Practice problems
Work through mathematics problems by topic and difficulty, with hints, worked solutions and a whiteboard beside every question.
Problems
- Chain Rule with Differential Equation
- Radical Limit with Unknown Constants
- Functional Equation with two variables I
- Cubic Polynomial Limit
- Functional Equation with two variables II
- Integral Function Limit via Leibniz Rule
- Limit of Logarithmic Hyperbolic Cosine
- Antiderivative of Reciprocal Sine Addition
- Limit of Rational Power
- Polynomial Limit I
- Continuity of a Piecewise Function
- Derivatives of an Inverse Function
- Sum over Sine-Power Antiderivatives
- Limit of a Logarithmic Ratio
- Differentiability of a Piecewise Function
- Constants from an Exponential–Cosine Limit
- Limit of Arcsine minus Arctangent
- Polynomial from a Limit at Infinity
- Limit of a Product Raised to a Vanishing Power
- Equal Values of a Composed Quadratic
- Derivative from an Integral Equation
- Constants from a Sine-of-Sine Limit
- Telescoping Sum over a Quartic
- Polynomial from Two Limits
- Powers of Complex Roots
- Linear Piece Matched to a Power Function
- Solving a Functional Composition
- Integral of a Cotangent Ratio
- Integral Involving an Inverse Function
- Limit of an Integral over x³
- Slope of a Parametric Curve
- Maximum plus Minimum of a Rational Function
- A Limit as a Derivative
- Sum from an Integral Identity
- A First-Order Differential Equation
- Limit with the Inverse of ln(tan x)
- Limit of the Log of an Exponential Average
- Difference of Two Definite Integrals
- Derivative of an Inverse at Zero
- Area from a Functional Equation
- Equal Ratios with Logarithms
- Riemann Sum of a Rational Expression
- Remainder by a Product of Divisors
- Constants from a Trigonometric–Log Limit
- Limit of an Integral of an Absolute Value
- Polynomial from Limits at Zero and One
- Limit of n² Times a Root Difference
- Riemann Sum with a Squared Factor
- Integrals over Unit Intervals
- Inverse of a Piecewise Function
- Constants from a Sequence Limit
- Integrals of a Function and Its Inverse
- Difference of Roots of Arithmetic Sums
- An Integral Identity with Unknown Constants
- Solving x f(x) through an Integral
- A Polynomial Defined by Its Own Derivative
- Extremes of a Linear Expression on an Ellipse
- A Linear Differential Equation
- Choosing Continuous Branches
- Integral from Two Functional Equations
- Area Enclosed by Four Curves
- Riemann Sum Giving an Arctangent
- Recovering f from a Double Integral
- Counting Functions with Logarithmic Conditions
- An Exponential Tangent to a Line
- Derivative of a Self-Composition
- Integral of a Periodic Function
- Limit of an Integral with a Logarithm
- Root Gap from a Limit Ratio
- An Integral Equation with a Sine Kernel
- Limit of an n-th Root of a Product
- n-th Root of the Central Binomial Coefficient
- Limit of a Sum of Floor Functions
- Limit of a Ratio of Power Sums
- Derivative of an Inverse with an Integral
- A Periodic Trigonometric Sum
- Three Diverging Terms That Balance
- Volume by Shells about x = 5
- A Polynomial from a Differential Identity
- Derivative of a Ratio of Geometric Sums
- An Arithmetico-Geometric Series
- Integral of a Log of Root Sums
- Area Between Two Parabolas
- An Integral Identity with a Scaled Argument
- A Circle Tangent Externally
- A Polynomial from Derivative Limits
- Inverse Derivative from an Integral Equation
- Limit from a Weighted Sum of Squares
- The King Property with x sin x
- A Trigonometric 0/0 Limit
- Integral of an Absolute Value
- A Reciprocal Pair of Log Integrals
- A Symmetric Integral with ln 6
- Integral with sin x + cos x over sin 2x
- Remainder by a Squared Linear Factor
- n-th Root of the Trace of a Matrix Power
- Harmonic Sum from n+1 to 6n
- An Integral of the Inverse of eˣ + x − 1
- A Chain of Polynomial Remainders
- An Alternating Sine Sum
- A Riemann Sum with a Parameter
- Minimum of a Quadratic Form on a Plane
- Scaling an Integral of f(x)/x
- Two Integrals That Nearly Cancel
- Cube Roots of Unity
- When an Integral Function Has One Extremum
- A Multiplicative Functional Equation
- A Linear Combination of Two Series
- First and Last Three Terms
- Logarithmic Derivative of an Even Function
- Length of a Parametric Circle
- Differentiating an Integral with x Inside
- Constants Hidden in an Integral Identity
- An Integral Equation with an Exponential Kernel
- An Asymptote Crossing Two Log Curves
- A Matrix That Repeats Its Powers
- A Double-Factorial Limit
- Absolute Values in an Arithmetic Sum
- A Log Curve Through a Circle's Diameter
- Derivative Ratio of a Product Polynomial
- Maximising f(2) under a Constraint
- Constants from a Limit at x = 1
- Local Extremes of an Integral Function
- A Line Crossing Two Shifted Exponentials
- Undoing Derivatives and Integrals
- Limit from a Sum of Matrix Powers
- A Cube-Root Limit and a Double Root
- A Polynomial from a Reciprocal Limit
- A Recurrence Whose Series Converges
- A Cubic Recurrence in Closed Form
- A Polynomial from Two Limits
- x over 1 + sin x on [0, π]
- n-th Root of (2n)! over (2n)²ⁿ
- Sum of Fifth Powers over n⁶
- Sum of Solutions of cos(π sin x) = −1/2
- Composing a Logistic Integral
- Cube Root of a Shifted Cubic
- A Dirichlet-Type Integral
- Integrating g′ of an Inverse Tangent
- e^x sin²x on [0, π]
- Closest Point on a Semicubical Parabola
- A Quadratic from Reciprocal Limits
- An Integral Equation with Two Unknown Integrals
- Inverse Derivative of x³ + 3x
- An Equation at Its Maximum
- A Basic Radical Limit
- A Polynomial Divided by x + 1
- Limit of 1 − cos²(3x)
- Powers of a Shear Matrix
- The 100th Power of a Matrix
- Area Under a Quarter Circle
- Minimum by AM–GM
- A Quotient Derivative at √π
- Constants from a Quadratic Limit
- Substitution then Parts
- Dividing Complex Numbers
- An Odd Integrand
- A Sum That Telescopes to a Cube
- Difference of Two Square Roots
- Three Equal Logarithms
- A Telescoping Sum of Square Roots
- Integral of a Maximum
- sin 5x over sin x on [0, 2π]
- Minimum as a Shortest Path
- An Exact Derivative in Disguise
- A Radical Equation in Disguise
- Divisible by x² − 1
- Composing a Function Five Times
- Splitting into Two Geometric Series
- A Cubic Through Four Points
- A Quadratic That Touches a Line
- Six Conditions on a Degree-5 Polynomial
- A Telescoping Product
- Powers from x + 1/x
- A Finite Limit at Infinity
- A Cubic from Two Root Limits
- A One-Sided Limit with the Floor Function
- Discontinuous at Exactly One Point
- Cancelling a Square Root at Infinity
- When a Floor-Function Limit Exists
- An Odd Cubic from Two Limits
- Making a Log Ratio Continuous
- Perpendicular Tangents to a Parabola
- Growth of an Inverse Function
- Shortest Distance to ln x
- When f and Its Inverse Meet
- Matching Orders in a Limit
- The Largest Base Wins
- An Arcsine Hidden in a Trig Integral
- Integrals of a Shifted Function
- Nested Absolute Values
- An Integral Built from an Inverse
- nth Roots of 2 and n!
