÷ height² (m²); for adults 18.5 to under 25 is a healthy range, but children are still growing, so their BMI is checked on charts for their age and sex
Choose a mass and a height and work out the body mass index, BMI = mass ÷ height². See which adult band it falls in, and learn why doctors use age charts, not these bands, for growing children.
Energy used (kcal) ≈ MET × body mass (kg) × time (hours), where the MET says how many times harder than resting an activity is; harder play, a heavier body and longer play all use more energy
Choose a game or exercise, your body mass and how long you play, and work out how much food energy your body uses, in kilocalories and in rotis.
First aid is the help given before a doctor arrives: it keeps the person safe, stops things getting worse and gets them to proper care; some common habits, like ice on burns or sucking snake venom, harm or waste precious time
Twelve things people do for burns, cuts, nosebleeds, snake and dog bites, diarrhoea and someone who has fainted. Find out why each helps or harms, and sort the good first aid from the old habits that do damage.
Each case of a disease infects R₀ people when no one is immune; when more than 1 − 1/R₀ of people are vaccinated, each case infects fewer than one other and the disease dies out, protecting even those who cannot be vaccinated
Choose a disease, from flu to measles, and how many people in a town are vaccinated. See how many people each case infects now, how many must be vaccinated for herd immunity, and whether the disease dies out or keeps spreading.
Bacteria multiply by splitting in two; in warm food they can double every 20 minutes, so after n doublings one germ becomes 2ⁿ, which grows faster and faster
Start with one germ on a plate of food left out in the warm. Choose how often it doubles and how many hours pass, and watch one germ become thousands, then millions, which is why cooked food should not sit out for long.
The body loses water in sweat, breath and urine, so a child needs about 35 mL of water for every kg of body mass each day, more when playing hard and on hot days
Choose a body mass, the minutes of hard play and whether it is a hot day, and work out how much water a child needs in a day, in millilitres and in glasses.
Inherited traits are set by genes passed from parents to children; acquired traits are learnt or gained during life, from practice, injury or the surroundings, and are not passed on in the genes
Look at twelve traits, from eye colour and dimples to a scar, a suntan and the language you speak. Find where each comes from and whether your children could get it from you, then sort them into inherited and acquired traits.
A lack of one nutrient causes a deficiency disease: vitamin A night blindness, vitamin C scurvy, vitamin D rickets, vitamin B1 beriberi, iron anaemia and iodine goitre; foods rich in that nutrient prevent it
Twelve cards show a sign of poor diet, such as night blindness or a swollen neck, or a food that prevents one, such as carrots or iodised salt. Find out what each tells you, then sort them by the vitamin or mineral they are about.
Red cells may carry A, B and Rh antigens; a patient's blood attacks any A or B it lacks, and an Rh-negative patient reacts to Rh, so O− can give to everyone and AB+ can take from everyone
Check twelve donor and patient pairs, from O− to A+ and AB+ to O−. Find the antigens on the donor's red cells and the antibodies in the patient's blood, and sort each transfusion as safe or not safe.
Magnitude is a logarithmic scale: each step of 1 means 10 times the shaking on a seismograph and about 32 times (10^1.5) the energy
Pick two earthquakes by their magnitude, from a small 3 to a great 8, and see how many times more the ground shakes and how many times more energy the bigger one lets out. Find why the 7.8 Gorkha earthquake of 2015 was so much worse than a 5.
1 mm of rain on 1 m² of roof is 1 litre of water, so a roof collects area × rain × 0.8 litres (some splashes or soaks away): even a small roof can store a lot of water in the monsoon
Choose the size of a roof and the rain in a month, and work out how many litres of rainwater a family could collect in a tank, and for how many days it would give a family of five their water.
Gravity pulls a satellite towards the Earth and keeps it moving in a circle; in an orbit of radius r its speed is v = √(GM ÷ r), so higher satellites move slower and take longer to go round
Put a satellite in a low orbit like the International Space Station, in a GPS orbit, or in a geostationary orbit used for TV. See how far it is from the Earth's centre, how fast gravity makes it go and how long one trip round the Earth takes.
Light travels at about 299 792 km every second, so the time it takes is distance ÷ speed: a fraction of a second from a satellite, 8 minutes from the Sun and more than 4 years from the nearest star
Send light to us from a GPS satellite, the Moon, the Sun, Mars and the nearest star, and work out how long each journey takes. Find out why we see the Sun as it was eight minutes ago.
Light travels in straight lines, so a stick's shadow is its height ÷ tan(the sun's height): the lower the sun, the longer the shadow, and with the sun straight overhead there is no shadow at all
Stand a stick in the sun and move the sun from low in the morning sky to straight overhead. See how long the shadow is each time, and why shadows are long in the morning and evening and shortest at noon.
