Follow shapes through turns, folds, reflections, and different points of view.
Library
Find a method. Follow an example. Put it into practice.
Browse by subject
Choose a subject and find your next lesson.
Group patterns, follow sequences, compare items a few steps back, and link names, faces and words.
Work with percentages, fractions, ratios, and estimates using clear examples.
Explore measurements, transformations, coordinates and geometric relationships through diagrams and worked examples.
Count outcomes, update chances with new information, and compare decisions through worked examples.
Read distributions, question samples, and explore how groups and outliers change a conclusion.
Find patterns, test rules, and plan a solution step by step.
Follow changing rules, spot differences, and track the next step.
Read the treble staff, compare pitches and melodies, count beats, and build chords.
Find main ideas, use context clues, and follow connections between words.
English and guides
Common English mistakes, each with the rule that fixes it.
Free English passages, with questions and pace mode when you sign in.
Tests, methods, and the research behind them.
Continue reading
Lessons, guides, grammar rules and reading passages you saved or started.
Follow a program
A little structure for what you want to learn.
Ready to try?
Take a method into a practice, or discover something new.
Browser storage is unavailable. Changes last only on this page.
Turn a shape, don’t flip it
Follow an uneven shape through a turn and a reflection.
Goes with Mental rotation
Unfold in reverse order
Undo the last fold first and mirror each hole across the crease you open.
Goes with Paper folding puzzles
Unfold through overlapping layers
Reverse diagonal and straight folds while checking which paper layers actually lie beneath a punch.
Goes with Paper folding puzzles
Unfold unequal folds and edge cuts
Only paper beneath a punch or cut is affected. Open the folds in reverse. Cut pieces that meet across a crease join into one opening.
Goes with Paper folding puzzles
Cut a solid
Find the flat shape where a plane cuts through a cube or rectangular block.
Goes with Cross sections
Move the cutting plane
Keep the plane at the same angle and predict its section at the new position.
Goes with Cross sections
Undo a turn
Learn how to put a cube back after one turn, and after a short run of them.
Goes with Turn the cube
Turn a corner
Learn what one corner turn of the pyramid puzzle moves, and why the tips can wait.
Goes with Turn the pyramid
Compare angle openings
Look past direction and arm length to compare how far two lines open.
Goes with Angle ranking
Build to match two views
Read the front view for each column and the right view for each row, then fill in the rest. See how few and how many cubes can cast the same two views.
Goes with Build from two views
Fit pieces to the outline
Use edges and corners to place every piece without gaps or overlaps.
Goes with Construction from parts
Count cubes by layers
Split a supported stack into horizontal layers and count each cube once.
Goes with Cube counting
Count cubes by painted faces
Use corners, edges, face centers and the interior to count without overlap.
Goes with Painted faces
Count exposed faces in a stack
Check each cube’s neighbours, exclude the ground, and count cubes with the requested number of painted faces.
Goes with Exposed faces
Count overlapping removals
Count regions A and B, then subtract their shared cubes only once. The stack does not fall or settle.
Goes with Cube counting
Read cube notation
Learn the six face letters, and what a prime mark and a 2 change.
Goes with Cube moves
Track a strip through several turns
Follow three known stickers through a scrambled cube, updating their position and order after each turn.
Goes with Cube moves
Follow the mark
Keep each mark on its face and follow its direction as a net folds into a cube.
Goes with Cube Nets
Match a marked cube
Use opposite faces and the order of marks around a corner to match a net to its cube.
Goes with Cube Nets
Read top and front views
Separate a stack’s footprint from its height seen from the front.
Goes with Cube views
Choose the extra clue
Two solids match the given views. Choose one observation that tells them apart.
Goes with Cube views
Find a missing view
Use the top and front views of stacked cubes to find the view from the right.
Goes with Cube views
Follow a gear train
Use gear ratios, shared shafts and direction changes to solve simple and compound gear trains.
Goes with Gears
Trace a hidden shape
Follow connected edges to find a shape among crossing lines.
Goes with Hidden figures
Project the whole solid
Combine every depth layer before choosing an opening.
Goes with Match the opening
Turn before passing
Turn the solid to the named approach, then choose its opening.
Goes with Match the opening
Use your viewpoint
Face a landmark in your imagination, then locate another using your own left and right.