- Area of a Looped Curve
- Counting Real Roots
- Tuning Two Square Roots
- Area Between a Cubic and Its Tangent
- A Separable Integral Equation
- Derivative of an Intersection Point
- A Line Touching a Parabola
- A Functional Equation for f′
- A Riemann Sum for x ln x
- Perpendicular Rays to Two Intersection Points
- An Averaged Integral Near Zero
- Two Coupled Sequences
- Sum of Cubes of a Geometric Series
- A Parametric Slope at Infinity
- A System of Product-Rule Equations
- Sum of Tangent-Power Integrals
- A Recurrence with a Hidden Invariant
- Recovering a Sequence from Its Sums
- A Constrained Minimum
- Area Cut by a Tangent from the Origin
- A Cubic and a Quadratic from Limits
- An Identity with ∫ t f′(t) dt
- Area Between a Hyperbola and Two Rays
- A Three-Term Product Recurrence
- Remainder by (x − 1)³
- Minimising AP² + BP² on a Hyperbola
- Ratio of Two Root Sums
- Integration by Parts with f·f′
- A Function Symmetric About x = a
- A Convolution Integral
- Using f′(0) > 0
- Parts with an Integral Function
- A Sequence Defined by Factorial Sums
- Area Between a Parabola and a Witch
- Largest Triangle with a Point on √(3x)
- Two Intersections with a Kinked Line
- A Broken Path Through Both Axes
- Area Between Two Root Curves
- A Substitution with cos³
- Differentiating an Integral with Variable Limit
- Maximum times Minimum of x e^(−x²)
- Arcsine Plus Its Partner
- Completing the Square under a Root
- A Telescoping Tangent Series
- cos x to the Power 1/x²
- n(a^(1/n) − 1)
- n(e^(a/n) − e^(b/n))
- (1/n + e^(1/n))ⁿ
- (√(x²+1) + x)^(1/x) near 0
- A Product of Two Logarithms
- x²(a^(1/x) − a^(1/(x+1)))
- n²(1 − cos(π/n))
- tan 2x · tan(π/4 − x)
- Sum of Squares over a Quadratic
- A Ratio of Root Differences
- nth Root of 1 + 2ⁿ + 3ⁿ
- √(n+1) − √n
- A Ratio of Quadratics
- A Large Constant Does Not Matter
- A Telescoping Product of 1 − 1/k²
- aⁿ/(2 + aⁿ) for |a| > 1
- A Quartic Root minus a Product
- √(n² + 4n + 5) − (n − 1)
- √n(√(n+2) − √(n+1))
- Sum minus Half of n²
- a²/n³ times a Sum of Squares
- Σ 1/(k(k+1))
- Σ 1/((2k−1)(2k+1))
- Three Consecutive Factors
- 1 + 2q + 3q² + ⋯
- Σ (2k − 1)/2ᵏ
- (5·3ⁿ + 3(−2)ⁿ)/3ⁿ
- Two Bases, the Larger Wins
- A Function Defined by a Limit I
- A Function Defined by a Series
- Σ k/(k + 1)!
- A Function Defined by a Limit II
- A Function Defined by a Series II
- x²[((x+1)/(x−1))^(1/x) − 1]
- (1 + x sin x)ˣ near 0
- (cos x)^(sin x) near 0
- Constants from a Root-Difference Limit
- A Limit at Minus Infinity
- x − x² ln(1 + 1/x)
- A 1^∞ Limit with an Integral
- √(1 + tan x) − √(1 + sin x)
- Σ n/(n² + k²)
- Constants from sin x/(eˣ − a)
- An Equivalent Infinitesimal
- A Convolution over eˣ² − cos x
- Continuity with arcsin and exp
- Continuity with (cos x)^(2/(x arctan x))
- Continuity with f′(0) = b
- Continuity with x − arctan x
- (cot x)^(sin 3x)
- ((2/π) arctan x)^(x²)
- (x + √(1 + x²))^(1/x) at Infinity
- Both Ends of ∞ with arctan
- An Exponentially Weighted Average
- An Infinite Product of Cosines
- Roots of Consecutive Triangular Numbers
- Taylor Expansions to Order 4
- An Integral over a Root Difference
- (1/x)(1/sin x − 1/tan x)
- (1 − x) tan(πx/2)
- 1/x − 1/ln(1 + x)
- ∛(x³ + 2x² + 1) − x
- (n sin(1/n))^(n²)
- Difference of Two Cube Roots
- The 8th Derivative at Zero
- A Parametric Second Derivative
- An Odd Part and a Semicircle
- Bounded Oscillations at Infinity
- sin(sin x) − x over x³
- ln(x/sin x) over x²
- A Functional Equation for a Polynomial
- A Quartic Divided by x² − 1
- Ratios of One-Sided Derivatives
- A Logarithmic Derivative of a Long Product
- An Increasing Function Meets Its Inverse
- A Cubic Whose Inverse Is Flat
- Polynomials with Natural Coefficients
- A Derivative Known Only Up to Sign
- Parametric Sums at t = 0
- A Derivative from the Definition
- The Chain Rule Through an Inverse
- x − 3/√x = 10
- A Variable in the Exponent
- Three Values of a Quadratic
- A Quartic Symmetric About x = 1
- A Separable Differential Equation
- Sum over Difference of Sines
- Counting Bijections
- A Self-Referential Polynomial
- A Cubic Bounded by a Line
- Powers of sin 18°
- 9²¹⁶ mod 11
- 6ⁿ + 8ⁿ mod 49
- Counting Injections with a Condition
- Three Exponentials Equal to 540
- A Minimum by AM–GM
- A Recursively Defined Function
- Two Exponential Equations
- Square Roots Near x = 1
- Growth of ∫e^(t²)
- Squared Integral over an Integral
- Σ i/(n² + i²)
- x⁴√(1 − x²) on [0, 1]
- A Classic: ln(1 + tan x)
- A Riemann Sum for x ln(1 + x)
- Recovering a Limit of f
- x − arcsin x over x² sin 3x
- ln(1 + 2ˣ)·ln(1 + 3/x)
- xˣ − x near x = 1
- |x² − 2x| on [−1, 2]
- Differentiating an Integral Identity
- An Integral to sin 2x over 1 − cos x
- An Even Part and an Odd Part
- A Logarithmic Arcsine Integral
- x²√(4x − x²) on [0, 4]
- 1/(2 + cos 2x) over [0, π]
- Integrating by Parts Twice
- 1 − cos t² up to √x
- A Nested Integral's Derivative
- A Substitution Inside a Limit
- An Integral Equation for f(e)
- t ln t from cos x to 1
- A Series from Tangent Integrals
- x⁷e^(−x²) on [0, ∞)
- A Function Containing Its Own Integral
- A Squeeze Between Two Sums
- (1 + 1/x − 1/x²)ˣ
- Logarithms of Binomial Coefficients
- A Ratio of Two Convolutions
- nth Root of an Exponential Sum
- Recovering f from a Limit
- A Second Difference of Square Roots
- The Inverse of an Integral Function
- An Odd Part Vanishes
- Four Limits of a Quartic
- A Branching Integer Recurrence
- An Exponential Meeting Its Inverse
- A Shifted Polynomial Glued Together
- √(aₙ² + n) − aₙ
- A Solid with Square Cross-Sections
- A Cubic Composed with eˣ
- A Tangent, a Curve and the y-Axis
- Doubling or Dividing by Three
- Circles Meeting a Fixed Line
- Combining Two Sums
- A Horizontal Line Across Two Exponentials
- A Polynomial from Two Limits III
- Sum of the Roots of a Sine Equation
- Two Curves Meeting Twice
- Equal Areas of |f′|
- A Product of Two Integrals
- A Multiplicative Function's Derivative
- A Quadratic from a Squared Limit
- A Parametric Second Derivative II
- A Quartic Through Five Fractions
- Composing Permutations 2015 Times
- The Largest Inscribed Trapezoid
- A Cubic Symmetric About a Point
- When a Nested Intersection Empties
- A Closed Path Touching Four Lines
- An Exponential and a Log Curve
- Local Maxima of an Integral Function
- Filling a Container
- A Bell Curve Minus a Line
- Making an Integral Function Increasing
- Two Geometric Series That Converge
- Reading One-Sided Limits from a Graph I
- Reading One-Sided Limits from a Graph II
- Reading One-Sided Limits from a Graph III
- Extrema of f(x) − x² from a Graph
- Three Limits from a Graph
- Composing a Function and Its Inverse
- A Line Missing a Composite Step Function
- A Centroid Sitting on the Focus
- A Focal Chord and Its Shadows
- A Perimeter That Knows the Major Axis
- Two Ellipses Sharing a Focus
- Shortest Path to a Hyperbola
- An Angle at the Focus
- Two Squares Standing on a Parabola
- A Midpoint Equidistant from Three Points
- A Tangent Line Through a Focus
- A Tangent Sloping at 30°
- The Largest Kite on an Ellipse
- A Tangent, Two Perpendiculars and a Trapezoid
- Two Tangents Meeting at a Known Angle
- Two Ellipses Inside a Right Triangle
- An Ellipse Built on a Tangent Line
- Two Points Roaming on Two Circles
- Nine Points Around a Triangle
- Two Series That Skip Terms
- Where a Sequence and a Series Agree
- A Series Built on a Recurrence
- From Partial Sums to a Series
- A Point Spiralling Inward
- Squares Climbing a Hypotenuse
- Quarter Circles Inside Squares
- What Convergence Says About the Terms
- A Shrinking Family of Circles
- Two-Digit Numbers a Sine Accepts
- A Curve Slid Sideways
- Partial Sums Multiplying to a Power of Two
- A Path Winding Around a Cone
- A Value Forced by Continuity
- A Product That Has to Stay Continuous
- The Gap Between Two Exponentials
- A Triangle Under a Logarithm
- Three Areas as an Angle Vanishes
- A Shrinking Integral, Magnified
- An Average Over a Vanishing Interval
- A Limit Function Made Continuous
- An Integral of an Absolute Difference
- A Symmetry Worth Ten
- A Function Hidden in an Integral Equation
- Three Conditions Fixing a Function
- Slicing a Trapezoid
- Distances to a Curve
- Areas Under a Fading Cosine
- A Cylinder Cut at 60°
- Triangles on a Semicircle
- A Line Cutting a Wave Evenly
- Even, Odd and Three Riemann Sums
- Equilateral Slices
- An Integral That Measures Length
- Counting the Surviving Terms
- A Tenth Derivative
- Rectangles Over a Shaded Cell
- Parallelograms in a Triangle
- Rectangles in a Cross
- A Nested Radical of Factorials
- A Function Defined by Its Own Integral
- Where a Root Curve Meets the Tangent
- Functions That Never Climb
- A Cubic Caught Between Two Bounds
- A Sequence With Two Moves
- A Vertex Roaming on a Circle
- Squares Built on Chords
- A Walk That Alternates Direction
- When Two Limits Agree
- Two Sequences From One Sine
- Matching the Order of Vanishing
- How Fast Does It Vanish?
