Learn by watching it work

Neighbours are compared and swapped until the largest values bubble to the end.

Take the next value and slide it left into the sorted part, like sorting cards in your hand.

Find the smallest value in the unsorted part and swap it into place, one position at a time.

Split the list in halves, sort each half, then merge the two sorted halves.

Pick a pivot, move smaller values to its left and bigger to its right, then sort each side.

Turn the list into a max-heap, then repeatedly move the largest value to the end.

Count how many times each value appears, then write the values back out in order.

Check each value from left to right until the target turns up.

On a sorted list, look at the middle and throw away the half that cannot hold the target.

Push every opening bracket; each closing bracket must match the one on top.

Players pass a potato around a queue; after every k passes the holder is out.

Walk the list once, turning every arrow around with three pointers: prev, curr and next.

Find the node to insert after, then splice a new node in by re-pointing two arrows.

Find the node before the target and point it past the target, which drops out of the chain.

Each key goes left if it is smaller than the node, right if bigger, until it finds an empty spot.

Compare with the node and go left or right — one path from the root, never the whole tree.

Visit left subtree, the node, then right subtree — which reads the keys out in sorted order.

Push values into an array-backed heap, then pop the two smallest — watch them sift up and down.

Find two numbers that add to a target by remembering every number seen so far in a map.

Each letter is a step down a tree of nested maps; words that share a prefix share a path.

Explore a graph in waves: all neighbours of the start first, then their neighbours, using a queue.

Go as deep as possible along one path, then back up and try the next unvisited neighbour.

From the start, always settle the nearest unsettled node next, relaxing the edges out of it.

Order tasks so every arrow points forward: repeatedly take a node with no incoming edges.

Mark nodes "visiting" while their DFS runs; reaching a "visiting" node again means a cycle.

Plain recursion recomputes fib(3) again and again; a memo dictionary answers repeats instantly.

How many ways to climb n steps taking 1 or 2 at a time? Build the table from the bottom.

Pick items with the most value that fit in a bag of capacity W — each item taken at most once.

The longest sequence of letters that appears, in order, in both strings — filled cell by cell.

For each position, the longest rising run that ends there — built from every earlier smaller value.

The fewest coins that make each amount from 1 up to the target, reusing smaller answers.

The fewest inserts, deletes and replacements to turn one word into another, as a table.

n! = n × (n−1)!: calls stack up until the base case, then the answers multiply back out.

x¹⁰ needs only 4 multiplications: square the half-power instead of multiplying ten times.

Move n disks between pegs, one at a time, never a big disk on a small one — in 2ⁿ − 1 moves.

Every ordering of a list: fix one item first, then permute the rest.

20 + 40 + 60 as hops on a number line, place-value blocks and column addition, exactly.

500 − 275: hop back along the number line and borrow across the columns.

156 ÷ 12 worked digit by digit: divide, multiply, subtract, bring down.

3/4 + 1/6: fraction bars are cut into twelfths so the pieces can be counted together.

(3 + 4) × 2 − 5 as an expression tree that evaluates brackets first, then ×, then −.

Two fractions drawn as bars, re-cut to a common denominator and added. Change the numbers.

A balance scale: the same operation happens on both sides until x stands alone.

Squares on each side of a 3-4-5 triangle show why a² + b² = c².

A circle is cut into wedges and unrolled into a near-rectangle: A = πr².

The parabola is plotted point by point; its roots, intercept and vertex are marked.

A cylinder of radius 7 cm and height 10 cm in 3D: the volume stacked up from its base, and the curved surface unrolled into a rectangle.

A cone of radius 3 cm and height 4 cm: the slant height by Pythagoras, one third of the matching cylinder, and the curved surface unrolled into a sector.

Archimedes' results in 3D: a sphere fills two thirds of the cylinder around it, and its surface covers exactly four great circles.

a × b by the determinant, its length as the area of the parallelogram a and b span, and its direction by the right-hand rule.

(1, 2, 3) · (4, 5, 6) = 32: multiply matching components, find the angle between the vectors, and see the dot product as a length times a shadow.

Points A(1, 2, 3) and B(4, 6, 3) in space: the distance from the box between them, the point dividing AB in the ratio 2 : 1, and the direction cosines.

A ball launched at 20 m/s and 30°: its path, velocity components, apex and range, with energy bars.

Drop from 20 m: position, speed and energy every instant until impact.

A 1 m pendulum released at 20°: the swing, its period and the trade between potential and kinetic energy.

A 2 kg mass on a 50 N/m spring pulled 0.2 m: F = −kx, the period and the energy exchange.

A 5 kg block on a 30° slope with friction 0.2: the forces, and whether it slides.

An object at 4 m/s on a 2 m circle: the velocity and centripetal acceleration vectors, and the period.

Two carts meet head-on; momentum and kinetic energy are tallied before and after.

A force on a 0.3 m lever: τ = r × F component by component, why only the part of F across the lever turns it, and which way τ points.

A 12 V battery with 2 Ω and 4 Ω in series: one current, and the voltage drop across each resistor.

A 12 V battery with 3 Ω and 6 Ω in parallel: branch currents and the equivalent resistance.

An object 25 cm from a 10 cm converging lens: the principal rays draw themselves and meet at the image.

A ray enters glass (n = 1.5) at 40°: how much it bends, and when it would reflect instead.

A proton entering a 0.1 T field at 60°: the force qv × B bends it into a helix, with its radius, period and pitch worked out as it moves.

Green light as a travelling wave: E and B at right angles and in step, B₀ = E₀/c, and one wavelength travelled every period.

Propane burning: C₃H₈ + O₂ → CO₂ + H₂O, balanced with atom counters on both sides.

Ethanol drawn from its SMILES string, every atom and bond labelled. Try another molecule.

One mole at 273.15 K in 22.4 L: solve PV = nRT for the pressure, with particles in a box.

Eight electrons fill 1s, 2s and 2p in Aufbau order, drawn as an orbital-box diagram.

Count the electron pairs around N, spread them to the corners of a tetrahedron and hide the lone pair: a trigonal pyramid with a 107° bond angle.

Five electron pairs around sulfur; the lone pair takes a roomier place round the middle, leaving the four F atoms in a seesaw.

Copper's unit cell: 8 × ⅛ + 6 × ½ = 4 atoms, 12 neighbours each, the radius where atoms touch, 74% packing and a density of 8.97 g/cm³.

Cl⁻ at the corners and faces, Na⁺ on the edges and in the centre: 4 formula units per cell, 6 : 6 coordination and the density of salt.

One strand’s bases pair A–T and G–C to build the complementary strand, which runs the other way; then the ladder twists, 10 base pairs to a turn.