Goes with Perspective taking
Set a frame for an angled viewpoint
Find your own front and right when the facing arrow falls between compass directions.
Goes with Perspective taking
Follow a rolling cube
Track the faces one quarter-turn at a time, even when the path changes direction.
Goes with Rolling cubes
Fit pieces together
Rotate pieces and check whether they can fill an outline without gaps or overlap.
Goes with Shape assembly
Track your direction
Separate where you are from the direction you face, then follow each turn.
Goes with Spatial orientation
Jigsaw edges first
Sort a jigsaw by straight edges, then by color, then by the shape of the tabs.
Goes with Jigsaw puzzles
Big triangles first
Use the areas of the tangram pieces and the outline’s corners to choose a useful starting point.
Goes with Tangram
Edges first for the knight
Learn why a knight visits the squares near the edge early, and how one rule from 1823 picks the next move.
Goes with Knight's tour
Compare exactly N steps back
Check the required earlier position, not just whether a square looks familiar.
Goes with N-back test
Read the recall direction
Keep the digits in order, then follow the forward or backward instruction for that run.
Goes with Digit span
Trace the route
Read the numbers as one path across the board, then cut the path into short groups.
Goes with Chimp test
Remember patterns in groups
Notice small groups and keep track of where they sit.
Goes with Visual memory
Follow the tile order
Keep the order of the flashes, including a tile that appears again.
Goes with Sequence memory
Place, then retrieve
Give each object a fixed place, recall it with the list hidden, and return after a delay.
Goes with Memory Palace
Recall the connections
Turn a short explanation into connected ideas, then recall both the ideas and their relationships.
Goes with Recall Map
Sort digits, then letters
Remember the characters, then put the digits in increasing order before the letters in alphabetical order.
Goes with Letter–number sequencing
Keep the newest digits
Update a short ordered group as new digits arrive, then report that group when the stream stops.
Goes with Running span
Listen, then order the digits
Listen to the full sequence, then recall its digits in the selected order.
Goes with Spoken sequence
Remember a card and its place
Attach a rank and suit to each position instead of remembering only red or black.
Goes with Card memory
Anchor the Go stones
Use a center or edge as an anchor, then remember each nearby stone and its color.
Goes with Go-board memory
Hold letters between sums
Judge each equation, then add its letter to an ordered list without keeping the numerical answer.
Goes with Operation span
Connect and recall word pairs
Make a connection between two words, then retrieve the paired word from its cue.
Goes with Learn and recall
Link a name to a face
Study a face and its name together, then recognize the pair when the order changes.
Goes with Names and faces
Recognize a studied word
Decide whether each word belonged to the original list, not just whether it looks familiar.
Goes with Word recognition
Work back from 24
Start at 24 and ask what makes it. Try pairs of cards, then explore other groupings and fractions.
Goes with Make 24
Multiply in parts
Split a number into useful parts, then add the products.
Goes with Mental math practice
Doubling and halving
Halve one factor and double the other to reach an easier pair.
Goes with Mental math practice
Multiply by 11 in your head
Move the two digits apart, then put their sum in the middle.
Goes with Mental math practice
Start where two runs cross
Learn the kakuro sums that allow only one set of digits, and how a crossing run picks the digit for the cell two runs share.
Goes with Kakuro
Compare two quantities
Write a fraction and a percentage in the same form, then compare their values.
Goes with Compare quantities
Estimate with useful benchmarks
Use easy nearby numbers, then check which option is closest to the original value.
Goes with Estimate and compare
Set up the equation
Name the unknown, write the thing that does not change, and solve one equation.
Goes with Exam word problems
Multiplication shortcuts
Split one number, multiply the parts, then add. Plus three shortcuts worth knowing.
Goes with Fast multiplication
Multiply around a base
Keep signed differences from 100 or 1000, multiply the base part, then add or subtract the correction.
Goes with Fast multiplication
Make equivalent fractions
Change the number of equal parts without changing the amount.
Goes with Fractions and decimals
Convert fractions and decimals
Use decimal place value and division to keep the same exact value.
Goes with Fractions and decimals
Use factors and remainders
See what is left after division, then find the numbers that divide exactly.
Goes with Number properties
Read a number sequence from its gaps
Write the gaps between the terms, then find the rule in them.