- Even Products and an Order
- A Quadratic Read Off Two Integrals
- Absolute Values Adding to Five
- A Quadratic and a Cubic
- A Series of Remainders
- A Sine Recursion That Settles
- Two Limits With Absolute Values
- A Chain of Three Sets
- Integers in a Union of Intervals
- Multiples of 2 and Multiples of 3
- Two Unions, One Intersection
- The Largest Two-Digit Union
- How Many Sets C?
- Inclusion–Exclusion for Three Sets
- Recovering a Set from a Difference
- The Smallest N That Fits
- A Difference With One Element
- Adding Two Complement Equations
- De Morgan Inside A
- Counting B Inside a Small Universe
- Listing a Complement
- The Part of B Outside Everything
- Three Regions That Cannot Be Empty
- When B Comes From an Intersection
- English, German, or Neither
- Only German, Only French
- Exactly Two Sports
- Who Came With Both Parents?
- A Day at the Appliance Store
- Three Kinds of Juice
- Basketball or Football
- Sea and Forest Paintings
- Reading a Venn Diagram
- The Set of 3m−1
- From Pairs to Triples
- Caught Between Two Sets
- A Ratio and Two Differences
- Three Counts, One Set
- Divisible by 3 but Not by 2
- Three Days a Week
- Simplify the Expression
- Three Differences and a Core
- Equal Rings
- Two Equations for One Union
- Three Claims About Sets
- An Unknown in the Union
- Five Elements, Three From the Front
- The Four-Digit Password
- Digits of a Page Group
- Cables on a Street
- How Big Can A Be?
- Who Comes on Day 21?
- Pages Without a 1 or a 2
- A Set Inside a Union
- Counting the Possible B
- Three Points on a Line
- Negatives Come in Blocks
- A Two-Digit Number of Subsets
- At Least Two From B
- A Prescribed Intersection
- Six and Its Divisors
- Pairs Sharing One Element
- Three Equal Regions
- Mysterious Sets
- Co-intersection Sets
- How Wide Can the Union Be?
- Spot the Faulty Step
- Three Steps to the Universe
- Letters in a Name
- Billiards or Darts
- Fig and Pomegranate Trees
- Planes and Helicopters
- A Night at the Camp
- Pens in Three Colours
- Two In, One Out
- No Digits Allowed
- Exactly One Vowel
- At Least One Letter
- Missing x or Missing y
- Strictly Above X
- Two Sums, One Universe
- Complement of a Ring
- Counting Proper Subsets
- Multiples of 3 and of 5
- Everything Collapses
- Guitar or Violin
- Two Languages on a Plane
- Escalators and Lifts
- At Least Two, At Most Two
- Three Languages on a Plane
- Girls, Boys and Two Sections
- Digits, Even and Small
- Two Absolute Values
- Three Relations, One Universe
- Even Numbers Meet Multiples of 3
- A Sum of Complements
- Simplify It
- Divisors of Two Numbers
- Element and Subset at Once
- A Combination of Complements
- Smallest Plus Largest
- Pinning Down A∩C
- Subsets and Proper Subsets
- Adding Up Every Subset
- At Least Thirteen
- Pairs Without a Prime
- Matching Subset Counts
- At Least Two Letters
- One Is In, x or y Is Out
- Soup, Meat and Vegetables
- Four Computer Games
- Three Parasites
- A Survey on Five Series
- Android or iOS
- A Complement Swallows a Set
- Nothing Is Left
- Three Sets, Two Empty Overlaps
- Two Arithmetic Progressions
- Subsets Against Proper Subsets
- Three More Elements
- Two Subset Counts That Agree
- Three Claims About x and y
- A Mean Above the Middle
- Small Subsets, Large Subsets
- All Subsets Against the Pairs
- Five From Seven
- Three Languages, Four Clues
- Six Desserts
- Leaving After Primary School
- Drums, Guitar and Violin
- Players, Boys and Girls
- One Progression Minus Another
- Chained Ratios
- How Big Can B Be?
- An Expression That Vanishes
- Belonging to Exactly One
- An Even Product
- A Difference of 4950
- Six Elements In Between
- The Missing Element
- A Mean Below Five
- None Inside Another
- At Most How Many Know Neither?
- Ages in a Group
- Land, Water and Air
- Course A or Course B
- Multiplying Every Subset
- A Union Through a Difference
- Element or Subset?
- Subsets That Contain 4 or 5
- A Set That Depends on p
- Odd Sums of Three
- The Smallest Possible Union
- Name the Shaded Region
- Counting B from the Outside
- Two Conditions on B − A
- Recovering A − B
- Three Regions in Proportion
- Integers in Both Intervals
- With −2 but Without −4
- Not Coprime to 10
- A Difference of a Difference
- From Bᶜ to (A ∩ B)ᶜ
- Elements in Exactly One Set
- Finding a and b
- Finding k from B − A
- A Square Hidden in A
- Sum of the Elements of B
- Sets Squeezed Between A and B
- Subsets of a Union
- Inside A, Containing A − B
- (B − A)ᶜ Rewritten
- Counting Aᶜ ∪ Bᶜ
- How Many Choices for B?
- (A ∩ B)ᶜ ∩ B
- Simplify Before You Choose
- Largest and Smallest Overlap
- Elements in Only One Set
- Outside the Overlap
- Counting A ∩ Bᶜ
- Between A ∩ B and A ∪ B
- Keeping Two Intervals Apart
- Recovering B
- Two Quadratics Sharing a Root
- Odd Images of A
- The Shaded Outside
- Bus and Subway
- Never Black with Blue
- Permutations That Use 3
- Three Friends Side by Side
- Parents at Both Ends
- Keeping the Letters Apart
- Tens Digit Not 2
- Start with T, Don't End with N
- Divisible by 5, No Repeats
- At Least Two Digits Equal
- Five-Letter Palindromes
- The 55th Number
- Where Does LKAEM Stand?
- Rearranging 520033
- Even Rearrangements of 100223
- Order of the Goals
- Every K Followed by I
- Lining Up by Height
- Ahead in the Queue
- Vowels in Alphabetical Order
- Odds and Evens in Order
- No Two Boys Together
- Alternating on the Shelf
- Hundreds Digit on Top
- Seats in Two Rows
- Digits from Two Sets
- Exactly Two in Between
- Two Courses at the Same Time
- The Top-Left Photo
- Strictly Increasing Digits
- At Least One Even
- At Least Three Boys
- At Most Two Different Digits
- Two Digits Nobody Picks
- Products of Pairs
- From C(n, 3) to P(n, 3)
- Pairs and Trips
- Which Groupings Are Right?