Trees take carbon dioxide from the air by photosynthesis and store the carbon in their wood: a grown tree takes about 21 kg of CO₂ a year, while a car gives out about 0.17 kg every kilometre
Plant 1, 10 or 100 trees and let them grow for 1 to 20 years. Find how much carbon dioxide they take from the air, and how many kilometres a car could drive to give out the same amount.
A scale model divides every size and distance by the same number: if the Earth were a 22 cm football, the Moon would be a 6 cm ball 6.6 m away and the Sun a 24 m ball 2.6 km away, so space is mostly empty
Shrink the Earth to a football, a cricket ball or a marble, and shrink everything else by the same scale. See how big the Moon and the Sun would be, and how far away they would sit.
Burning petrol and diesel gives carbon dioxide, a greenhouse gas: kg of CO₂ = km travelled × kg per km per person, so walking or cycling gives none, a shared bus gives little, and a car gives the most for each student
Choose how you get to school, how far it is, and for how many days. See how many kilograms of carbon dioxide the trips put into the air, and how many trees it would take a year to soak it up.
The Earth turns 360° in 24 hours, so 15° of longitude is 1 hour: places to the east see the sun first and their clocks are ahead; each country sets its standard time from a longitude near its middle
It is noon in Kathmandu. What time is it in London, Delhi, Tokyo or New York? Pick two cities and see how the turning Earth gives every country its own clock time, and why Nepal is 5 hours 45 minutes ahead of London.
Force = pressure difference × area: a fan that lowers the pressure under a car presses it onto the road
Switch on a fan that sucks the air out from under a sports car, and see how a small pressure difference over the car's floor gives a huge downforce, more grip and faster cornering. Then blow air in to make the car float, and fire thrusters to see when thrust beats weight and the car flies.
A stretched band stores elastic potential energy E = ½kx²; on release it becomes kinetic energy ½mv², so v = √(2E ÷ m), and at 45° the pellet flies about v² ÷ g: twice the stretch gives four times the energy and twice the speed
Stretch a rubber band and fire a paper pellet. Change the band, how far you stretch it and the pellet's mass, and see the energy stored in the band, the launch speed and how far the pellet flies.
A falling object speeds up until air drag equals its weight; then it falls at a steady (terminal) speed v = √(2mg ÷ (ρCA)), so a bigger canopy falls slower
Hang a 20 g, 50 g or 100 g toy under a canopy of 0.1 to 1 m², and find how fast it falls once air drag balances its weight, and how long it takes to drop 10 m.
Gear ratio = front teeth ÷ rear teeth: the back wheel turns that many times for each turn of the pedals, so a big ratio goes far and fast but is hard to push, and a small ratio is easy to push but slow
Change the front and rear gears of a bicycle and see how far one turn of the pedals takes you, and how fast you go at the same pedalling. Find the easy gear for a hill and the fast gear for a flat road.
Radioactive atoms decay at random, but in each half-life half of those left decay: after n half-lives 100% × (½)ⁿ remain, whether the half-life is hours or thousands of years
Watch a crowd of radioactive atoms decay. Choose an isotope used in hospitals or for dating old bones, let half-lives go by, and see what share of the atoms is left and how long that took.
Plants cannot move, so they spread their seeds away from the parent plant by wind, water, animals or bursting pods, and each seed or fruit has a shape that suits its way of travelling
Look closely at twelve seeds and fruits, from the dandelion's parachute to the coconut's husk and the burr's hooks, and sort them by how they travel: by wind, by water, by animals or by bursting pods.
Light arrives almost at once but sound travels only about 343 m a second, so the distance to lightning in km is the seconds from flash to thunder × 0.343
Count the seconds between a lightning flash and its thunder, and work out how far away the lightning struck. Find out why a short gap means you must go indoors at once.
Loudness is measured in decibels (dB): every 10 dB is ten times the sound energy; the level falls about 6 dB each time the distance doubles, and above 85 dB the safe listening time halves for every 3 dB more
Stand near a whisper, a conversation, busy traffic, a pressure horn or a firecracker. See the sound level in decibels, how it drops as you move away, and how long your ears can safely take it each day.
Hot things cool faster the hotter they are than the room: T = room + (start − room) × e^(−kt); a lid and a vacuum flask cut heat loss, giving a much smaller k
Pour tea at 90 °C into an open cup, a covered cup or a vacuum flask in a 25 °C room, and follow its temperature for an hour to see which keeps it hot and why.
Stopping distance = thinking distance (speed × reaction time) + braking distance (v² ÷ 2μg): doubling the speed doubles the first but four-times the second
A child steps onto the road 30 m ahead. Change the vehicle's speed, the driver's reaction time and the road, and find the thinking distance, the braking distance and whether it stops in time.
A scale shows how hard it pushes up on you: N = m(g + a), so you seem heavier when the lift speeds up upward, lighter when it speeds up downward, and weightless in free fall
Stand on a bathroom scale in a lift. See what it reads as the lift starts going up, moves steadily, starts going down, and, if the cable snapped, falls freely.