Goes with Number sequence puzzles
Find a percentage in your head
Find 10% first, then build the percentage you need from it.
Goes with Percentages
Reverse a percentage change
Use the original value as 100% to work backwards from an increase or decrease.
Goes with Percentages
Divide in a ratio
Turn a ratio into equal parts, then find the value of one part.
Goes with Ratios & averages
Weight the average
Use group counts to combine averages without losing the totals.
Goes with Ratios & averages
Combine work rates
Add the work completed per hour to find a shared finishing time.
Goes with Speed & work rates
Learn the tables
Many times-table facts repeat. Learn the patterns, then practice six useful facts.
Goes with Times tables
Undo the operation
Find the missing number by doing the opposite operation to the result.
Goes with Find the missing number
Read the places
A base is only place values. Read binary by adding the places, and read hex four binary digits at a time.
Goes with Number bases
Keep the pace
Skip what stalls you, learn the sign rules, anchor a decimal, and know the conversions by heart.
Goes with Quant interview arithmetic
Squares near round numbers
Square a number by working from the round number next to it, and estimate a root from the two nearest squares.
Goes with Squares and roots
Convert with anchors
Keep two known pairs for each set of units, and use them to check every answer.
Goes with Unit conversion
Preserve the solution set
Undo operations, track signs and check restrictions when an equation becomes an inequality.
Goes with Algebra: Equations & inequalities
Simplify without losing restrictions
Choose the right identity, cancel complete factors and keep the original domain.
Goes with Algebraic expressions
Prove a comparison or find a counterexample
Use real-solution constraints to prove a bound, then test whether equality or a counterexample changes the comparison.
Goes with Compare quantities: Variables
Read functions and graphs
Connect inputs and outputs, combine graphs and compositions, and keep every restriction when finding a domain or inverse.
Goes with Functions & graphs
Reconstruct from number clues
Use digit totals to find the digits, position weights to order them, and nearby powers to fill missing places.
Goes with Number challenges
Use base and height
Measure perpendicular height to compare triangle areas without relying on shape.
Goes with Area
Count area and perimeter
Count filled unit squares for area and exposed edges for perimeter.
Goes with Area builder
Scale similar triangles
Distinguish the scale factor for side lengths from the scale factor for area.
Goes with Triangles & shapes
Read rise over run
Find a line’s slope from coordinate differences and keep the sign.
Goes with Slopes
Scale length, area and volume
Predict how a solid changes when every length is multiplied by the same factor.
Goes with 3D solids
Connect the circle centers
Turn circle tangency into distances between centers, then use a right triangle to find a radius.
Goes with Circle geometry
Follow circle tangency
Build equations from touching circles, then use the stated region to choose the circle that fits.
Goes with Circle geometry
Choose a triangle ratio
Identify opposite, adjacent and hypotenuse before using sine, cosine or tangent.
Goes with Trigonometry
Reason around the unit circle
Use coordinates, signs and identities to compare trigonometric expressions beyond a right triangle.
Goes with Trigonometry
Track a transformation
Follow a rotation around a stated center and check what stays unchanged.
Goes with Transformations
Use coordinate differences
Separate distance from midpoint by using differences for length and averages for location.
Goes with Coordinate geometry
Sum polygon angles
Connect triangle totals to interior angles and one complete exterior turn.
Goes with Polygons
Measure a circle sector
Use the central angle to take the same fraction of circumference or circle area.
Goes with Circle measures
Classify by relationships
Find the relationship that survives changes of shape, size and direction, then check a new member.
Goes with Shape classification
Count complete outlines
Include the larger triangles and rectangles formed by joining small regions, and avoid counting an outline twice.
Goes with Figure counting
Trace every edge once
Check that the drawing is connected, then count the dots with an odd number of lines.
Goes with One-stroke figures
Unfold a surface route
Unfold the permitted faces, check that the straight segment stays inside them, then use the perpendicular offsets.
Goes with Shortest surface routes
Count equally likely outcomes
Keep outcomes equally likely before dividing favorable cases by all cases.
Goes with Reason with probability
Decide whether order matters
Separate an ordered arrangement from a group that contains the same items.
Goes with Counting combinations
Condition on information
Use natural counts to choose the right conditional denominator.
Goes with Conditional probability
Update after a draw
Track the changing counts when an object is not replaced.