- Three Toys, Four Siblings
- At Most One Toy Each
- At Least One Toy Each
- Identical Boxes
- Identical Balls, Different Boxes
- Sums to Eight
- Triangles from Seven Points
- Avoiding One Street
- When C(x, y) = 1
- One or Two Boxes per Trip
- Two More Elements
- Subsets They Share
- Only Vowels
- Two or Three Elements
- Between {a, b} and {a, …, f}
- At Least Six Elements
- With a, Without c
- Sets Inside Sets
- Proper Subsets and the Union
- Subsets of the Overlap
- The Largest Overlap
- Elements That Are Sets
- As Much of A as Possible
- Three Equal Ratios
- Sums of Subsets
- Counting Subsets of A − B
- Multiples of 4 Meet Multiples of 6
- Neither 5 nor 6
- Squares Under Control
- When the Union Is Smaller
- The Middle Set Alone
- The Smallest x
- Subsets of the Overlap Again
- Biggest and Smallest Union
- Finding n(A − B)
- Inside the Ring
- Difference of Intervals
- Intersection of Intervals
- Multiples of 3 or 4
- Counting in a Three-Circle Diagram
- A Inside B
- Differences of Complements
- Complements Only
- Difference from Complements
- Union from a Chain of Ratios
- Complements on the Real Line
- The Largest Possible Class
- Fewest Who Play Nothing
- Only Football
- Only English
- Eyes and Glasses
- All Three versus None
- Passing Math or English
- Exactly One Subject
- Subset Counts Adding to 49
- Every Subset of {1, 2, 3, 4}
- Only English in a Class of 34
- How Big Is the Group?
- Failing Exactly One
- Women Who Don't Speak English
- Only French
- Fewest Who Play Both
- President, Then Vice-President
- Not with the Blue Hat
- Roads from A to D
- Not the Same Way Back
- Letters into Postboxes
- Runs of Six
- The Forgotten Code
- Pass or Fail
- Below 700
- Between 2000 and 6000
- At Most Three Digits in Base 5
- Below 500
- Odd Three-Digit Numbers
- No Two Answers in a Row
- Three Digits from 1 to 6
- Three Digits with a Zero
- Multiples of 5
- Even and Distinct
- Between 3000 and 5000
- Greater Than 350
- Odd, with a 4
- At Least One 6
- Divisible by 9
- Divisible by 4
- Adding Every Two-Digit Number
- At Least Two Digits Alike
- From A to F
- Decimals Above a Fraction
- Gluing Numbers Together
- Paying 95
- A Factorial Equation
- Two Factorials, One Equation
- Four People, Six Chairs
- Two Shelves of Books
- Keeping Subjects Together
- a and b Side by Side
- With 1, Without 2 and 3
- In Alphabetical Order
- A Physics Book Between
- No Two Students Together
- Exactly One Physics Book Between
- Who Can Drive?
- One Player Between Them
- Choose, Then Shelve
- Only Nazmiye Between
- Girls Together, Boys Together
- At Both Ends of the Row
- Girls in Front
- Zero Before Three
- Adding All the Numbers
- V Between S and Y
- Where Is 406?
- No Two Girls Together
- At Least Two in Between
- 179 as a Block
- The 26th Number
- The 150th Number
- Only Two Fives Together
- 2 Just Before 1
- Five Pens, Three Children
- Even Palindromes
- Kaya's Seat Is Fixed
- Demir on Tuğba's Right
- Öykü or Aykut on the Right
- Parents Side by Side
- Glasses Together
- Facing Each Other
- Couples Together
- Two Boys on Each Side
- Semra and Sinem Apart
- No Two Girls Side by Side at the Table
- Mothers Between Their Children
- Doctors and Nurses Alternate
- Three Together, 720 Ways
- A Girl After Every Two Boys
- Keys on a Ring
- Big Squares in a Grid
- Starting with F
- Start with A, Don't End with K
- Rearranging 11224
- Rearranging 1001133
- Even Rearrangements of 1100022
- Reading TONYALI
- The Ant Avoids K, L and M
- Ending in 1
- Reading ARMUT
- Pegs on a Line
- Emptying the Car Parks
- Digits Multiplying to 48
- Words from FİLİZ
- Sums to Seven
- Through B, Around the Closed Road
- Through K or L
- P(n, 4) and C(n, n − 3)
- An Equation in C and P
- Two Binomials Make 70
- Most of Row 7
- Permutations Meet Combinations
- Choosing Two Hats
- Equal Binomial Coefficients
- Eleven Small Subsets
- From 140 to 3-Element Subsets
- At Least Six of Seven
- A Team of Three
- Exactly Two Girls
- Leaders Must Play
- Three Clashing Electives
- A Group and Its Leader
- Threes and Sevens
- At Most Two Doctors
- A Committee and a Chair
- Three Cities, Five Travellers
- Two Already Chosen
- Three Vowels, Two Consonants
- At Most Two Pass
- At Least Three from the First Six
- Choose, Then Seat Around a Table
- Handan and Aykut Together
- The First Five Are Compulsory
- Captains Already Known
- Subsets Containing 2 and 3
- Murat and Handan Apart
- At Least One Couple
- Lines Through Eight Points
- Six Circles Crossing
- Triangles on a Circle
- Counting Rectangles
- Three Countries, Two Each
- Eggs in Every Basket
- A ≥ B > C
- At Least One Banknote
- Eggs in Baskets
- Circle Points and Line Points
- Lines from Seven Points
- Triangles from Nine Points
- Eight Lines, Two Special Points
- Lines Through a Triangle's Points
- Counting Quadrilaterals
- The Fourth Term
- Finding a from a Term
- The x³ Term
- The Constant Term
- Two Middle Coefficients
- Reading a and b
- When the 4th Term Is Constant
- The Middle Term
- The y²/x⁴ Term
- Exponents Adding to 8
- A Trinomial Coefficient
- Terms Containing z⁶
- When Is There a Constant Term?
- Adding the Rational Terms
- Every Other Coefficient
- A Hidden Binomial
- At Least 2 on a Die
- Odd or Prime
- Even or at Least 4
- A Sum of 7
- A Sum Above 8
- A Sum of at Most 3
- An Odd Product
- A Difference of 3
- An Even Sum
- Three Coins Alike
- At Least Two Tails
- One Girl, One Boy
- Three Disjoint Events
- A Valid Probability Function
- Skipping City B
- An 18-Year-Old
- Even or Greater Than 4
- A Girl or Wearing Glasses
- An Odd Value in the Image
- Two Four-Sided Dice
- Both at Least 5
- Two More Than Before
- A Sum of 10 or 11
- An Even Product of Three Dice
- A Biased Coin
- At Least One Tail
- Three Tails in Five
- Three People Between Them
- Three Different Numbers
- All the Same in Twenty
- Red or White
- An Odd Number on the Card
- A Subset with 2 and 3
- Divisible by 4 and 6 Too
- Two Even, One Odd
- A Square Ending in 4
- The Same Number Three Times
- Late to the Theatre
- P(A ∪ B)
- Chair and Deputy Side by Side
- Starts with 2, Ends with 1
- Given at Least One 3
- A Triangle Through A
- Red Once, White Once
- At Least Two Girls
- Two Couples Chosen
- An Even Two-Digit Card
- An Even Three-Digit Card
- A Triangle Using A and L
- Only Samarkand
- In Different Teams
- Passing Through B
- A Positive Product
- At Most How Likely?
- Neither Subject
- A Pair of Opposites
- İsmail and Merve Both Chosen
- Divisible by 15 or 25
- Three Exclusive Outcomes
- Landing in the Ring
- Which Unit Made It?
- Glasses or Boys
- Odd and Heads
- Tails or a Prime
- Finding P(B)
- A Base on d₂
- Matching Coins or Matching Dice
- A Big Square
- Seeing the Blue Face
- A Vertex at the Crossing
- Two Different Subjects
- Also a Multiple of 6
- A Reflexive Relation
- First to Heads Wins
- By 3 but Not 5
- Neither A nor B
- Passed Maths, Failed Literature
- Three Tails, Given One
- Two Reds
- Comparing Two Probabilities
- The Same Colour
- Exactly One White
- A Wearable Pair
- At Least One Yellow
- How Many Marbles?
- Twice as Many Reds
- Two Working Bulbs
- Three Reds
- One Faulty Shirt
- White, Red, White
- At Least One White
- Swapping Between Bags
- Three Different Colours
- At Most Two White
- At Least Three White
- The Second White on Draw Four
- Down the Peg Board
- Evens Go Back
- Will the Lamp Light?
- A Trick Coin in the Bag
- Finding P(B − A)
- Odd, Then Prime?