Goes with Conditional probability
Follow the host rule
Explain why the same empty door can lead to different odds.
Goes with Switch or stay
Compare expected rewards
Subtract the cost and weight every net outcome by its probability.
Goes with Risk and reward
Count winning sequences
Use complements and disjoint cases to compare multi-roll targets.
Goes with Risk and reward
Count birthday collisions
Use the chance of no match to understand why shared birthdays arrive surprisingly soon.
Goes with Birthday collisions
Why patterns overlap
Separate a pattern’s chance in one window from the time you wait for it.
Goes with Pattern races
Race patterns with a biased coin
Track both patterns on one sequence and count unfinished races.
Goes with Pattern races
Compare dice in pairs
Discover why a die can beat one rival and lose to another.
Goes with Nontransitive dice
Choose a die after two rolls
Build the distribution of sums before choosing a counter to an opponent.
Goes with Nontransitive dice
Bound the missing overlap
Use two valid extreme arrangements to separate what is guaranteed from what is merely possible.
Goes with Chance bounds
Notice the shared gate
Separate independent components from routes that share a bottleneck, then compare two upgrades.
Goes with Probability paths
Compare stopping values
Compare the guaranteed bank with the average final score. The best single card does not determine the average.
Goes with Stop or continue
Check equal outcome chances
Balance sequences within each equal-probability group. Repeating the same independent procedure preserves that balance.
Goes with Make chance fair
Read data before calculating
Read labels and units, then combine two series to find totals, differences, shares and percentage changes.
Goes with Data interpretation
Choose a useful center
Compare the mean and median, then explain what an unusually large value changes.
Goes with Distributions
Compare the spread
See why equal means can hide different variability and what standard deviation summarizes.
Goes with Distributions
Question the sample
Distinguish chance differences between random samples from systematic selection bias.
Goes with Sampling
Weight the groups
Combine totals correctly and see how changing group proportions can reverse a comparison.
Goes with When averages mislead
Inspect the scatter
Investigate influential observations while keeping correlation separate from causation.
Goes with Correlation detective
Shuffle to test a difference
Use random assignments to understand how surprising a difference is.
Goes with Signal or chance?
Read confidence as coverage
Understand what confidence intervals cover, what more data changes and what bias leaves behind.
Goes with Confidence intervals
Separate change from selection
Distinguish a real effect from the change expected after selecting unusual first results.
Goes with The comeback illusion
Check what the data prove
Use ranges and totals together to distinguish a proof from a possible example.
Goes with Data conclusions
Match labels before shapes
Preserve labels and units, then use each group’s own total when converting counts to percentages.
Goes with Match the chart
Isolate the changing condition
Change the target condition and hold the other listed condition constant.
Goes with Choose the comparison
Compare forecasts with outcomes
Compare absolute percentage-point gaps from the observed frequency. This completed batch does not establish future accuracy.
Goes with Check the forecast
Follow the whole story
Connect the beginning, change, and ending in a short summary.
Goes with Reading skills
Ask why the writer included an example
Explain what an example helps the writer show.
Goes with Reading skills
Find the point of a comparison
Explain how two options differ and why the writer compares them.
Goes with Reading skills
Follow a recommendation and its reasons
Identify what the writer recommends and the reasons that support it.
Goes with Reading skills
Keep benefits and limits in a summary
Summarize a useful idea without dropping the writer's qualifications.
Goes with Reading skills
Keep times and events straight
Match each time or sequence word to the event it describes.
Goes with Reading skills
Notice conditions and requirements
Identify what a reader must do or provide for an offer or instruction to apply.
Goes with Reading skills
Revise an idea when new facts appear
Explain how later information changes the most likely meaning of an action.
Goes with Reading skills
Spot a definition in the sentence
Recognize when the writer explains an unfamiliar word directly.
Goes with Reading skills
Track a change in attitude
Use words and actions to compare how someone feels at two points in a text.
Goes with Reading skills
Use a word part as a clue
Use a familiar word part to suggest a meaning, then check the sentence.
Goes with Reading skills
Work out what a short phrase refers to
Connect a short phrase with the specific situation it describes.
Goes with Reading skills
Choose the meaning that fits
Choose the meaning that fits this sentence, not another use of the word.
Goes with Reading comprehension
Connect a cause to its effect
Find the stated reason for an event or result.
Goes with Reading comprehension
Do not infer too much
Choose what the clues support, without assuming more.
Goes with Reading comprehension
Locate the right detail
Find the sentence that answers the exact question.
Goes with Reading comprehension
Recognize the same idea in new words
Match an answer to the text even when it uses different words.
Goes with Reading comprehension
Use clues, not guesses
Use clues to understand something the writer does not say directly.
Goes with Reading comprehension
Use context clues
Use an example, explanation, or contrast to work out a likely meaning.
Goes with Reading comprehension
Change one letter
Turn one word into another one letter at a time, and find the short way through.
Goes with Word ladder
Start from a corner
Find where a letter path starts: the ends are on the corners or the middle, and a corner has only two neighbors.
Goes with Letter path
Follow a word in one direction
Find the first two letters, then keep the same direction through the whole word.
Goes with Word search
Carry the word relationship
Put the first word pair into a sentence, then keep that relationship and direction in the second pair.
Goes with Verbal analogies
Name the shared property
Describe what four words share, then check why the remaining word falls outside that group.
Goes with Word classification
Match the definition
Use a word’s meaning and grammatical role to choose a precise definition.
Goes with Word meaning
Split a word into parts
Read a long word part by part, then put the meanings back together.
Goes with Word parts
Find the only place
Learn the two steps that start most sudoku puzzles. Scanning by symbol beats scanning by square.
Goes with Sudoku
Move the smaller tower first
Learn how a tower of disks comes apart, and which peg the first move goes to.
Goes with Tower of Hanoi
Clear the lane
Learn how to unblock a sliding puzzle. Name what blocks the target, then name what blocks the blocker.
Goes with Sliding puzzles
One queen in each row
Learn why every row holds exactly one queen, how to work down the board, and why 92 answers are 12 shapes.
Goes with Eight queens puzzle
Clear the board in packages
Plan three jumps that clear a group of pegs and bring the first peg home, and see why an isolated peg cannot jump immediately.
Goes with Peg solitaire
Look at the checks first
Learn the order to search a position in: every check, then every capture, then every threat, and why counting the king's escape squares is what finds a mate in two.
Goes with Checkmate puzzles
Think in states
Learn to write a river crossing position down, and why the move that matters is the one that comes back.
Goes with River crossing puzzles
Start with the small cages
Learn which square of a calcudoku tray you can fill first, and why the rest follows from it.
Goes with Calcudoku
Read the rows, then the columns
Find the rule that runs across a grid before you look at the answer options.
Goes with Missing pattern puzzles
Combine the marks
Distinguish four ways of combining two patterns, especially when their marks overlap.
Goes with Missing pattern puzzles
Turn before combining
Track a quarter-turn of the first pattern, then apply the combining rule.
Goes with Missing pattern puzzles
Find the rule in a figure series
Check one property at a time until you find the one that moves.
Goes with Figure series
Cross off impossible pairs
Use exclusions and shared clues to connect people with their items.
Goes with Logic grids
Assume one person, then check
Start with the person everyone talks about, and follow the statements until something breaks.
Goes with Truth tellers
Two lines or none
Learn the two counts behind Make one loop, the sides around each number and the lines at each dot, and why a loop that closes early is wrong.
Goes with Make one loop
Find the overlap
Learn the first sure marks in a nonogram line: the squares a run always covers, and the squares no run can reach.
Goes with Nonograms
Combine the balances
Replace equal groups and cancel matching shapes to find what balances the target pan.
Goes with Figure weights
Use two complete lines
Check an arithmetic relationship against two complete rows before filling the missing number.
Goes with Number matrices
Find the rule from the examples
Test one rule on every solved figure before you answer, so a rule that fits only the first figure does not catch you out.
Goes with Rule finder
Check the fast answer
Notice the answer that arrives at once, then test it against the words of the question.
Goes with Cognitive reflection
Carry the same change
Name what changes in the first pair, then apply it to the second pair.
Goes with Figure analogy puzzles
Follow only certain links
Trace what the premises guarantee without adding assumptions about the groups.
Goes with Logical deduction
Draw the statements as a Venn diagram
Turn All, No and Some into a Venn diagram, then read the conclusion off the picture.
Goes with Logical deduction
Place the fixed clues first
Anchor one position, then use before and next-to clues to fill the row.