- Small Product, Even Sum
- Who Wins the Contest?
- Exactly Two Heads
- Two Tails and an Even Number
- Hit by Exactly One
- A New Colour Each Roll
- Two Prime Numbers
- A Vertex at A
- Spin to Win a TV
- Guessing Three Answers
- At Most One Hit
- Finding the Right Key
- A 3 in a Sum of 5
- Tails on the Second Toss
- A Quadrilateral Using A
- At Least One Passes
- Same Colour — Both Blue?
- Which Bag Was It?
- Different Colours from Two Bags
- How Big Was Bag B?
- A Different Colour
- Twice as Many Blues
- Keep the Small Ones
- Through K, L or M
- The Right Box Plot
- Which Ages Fit?
- Bills Above the Median
- Girls' and Boys' Scores
- Finding the Median Salary
- Two Swimming Teams
- Passing Both Exams
- A Sleep Survey
- Salaries and Rents
- Walking and Running
- Planning the Land
- The Angle for Option E
- Weekend Readers
- Ages at Work
- Brand B Sales
- Six Exam Scores
- Six Months of Gas Bills
- Can and Aslı's Sales
- A Share Price Over a Week
- Wednesday's Sweaters
- Matching Unit Prices
- Women at Two Sites
- The Most Popular Colour
- A Quarter Civil Servants
- Which Bar Is x?
- Flour Packets in a Box Plot
- Finding x from Two Charts
- The Most Common Score
- The Median Score
- Baskets Scored
- Survey Ratings
- When Some Leave
- Three New Members
- Mode 4, Median 4.5
- Mean of Grouped Scores
- A Mistake in Two Salaries
- Two Car Dealers
- The Most Successful Subject
- The Weakest Test
- All Three Equal
- The Smallest Median
- Three Months of Sales
- Deniz's Mock Exams
- Watches Sold in Four Days
- A Farmer's Wheat
- Always True
- Erol's Darts
- A Larger Spread
- The Most Uneven Student
- The Most Spread-Out Class
- The Largest Possible Mean
- The Best Topic
- Successful and Spread Out
- A z-Score
- Finding the Mean
- Finding the Standard Deviation
- Six Points Below
- Three Times the Mean
- Below Zero
- What z-Scores Tell Us
- Elif's z-Score
- Fifty Points in Five Branches
- Erdem and Aslı
- Three z-Scores
- Same Score, Different Classes
- From z to T
- A T-Score of 60
- T-Score of a 70
- T Is Twenty Times z
- Sedef's T-Score
- A Grade of 4
- Deniz's T-Scores
- Values from a Set of Pairs
- Range from an Arrow Diagram
- Which Set of Pairs Is a Function?
- Two Values of a Quadratic
- Not a Function from Z to Z
- The Vertical Line That Touches Once
- Equal on a Two-Point Domain
- Value from an Inner Expression
- A Constant Hidden in the Input
- One Input, One Root
- Find the Shift from the Range
- How Many Domains Make Them Equal?
- Three Pieces, Three Inputs
- Floor of a Double
- Fractional Parts
- Defined for Every Real Number
- Landing in the Natural Numbers
- A Line from Two Values
- Equal Values at 1 and 5
- Remainders of Squares
- When the Floor Equals Three
- Fractional Part Meets a Line
- Domain of a Root over a Line
- Integers Inside a Root
- Range of an Upward Parabola
- Range of a Shifted V
- Range on Five Points
- A Logarithm and a Root
- The Value That Is Never Reached
- Maximum Plus Minimum on an Interval
- From the Minimum to the Maximum
- Integers in the Range of a Root
- Largest Possible x + y
- A Substitution with a Hidden Limit
- Maximum of an Expression in f
- Excluded Values
- How Wide Is the Range?
- A Root Defined Everywhere
- Integer Values of a Difference
- Maximum and Minimum Under a Constraint
- Root of a Logarithm
- Integer Values of a Bounded Fraction
- A Minimum That Moves with a
- Range of a Sum of Two Roots
- Integers Between Two Roots
- Onto a Closed Interval
- Read Three Values
- How Often Does f Reach 2?
- Values of a Shifted Graph
- Solutions of |f(x)| = 1
- Moving a Parabola Home
- A V from Its Corner
- Triangle Under y = 4
- A Square Between Two V's
- Absolute Value Inside Absolute Value
- Exactly Three Solutions
- How Many k Give Four Solutions?
- Four Crossings of a W
- Open Dot or Closed Dot?
- A Point on a Moved Graph
- Stretched and Shifted
- Where the Vertex Lands
- Infinitely Many Solutions
- A Parabola from Its Graph
- A Hyperbola from Its Asymptotes
- A Root Graph from Its Starting Point
- Sine Meets a Gentle Line
- Three Crossings of x|x| − 4x
- Using the Axis of Symmetry
- Match the Formula
- Where Is the Sum of Distances Least?
- A Trapezoid Above a Flat Bottom
- Three Solutions of f(|x|) = k
- f(|x|) and |f(x)| from One Graph
- An Exponential Graph
- A Moved Hyperbola
- Which One Is One-to-One?
- Coefficients of the Identity
- A Fraction That Never Changes
- Pick the Odd Function
- Split into Odd and Even Parts
- An Odd Polynomial
- Far Away, Still Known
- A Sign Flip Every Three Steps
- Twelve More Every Four Steps
- Two Step Rules Together
- An Inequality Through an Increasing Function
- An Inequality Through a Decreasing Function
- Onto from a Parabola Piece
- Making a Parabola a Bijection
- Gluing Two Pieces into a Bijection
- When Is a Bent Line One-to-One?
- Numbers a Shift Misses
- Integers a Split Shift Misses
- An Odd Function from Its Right Half
- Odd and Periodic
- Which Combination Must Be Odd?
- A Cubic Inequality Without Expanding
- The Hidden Period
- An Even Quadratic
- Inside First
- Both Orders at x = 1
- Two Lines That Commute
- Find the Inner Function
- Find the Outer Function
- A Quadratic Outer Function
- Composing Two Drawn Graphs
- Applied 2027 Times
- Every Third Step Comes Home
- How Many Roots Does f(f(x)) Have?
- A Piecewise Function Inside Itself
- Sum of All Roots of f(f(x)) = 3
- Where the Compositions Have Opposite Signs
- A Line Applied Twice
- A Decreasing Line Applied Twice
- Following the Arrows Twice
- Two Bijections from Four Clues
- Three Layers Deep
- Two Compositions, One Known Value
- The Two Orders Differ
- |x − 2| Inside Itself
- A Tent Applied Three Times
- Undo a Line
- A Constant from One Inverse Value
- Inverse of a Fraction
- Inverse Values of a Cubic
- An Inverse Value Through a Substitution
- An Inverse after a Shift
- Inverse Values from a Graph
- A Point on a Moved Inverse
- A Fraction That Is Its Own Inverse
- Inverse on Half a Parabola
- Where a Curve Meets Its Inverse
- Two Meeting Points with the Inverse
- Inverse of a Composition
- Undo, Then Apply
- Two Inverses in a Row
- Inverse of a Two-Piece Function
- Its Own Inverse, Applied 2025 Times
- An Equation with an Inverse Inside
- f(2) Minus Its Inverse
- Inverse of a Bent Line
- Which Has an Inverse on ℝ?
- Cut at the Vertex, Then Invert
- Careful: the Graph Is Shifted
- Smallest Sum That Keeps It One-to-One
- Undo the Shift Inside
- A Double Inside
- A Quadratic with a Shifted Input
- A Reciprocal Pair
- A Mirror Pair
- An Additive Function
- Additive with a Cross Term
- Turning Products into Sums
- Step by Step
- Jumps of Two
- A Three-Step Cycle
- Swap x and 1 − x
- Spot the Square of the Inside
- Fraction Inside, Fraction Outside
- Ratios on a Line
- Find f(2) Without Knowing f(1)
- A Quadratic from Its Differences
- Additive, Plus a Product, Minus One
- Turning Sums into Products
- A Chain of Doubling Inputs
- A Quadratic from Two Values
- Double and Subtract One
- How Many Functions?
- How Many One-to-One Functions?
- How Big Is the Codomain?
- How Big Is the Domain?
- Onto a Two-Element Set
- Onto a Three-Element Set
- One Forbidden Value
- Two Forbidden Values
- Strictly Increasing Functions
- Non-Decreasing Functions
- Three Values Adding to 8
- Two Pairs with Equal Sums
- An Even Product
- An Even Sum
- Never Below the Input
- No Input Stays Put
- Everyone Moves
- Exactly Two Fixed Points
- Its Own Inverse on Six Elements
- When f ∘ f = f
- Back Home After Three Steps
- A Range of Exactly Two
- A Range of Exactly Three
- Increasing with a Fixed Value
- Up, Then Down
- Two Order Conditions
- Six Students, Three Clubs
- Letters into Mailboxes
- One-to-One with an Odd Sum
- Back to 1 in Two Steps
- When f ∘ f Is Constant
- Odd Inputs Need Even Outputs
- Which One Is a Polynomial?
- Simplify Before You Decide
- Which n Make It a Polynomial?
- Degree Two Fixes the Constants
- Three Degrees Added
- When the Leading Terms Cancel
- Degree of a Quotient
- Two Equations in Two Degrees
- Degree from a Composition
- Leading Terms Cancel by Design
- Divisible by x³
- First and Last Coefficients
- Composition Minus a Square
- How Many Terms Survive?
- Rewrite Around x = 1
- Sum of All Coefficients
- Coefficients of the Even Powers
- All but the Constant Term
- In Powers of x − 2
- A Linear Polynomial and Its Twin
- A Linear Polynomial Applied Twice
- A Quadratic from a Difference Rule
- A Polynomial of x² − 1
- Neighbours Add Up
- Odd Powers of a Product
- A Quadratic with Jumps of Three
- One Coefficient, Smartly
- Factored Form Meets Expanded Form
- Long Division by a Quadratic
- Synthetic Division by x − 2
- Fill In the Synthetic Division
- Dividing by 2x + 1
- Rebuild the Dividend
- x⁴ + 1 Divided by x² + x + 1
- A Quotient Made of Ones
- Make the Division Exact
- Degrees After Division
- A Quotient Evaluated at 2
- Remainder Without Dividing
- Remainder on Division by x + 1
- Remainder on Division by 2x − 1
- A Huge Exponent
- Find k for a Factor
- Two Factors Known, Find the Third
- Remainder by a Factorable Quadratic
- Divisible by x² − 1
- Remainder by x² + 1
- Divisible by (x − 2)²
- An Eighth Power with a Double Root
- Divisible by x² + x − 1
- Divisible by (x + 1)³
- Factor, Then Square the Roots
- The Largest Root
- Remainder After a Substitution
- Remainder of a Combination
- The Same Remainder Three Times
- Big Powers Divided by x² − 1
- x⁴⁰ Divided by (x + 1)²
- Using x³ = 1
- Combine Two Remainders
- The Quotient Has a Remainder Too
- A Linear and a Quadratic Remainder
- A Square Divisor and a Linear One
- The Quotient at x = 1
- Multiply by x³, Then Divide
- Multiply, Then Divide
- A Binomial Power Divided by x²
- The Quotient of an Odd Power
- Sum of the Squares of the Roots
- When There Are No Real Roots
- Sum of the Cubes of the Roots
- How Far Apart Are the Roots?
- Reciprocals of Three Roots
- Product of Shifted Roots
- Sum of Squares of Three Roots
- Roots Squared
- Roots One Apart
- One Root Twice the Other
- Roots in Arithmetic Progression
- Fifth Powers of the Roots
- A Cubic from Its Roots
- A Cubic Read from Its Graph
- A Monic Cubic Through Three Points on a Line
- The Same Value at Three Points
- A Quartic from Five Values
- A Quadratic That Is Never Negative
- A Quartic That Touches the Axis
- A Linear Quotient and One Value
- A Quadratic in x²
- A Palindromic Quartic
- A Sign Chart for a Cubic
- A Squared Factor Removes a Point
- Even and Odd Multiplicities
- Factor First, Then Solve
- Integers from −10 to 10
- A Factor That Is Always Positive
- Read the Sign from the Graph
- Multiply the Graph by x
- Positive for Every x
- Four Consecutive Factors
- Doubling the Volume
- From Glass into Diamond
- The Odd Wave Out
- How Many Lines?
- Height of a Mirror Image
- Kinetic Energy from a Field
- Energy of a UV Photon
- How High Does A Climb?
- Index and Wavelength
- Shortest Wavelength Transition
- Comparing Two Media
- Pushed Apart from the Inside
- Twice the Threshold Frequency
- Fastest Point on the String
- Capacitance from the Field
- The Hottest Point on the Line
- Period of a Travelling Wave
- Pulling the Rod: Numbers
- A Simple Perpendicular Wire
- Inside the Focal Point
- Short-Wave UV on a Metal
- Only Waves Do This
- An Electron Climbing the Potential
- Power Used by a Resistor
- Ratio of de Broglie Wavelengths
- From Glass into Water
- The Wave Equation
- Focal Length of a Convex Mirror
- A Warm Tyre
- Period in Numbers
- Constant-Pressure Expansion
- Charge on a Capacitor
- Orbit Size
- Work in an Isothermal Expansion
- Bring Them Closer
- Momentum of a Short-Wavelength Photon
- Resistance Between X and Y
- Energy in the 3 μF Capacitor
- Arrival Speed
- Volume after Heating and Squeezing
- Squeezing at Constant Temperature
- Drifting Downstream
- Which Is a Vector?
- Photoionisation from an Excited State
- Alpha Particle Versus Proton
- How High Was the Cliff?
- Two Series Resistors
- A Gentler Slope
- Charge on a Capacitor in a Circuit
- Work Along the Slanted Side
- Induced Current in Numbers
- Bending Toward the Normal
- Seal It, Then Cool It
- A Hotter Ball
- Heating Along a Straight Line
- Two Like Charges
- A Rod Along the y-Axis
- Braking Past Zero
- Equal Masses
- Speed Through the Middle
- Two Trains Leaving a Station
- Velocity of the Image
- Without Wire C
- Bringing in the Third Charge
- Two in Series
- Half the Thickness Filled
- From Field Back to Charge
- Speed After Half a Second
- Work in Units of P₀V₀
- Falling Toward a Fixed Mass
- Two Lenses in a Row
- There and Back
- Sharing Charge Through a Wire
- Second Harmonic
- Average Force of the Bat
- How Much Bigger?
- A Square Frame
- Net Heat over a Cycle
- Acceleration on the Steep Part
- How Fast Does the Rain Seem to Fall?
- Two Different Samples
- Index from Frequency and Wavelength
- Pulling Against a Hanging Block
- Final Speed from a Force Graph
- Three Loops on a Rope
- How Tall Is the Image?
- Power at 3 A
- A Pendulum on the Moon
- An Egg Tossed in a Moving Car
- Jumping off a Cart
- Field Along z
- Speed at Half Amplitude
- How Deep Was It?
- The Mirror Moves Instead
- An Electron Instead
- Double Everything
- An Explosion from Rest
- Which Ion Turns Widest?
- Stopping Potential
- Shortest Wavelength
- Power in the 4 Ω Resistor
- Equal Masses
- Stopping Potential
- Reverse the 50 A Current
- Energy After Halving the Gap
- Properties of Electromagnetic Waves
- Moving the Screen Back
- Air into Water
- How Long Does It Take?
- Frequency Limit
- Tension in an Atwood Machine
- A Fast Wave
- Where Does It Balance?
- Crossing Empty Space
- Finding the Inductance
- Landing on Cart B
- Distance While Speeding Up
- Energy of n = 3
- Two Springs in a Rising Elevator
- Fastest Point of the Fall
- A Dielectric in an Isolated Capacitor
- Which Plate Is Positive?
- Current in the 30 Ω Resistor
- How Long to Pass?
- Timing the Second Rock
- Mass from a Rebound
- Speed Just After the Bullet Embeds
- Friction That Is Not at Its Limit
- Net Work Around the Cycle
- Power Needed to Keep It Moving
- Doubling the Period
- Find the Acceleration
- Heating a Copper Calorimeter
- Distance in Six Seconds
- A Hotter Blackbody
- Image Distance at a = 15 cm
- Throwing a Ball off a Cart
- A Movable Pulley
- Balanced in the Middle
- Energy Stored in the Pair
- A 37°–53° Split
- Pushing Against Friction
- Resistance from a Bulb's Rating
- Which Statement About Mirrors Is Correct?