Goes with Ordering logic puzzles
Keep both sides equal
Find one shape's value, then use it in the next balance.
Goes with Shape equation puzzles
Balance the magic square
Use complete lines, missing totals and unused numbers to finish a magic square.
Goes with Magic squares
Find the argument gap
Separate a claim from its support, then test the missing link.
Goes with Argument analysis
Seat around a circle
Start from the top anchor and count clockwise seat moves using the stated clues.
Goes with Circular seating
Decode the letter rule
Check a letter rule against every example, including where the alphabet wraps around.
Goes with Coding-decoding
Test each statement
Decide whether the information determines an answer, testing each statement on its own first.
Goes with Data sufficiency
Trace family links
Follow the stated parent links, then read the relationship in the direction the question asks.
Goes with Family relations
Compare the pair rule
Find the pair whose transformation differs, even when the starting figures look unlike one another.
Goes with Figural odd one out
Read set regions
Match and, or, but not and not to selected regions, including the space outside the circles.
Goes with Set operations
Follow each command
Use the visible rules in order, carrying each change into the next step.
Goes with Symbol commands
Wait for the signal
Respond to the visible signal, not to a guess about when it should appear.
Goes with Reaction time
Name the ink
Choose the color of the letters, even when the word names another color.
Goes with Stroop test
Read the symbol key
Match the current symbol to this round’s key and enter its paired digit.
Goes with Symbol coding practice
Check both features
Find the named shape by checking its color and outline together.
Goes with Visual search
Scan, then compute
Read the instruction twice, mark the numbers it names, then work it out.
Goes with Grid arithmetic
See the difference
Compare in pairs from the left, and stop at the first one that does not match.
Goes with Perceptual speed practice
Watch for the target
Check each item against a visible rule and respond only when it matches.
Goes with Sustained attention
Keep a steady pace
Add digits that sit next to each other, keep only the last digit, and hold one pace across the row.
Goes with Continuous addition
Hold the rule
Say the rule before the card, and say the new one again the moment the banner changes.
Goes with Task switching
Follow the moving group
Keep track of the original balls as they move, rather than remembering their starting places.
Goes with Multiple-object tracking
Scan a number grid
Find numbers in ascending order while keeping your place in an unchanged grid.
Goes with Schulte table
Focus on the middle
Find the center arrow and answer its direction, even when the surrounding arrows disagree.
Goes with Flanker
See a small group
Recognize a few dots as a group, then choose their number after they disappear.
Goes with Quick dot count
Follow the shape, not the side
Follow the shape’s response, even on the opposite side.
Goes with Simon task
Notice what changed
Remember each figure and its direction, then find the one change in the same positions.
Goes with Spot the change
Compare two pitches
Separate high and low pitch from loud and quiet volume.
Goes with Pitch direction
Read the treble staff
Count the lines and spaces from the bottom, and find middle C.
Goes with Note reading
Build a triad
Use the root, third and fifth to build a major or minor chord.
Goes with Chord builder
Spatial reasoning tests: the six question types
What a spatial reasoning test asks, from the six question types of the Dental Admission Test.
Goes with Match the opening
Paper folding test: how to unfold the holes
A paper folding test folds a square, punches it and asks where the holes land.
Goes with Paper folding puzzles
Cross section test: how to find the shape of a cut
A cross section test shows a solid with a flat cut through it and asks for the shape of the cut.
Goes with Cross sections
Mental rotation test: turned or mirrored
A mental rotation test asks whether two block figures are the same shape turned, or mirror images.
Goes with Mental rotation
Reading comprehension: find the line that answers it
Find the evidence for main idea, inference, detail and word meaning questions.
Goes with Reading skills
Reading speed: what words per minute means
What words per minute measures, why understanding comes first and what research says about speed reading.
Goes with Reading comprehension
Reaction time test: what the timing means
A reaction time test measures the milliseconds between a signal and your press.
Goes with Reaction time
Stroop test: what the slowdown means
A Stroop test times how long you take to name an ink color when the word says a different color.
Goes with Stroop test
N-back test: what one score can tell you
An n-back test asks you to mark the moment an item repeats one from N steps earlier.
Goes with N-back test
Digit span test: what your span means
A digit span test shows digits one at a time and asks you to type them back.