- Perpendicular Rays, Unknown Liquid
- Identical Rods
- Pendulum in an Accelerating Train
- Lifting to a Height of One Radius
- Counting Photons
- From 300 K to 400 K
- A Rising Bubble
- EMF of the Moving Rod
- Swapping the Masses
- Same Kinetic Energy
- Same Direction This Time
- Aiming Upstream
- A Different Separation
- Heat in Thirty Seconds
- Released from an Unstretched Spring
- Heat Released During the Rest of the Cycle
- Heat Given Off by the Ball
- Speed of an Image in a Moving Mirror
- Pulling from the Other End
- Average Acceleration, 2 s to 4 s
- What Holds Up the Top Two Books?
- Force per Metre
- Which Wave Is Different?
- An Electron Circling in a Field
- Electron Versus Proton
- Force Between Two Small Charges
- Threshold Wavelength
- Heat at Constant Pressure
- Image in a Flat Mirror
- Heating a Sealed Can
- Work During 3 → 1
- Temperature of a Blue Star
- Unequal Masses on Two Springs
- Three Charges on a Triangle
- Find the Spring Constant
- How Far Does It Land?
- Bullet Speed from Spring Compression
- Light Slows Down in Glass
- Radius of the Spiral
- Highest Potential
- A Spring in a Rising Elevator
- Speed from Two Graphs
- A Glancing Elastic Collision
- Atwood Machine
- Three Dielectrics in One Capacitor
- X to Z
- Total Charge Through the Loop
- Current Through the 3 Ω Resistor
- Einstein's Two Postulates
- Internal Energy Change at Constant Pressure
- A Much Better Conductor
- Cruising Twice as Far
- Charge and Field in a Capacitor
- Three Steps of Force
- A Short Conical Pendulum
- Last Second Gives the Height
- A Block Dragged by a Falling Weight
- Mixing Warm and Cool Water
- Train Acceleration in Numbers
- Falling Versus Sliding
- Energy Stored in Two Springs
- Heat in an Isothermal Expansion
- Which Way to Point the Nose?
- Three Bodies, Two Pulleys
- The Electric Force Alone
- Size of the Electric Force
- Midway Between Two Equal Currents
- Ionising Hydrogen from n = 4
- Pushing a Moving Block
- Heat Thrown Away
- Two Equal Masses
- Force After Touching
- The First Excited State
- Speed from Two Wave Graphs
- One End Closed
- Quadruple the Voltage
- A Wire at 60°
- Largest Angular Speed
- Object Inside the Focal Point
- Acceleration in the Gap
- Equal Speeds
- Where Is the Candle's Image?
- Average Velocity over Six Seconds
- Two Capacitors in Parallel
- Frequency of a Stiff Spring
- Reverse the Current
- Net Force on the Middle Book
- A Real Battery
- Four Times as Long
- Time from A to C
- Doubling the Charges
- Speed from a Force–Time Graph
- Pull, Then Let Go
- How Far Does It Tilt?
- This Time It Sticks
- Speed at the Bottom
- How Far from the Axis?
- Force on a Wire Between Magnet Poles
- A Shorter Spaceship
- Potential Difference Between A and B
- Wavelength Under Water
- An Accelerating Car
- Energy of a Photon
- Double the Radius
- Balanced Pulleys
- Energy of the λ₁ Photon
- The Red Hydrogen Line
- Angle Between the Two Rays
- Half a Kilogram of Copper
- A Bigger Resistor Beside the Capacitor
- Where Kinetic Equals Potential
- A Heavier Right-Hand Block
- Friction Between Stacked Blocks
- Highest Frequency
- Smallest Critical Angle
- Speed in the Liquid
- Into a Dense Liquid
- Doubling the Current
- Density of the Moon
- Speed Right After the Collision
- Potential at the Midpoint
- Jumping Straight Down
- Heading Toward the Wire
- Energy Stored in the Pair
- Swap the Capacitors
- Force at 10 cm
- Was Any Energy Lost?
- What Is Conserved?
- Energy in the Coil
- A Far Charge
- A Three-to-One Mass Ratio
- When Is x Largest?
- Find the Mass of B
- A Collision That Splits an Object
- A Softer Spring
- Halving the Gap After Charging
- Field Between Capacitor Plates
- Running It Backwards
- Energy of an Orange Photon
- Force from a Velocity Graph
- Another Floating Charge
- Opposite Charges Touch
- Heat for Warm Water
- Net Heat Absorbed
- How Fast Does It Go?
- Find the Scalar
- All the Way Out
- Where Is the Image?
- An Electron in the Same Field
- Pushed Apart: Speed of B
- Where Do They Cancel?
- Equal Shares
- Equal Stretches, Different Springs
- Escaping the Moon
- Hit Straight Back
- Net Work on the Gas
- Heat for a Monatomic Gas
- Opposite Currents at the Midpoint
- Equal Pendulum Bobs Collide
- Scattered Wavelength
- Speed After the First Push
- Power in C Compared with A
- Pitch of a Helix
- Momentum of Red Light
- How Far Does the Train Move?
- Same Energy, Different Mass
- Pressure and Temperature at D
- A Frictionless Table
- In Numbers
- Speed on Reaching the Spring
- Jumping Between Carts
- Critical Angle for n = 2
- Which Arrow Gives the Shortest Wavelength?
- Period of a Spring Oscillator
- A Heavier Cart Hits a Lighter One
- Net Force on the Whole Loop
- Thrown Up, Lands Below
- A Spring Pushes Two Blocks Apart
- Ranking the Speeds
- An Electron in Crossed Fields
- How Much Does the Ray Bend in Water?
- Radius of Its Circle
- Ranking Three Indices
- Two Resistors in Series
- Work from a Pressure–Temperature Graph
- What Doesn't Change the Speed?
- How Far Apart Are the Slits?
- A Smaller Force
- Two Capacitors in Series
- Stick Together and Climb
- A Proton Through the Same Fields
- Object at the Centre of Curvature
- A Lighter Sample A
- Along an Equipotential
- Arrangement (A) in Numbers
- Two Carts Couple Together
- Tension in Each String
- Compress It Twice as Far
- Voltage Across the 80 Ω
- One Charge Four Times the Other
- A Lighter Lift
- Area of the Window
- How Far in x?
- Same Mass
- Tension in a Swinging String
- A Smaller Temperature Rise
- Three Objects, One Unknown
- Speed at Half the Amplitude
- Nearly Equal Charges
- A Charge Floating Between Plates
- Completely Inside
- A Deeper Fish
- From Glass into Air
- How Much Energy Was in the Spring?
- How Heavy Was the Ball?
- Four Times the Voltage
- Both Must Be Vectors
- Which Way Is the Right Wire Pushed?
- Stretch of the Lower Spring
- Unequal Currents
- Speed in Numbers
- What Stays the Same?
- Which Ion Turns Tightest?
- Exciting Atoms by Electron Impact
- Index of a Liquid
- Compton Scattering
- Force on a Charge Beside a Wire
- Distance in the Third Second
- Which Is Transverse?
- Selecting a Speed
- A Wire at 30° to the Field
- The Elevator at Rest
- How Much Bigger?
- Where Is the Field Zero?
- Period of the Spiral
- A Bigger Middle Resistor
- Half the Area Filled
- Force from a Speed Change
- Ranking the Speeds
- Double the Wavelength
- Work During 1 → 2
- Comparing the Kinetic Energies
- Starting from Rest
- Sharing with an Empty Capacitor
- Kinetic Energy Three Times Potential
- Equal Amounts
- Running the Cycle Backwards
- Focal Length from an Image
- Energy in Five Minutes
- Down a Sloping Line
- Twice as Fast
- Current in the Loop
- How Far Apart?
- A Lopsided Atwood Machine
- Potential at Point A
- Forces on Two Sides of a Loop
- A Slower Glancing Collision
- Equivalent Capacitance
- A Pendulum Bounces Back
- Up at Right Angles
- Two Perpendicular Wires
- Closing the Gap with the Battery Connected
- Three Times the Threshold
- A Two-Second Pendulum
- Crossing Time
- Books in a Rising Elevator
- Doubling the Volume Isothermally
- A Gentler Toss
- Does B Turn Back?
- Equivalent Resistance
- Twice as Far Apart
- Rain Seen from a Moving Car
- Speed from a Snapshot
- Gravity One Radius Up
- Power in One Parallel Branch
- A Candle Inside the Focal Length
- Voltage on the Smaller Capacitor
- Light Seen from a Rocket
- No Force on the End Charge
- Spot the False Statement
- Comparing Three Arrangements
- Catching Up
- Photon Energy in Electronvolts
- Speed Halfway Along
- Starting from Rest
- The Heavier Block Wins
- A Heavier Lower Block
- Describing Hydrogen's Energy Levels
- Critical Angle of Water
- Head-On Collision: Speed of A
- Time to Fall 80 m
- The Lighter Block Stops
- How Much Smaller?
- Just Two Charges
- Object Beyond F
- Kinetic Energies from Wavelengths
- The Convex Lens Alone
- Energy Lost When the Bobs Stick
- A Ball Dropped onto a Spring
- How Fast Can It Spin?
- Twice the Mass, Twice the Radius
- Object Beyond C
- Changing the Laser Colour
- Moving Along the Field
- How Fast Does the Heavy One Move?
- Pulling the Plates Apart
- Which Bends More?
- A Train Passing a Post
- Velocity at the Furthest Point
- Just Enough Friction
- Bigger Charges, Further Apart
- Index of Refraction from Two Angles
- Period from a Stretch Test
- Two Blocks and One Spring
- Photons per Second from a Laser
- Putting the Spectrum in Order
- Stretching a Spring Further
- Work Along the Top
- How Fast Does the Image Move?
- Throwing Backward from a Cart
- Speed from a Wave Function
- Specific Heat Ratio from a Graph
- A Battery with Internal Resistance
- A 45° Tilt
- Equal Blocks, One Spring
- Walking Toward a Mirror
- Expanding While Pressure Falls
- Kinetic Energy of the Ejected Electron
- Steady Current from the Battery
- A Convex Mirror Shrinks the Image
- Impulse in the First Second
- Work Function from a Graph
- Exciting Hydrogen to n = 3
- A Small Resistor on 12 V
- The Last 25 Metres
- Radius in Numbers
- Time to the Bottom, in Numbers
- How Much of the Bullet's Energy Reaches the Spring?
- Tension While Pulling
- Work from a Stepped Force Graph
- When Does It Start to Slide?
- Force on the Left Side
- Two Springs, Two Masses
- Internal Energy Along the Adiabat
- Two Charged Spheres on Strings
- Lifting a Block
- Same Heat, Different Metal
- Three Spheres, Two Contacts
- Work Through 6 V
- Another Wave Function
- Temperature from the Work Done
- An Elastic Bounce
- Average Acceleration from a Velocity Graph
- Temperature at the Joint
- At 30° to the Field
- Accelerating Downward
- Radius of the Circle
- What Stays Constant?
- How Long a Pipe?
- Still Connected
- Equal Charges at the Ends
- Wavelength in Vacuum
- With a Tailwind
- Current Through the 12 Ω Resistor
- Warming a Balloon
- When C Matches A and B
- Running Inside a Train
- Net Work per Cycle
- A Heavier Target Block
- Back the Other Way
- Average Speed in Numbers
- Total Stored Energy
- Induced Current While the Field Changes
- Image Speed When the Object Is Farther
- Find the Launch Speed
- Fastest Point on the String
- What Stays Inside
- Speed at the End
- Height Through a Diverging Lens
- Spring, Collision, Slope
- Time Down a 30° Slope
- When Does It Stop?
- Field at Q from Three Wires
- A Fixed Mirror
- Three Times Hotter
- Heat from a Resistor in One Minute
- Which Way Does the Fragment Go?
- Speed in the Circle
- Mirror Coming Toward Him
- A Photon That Cannot Be Emitted
- How Long to Warm the Water?
- Moving to a Lower Potential
- A Short-Focus Lens
- Power in a 20 Ω Resistor
- Reversing Wire C
- Work Under a Sloped Line
- Field from the Vertical Wire
- Force on One Side in Numbers
- Work Done by the Force
- Mass Ratio from Energy and Wavelength
- Who Lands Faster?
- Speed in That Material
- Moved Twice as Far Out
- A Slower Push
- A Heavier Target
- Which Is Not a Vector?
- Speed After Two Seconds
- Which Cannot Be Polarized?
- Four Oscillations a Second
- A Zero Point Outside
- Total Distance Covered
- One Full Slab
- Work on the Gas Around a Cycle
- Four Charges on a Square
- When the Middle Goes By
- Distance Is Not Displacement
- Speed Given to the Block
- Power from a Hotter Body
- Tension on C's Side
- Fastest Moment
- Falling Toward Each Other
- Charge on the Series Capacitors
- Another Spring Period
- From 10 cm to 15 cm
- Potential Difference from Work
- How Far Does It Drop?
- A Taller Image
- A Late but Faster Train
- Tension in a Charged Pendulum
- Which Path Is Isothermal?
- Largest Shift
- Free Fall: The Next Second
- Energy Lost in Coupling
- The Lorentz Factor
- A Parallelogram Entering a Field
- A Symmetric Ramp
- A Stops Dead
- Doubling Both Charges
- Frequency of the Wave
- How Fast Do They Separate?
- Efficiency of a Heat Engine
- Energy in That Capacitor
- Tension in the Cord
- A Bat Sends the Ball Straight Up
- Finding the Angle
- What the Slope Means
- Double the Voltage
- Only 7 V
- Recoil of Cart A
- Halfway in Time Is Not Halfway in Distance
- From Attraction to Repulsion
- Temperature at the End of a Curve
- How Long Does the Heater Take?
- Heat in the Constant-Pressure Step
- Same Energy Instead of Same Speed
- Flying Through a Crosswind
- Speed at the Floor
- Where Could It Be Trapped?
- Throwing Forward Instead
- If the 6 Ω Resistor Burns Out
- Closing Speed with Your Image
- A Smaller Coil
- Heat Released During W → X
- Light Versus Sound
- Heat in One Branch of a Parallel Pair
- Same Density, Twice the Size
- Force in the Second Spring
- A Piece Thrown Backward
- Frequency of a Standing Wave
- Energy of a Blue Photon
- Zero Net Force on the Middle Charge
- Y to Z with Small R
- A Mirror for a Wide View
- Seen from Inside the Car
- Pushing Down in a Rising Elevator
- One Spring, One Mass
- Field Between Opposite Charges
- Opposite Currents
- How Fast Does A Speed Up?
- Kinetic Energy Eight Times Potential
- Longest Wavelength
- Force on a Moving Charge
- Charge on the Single Capacitor
- Touch, Then Pull Apart
- Current in the Outer 70 Ω
- Two Pendulums Stick Together
- The Angle Between Them
- A Fish's Window to the Sky
- Speed After Acceleration Through V
- Radius at the Surface
- Moving Along the Field
- Index from Frequency and Wavelength
- Twice as Fast
- Momentum of a Photon
- Period of a Conical Pendulum
- Position and Size of the Image
- Comparing Work Between Equipotentials
- Temperature from the Combined Gas Law
- Half Amplitude, in Numbers
- Finding the Work Function
- Specific Heat of an Oil
- Fastest While Squashing
- An Electron That Is Too Fast
- Work Done by the Field
- Force on a Moving Charge
- How Hard to Pull?
- A Longer Aluminum Rod
- Water, Glass and Diamond
- A Slower Turntable
- Magnification at a = 15 cm
- Combined Capacitance
- Force to Pull a Rod Along Rails
- Standing Wave in an Open Tube
- Ionising Hydrogen from n = 2
- Charge on the 2 μF Capacitor
- Speed on Landing
- Displacement from the Same Graph
- Air Escaping from a Heated Flask
- Work Under a Ramp-Shaped Force
- Parallel to the Field
- A Horizontal Line Instead
- Rod Below the Plane
- Charge When Still Connected
- Two Resistors in Series
- Twice as High
- Removing Resistor B
- Tension in the Tilted String
- Two Rocks Landing Together
- EMF Across a Coil
- Which Way Does the Compass Point?
- Distance from an Acceleration Graph
- A Closer Second Charge
- Reflected and Refracted Rays at Right Angles
- Bullet, Block and Spring
- Metal into Water
- Speed Relative to the Train
- Quadrupling the Accelerating Voltage
- A Rectangular Cycle
- Now in Series
- Speed Off the Spring
- Reading the Elevator's Acceleration
- Bigger Plates, Closer Together
- Tension in the String
- Power in the Series Resistor
- As Seen by the Boy
- Bringing the Slits Closer
- The Lyman-Alpha Photon
- Work in a Constant-Pressure Heating
- Wavelength in the √3 Liquid
- Braking with a Constant Force
- Longer String, Heavier Bob
- A Chain of Three Blocks
- Back at x = 0
- A Diverging Lens Makes It Smaller
- A Battery Across the Plates
- Crossing the Gap
- Speed Up, Then Cruise
- The Concave Lens Alone
- Moving Against the Current
- Longest Wavelength
- Equal Masses on the Graph
- Critical Angle