Goes with Digit span
Visual memory test: what the score means
A memory test on a website measures one kind of memory at a time.
Goes with Visual memory
Why QuickenBrain does not give you an IQ number
Why an online quiz cannot give an IQ score and how validated tests use reference samples.
Goes with Missing pattern puzzles
Quant interview math test: how to prepare
Trading firms often start hiring with an online assessment that candidates say includes timed arithmetic.
Goes with Quant interview arithmetic
Mental multiplication: six useful methods
Six ways to multiply in your head.
Goes with Fast multiplication
Chimp Test: what the longest-board score means
How the Chimp Test works and what its longest-board score records.
Goes with Chimp test
Tower of Hanoi: the method behind the fewest moves
The Tower of Hanoi rules and a three-disk solution.
Goes with Tower of Hanoi
Go/no-go test: how to read the result
How go/no-go and stop-signal tasks differ and what false presses and missed signals mean.
Goes with Go/no-go test
Cognitive reflection: how to check the first answer
What cognitive-reflection questions ask and where the original three-item test came from.
Goes with Cognitive reflection
Verb tense
Match the verb to the time.
Subject-verb agreement
The verb agrees with the subject in number.
Since, for, ago
Use "since" with a starting point ("since 2019") and "for" with a length of time ("for three years").
Present perfect vs. simple past
If the sentence names a finished time (yesterday, last year, in 2023), use the simple past.
Used to
"Used to" + base verb describes a past habit that is over.
Gerund or infinitive
Some verbs take -ing and some take to + verb.
If-sentences: real vs. unreal
For a real future, use the present after if and "will" in the other half.
Modal verbs: can, could, must, have to
A modal verb takes the base verb, with no "to" and no -s: "She can swim", never "can to swim" or "cans swim".
Passive voice
The passive is be + past participle.
Phrasal verbs: turn it off
With separable phrasal verbs, a pronoun must go between the verb and the particle.
Articles
Use "a" before a consonant sound and "an" before a vowel sound.
Plurals
Most nouns add -s.
Countable vs. uncountable
Some nouns have no plural form: advice, information, news, furniture, luggage, feedback.
Much vs. many
Use "many" with countable nouns and "much" with uncountable nouns: "many emails", "much time".
A few vs. a little
"A few" goes with countable nouns and "a little" with uncountable ones: "a few emails", "a little time".
This, that, these, those
Match the word to distance and number.
Some vs. any
Use "some" in positive sentences and "any" in negatives.
Other vs. another
"Another" is one word and goes with a singular noun: "another cup".
Of vs. possessive -s
For people, use 's: "my brother's book", not "the book of my brother".
All, every, each
"Every" and "each" take a singular noun and a singular verb: "every employee gets a badge".
Comparatives and superlatives
Use one comparison marker, not two.
So vs. such
"So" goes before an adjective or adverb alone: "so expensive", "so quickly".
Too vs. very
"Very" makes an adjective stronger and stays neutral: "very tall".
Prepositions
Prepositions rarely translate word-for-word, so learn them with the phrase.
Word choice
Pick the word that fits the meaning and the usual pairing (collocation).
Make vs. do
Use "do" for work and general tasks: "do the dishes", "do your homework".
Say vs. tell
"Tell" needs a person right after it: "tell me", "tell the team".
Affect vs. effect
"Affect" is usually the verb, meaning one thing changes another: "the delay affects the launch".
Lie vs. lay
"Lie" means to rest yourself, and takes no object: "I lie down".
Borrow vs. lend
"Borrow" means to take something for a while: "I borrowed his charger".
Word order
English is subject–verb–object.
Clarity
One idea per sentence.
Forming questions
Most questions need do, does, or did before the subject.
Reported speech
With a past reporting verb, you usually shift the tense back.
Despite vs. although
"Despite" is followed by a noun or an -ing form: "despite the rain".
During vs. while
"During" is followed by a noun: "during the meeting".
Punctuation
End sentences with a period.
Spelling
Watch commonly confused spellings and the i-before-e pattern.
Capitalization
Capitalize the first word of a sentence, the pronoun "I", proper nouns, and days, months, and languages.
Possessive apostrophes
Add 's to show who owns something: "the manager's office".
Its vs. it's
"It's" with the apostrophe always means "it is" or "it has".