← Reasoning

Invariant

75 problems

Compute one expression several ways
Calculating Cleverly
Simplify long add-subtract chains
Calculating Cleverly
Fast sums using pattern and symmetry
Applying the Multiplication Facts
Flipped digital clock readings
Exploring Clocks
Clock seen in a mirror
Exploring Clocks
Perimeter of a figure built from squares
Shapes and Length
Rectangle side and perimeter from equal squares
Shapes and Length
Which glyphs survive a turn or flip
Transformations of Figures
Spot rotated copies of a given shape
Transformations of Figures
Draw the figure a mirror doubles
Transformations of Figures
Sum of every digit written
Numbers and Digit Counts
Find angles by grouping equal pairs
Triangles and Angles
Angles from overlapping right-triangle set squares
Triangles and Angles
Chase angles through triangles in a crossing figure
Polygons and Angles
Angle made by folding rectangular paper
Polygons and Angles
Sum of a polygon's exterior angles
Polygons and Angles
Times whose clock-hand angle is acute
Clocks and Angles
Where the hour hand sits within the hour
Clocks and Angles
Surplus and shortage when sharing equally
Division
One interior angle of a regular polygon
Tessellation
Cross every bridge exactly once
One-Stroke Drawing
Draw a figure in one stroke
One-Stroke Drawing
Move first or second in a blocking game
Nim Games
Which marbles you must take to win
Nim Games
Mirror the opponent to keep piles equal
Nim Games
Equal sides force equal base angles
Triangles and Angles
Perimeter after cutting equilateral corners
Triangles and Angles
Find the hidden isosceles triangle
Triangles and Angles
Draw an auxiliary parallel through the bend
Parallel Lines and Angles
Chase angles using a rectangle's parallel sides
Parallel Lines and Angles
Prove triangle angle sum with a parallel line
Parallel Lines and Angles
Angles of a star built from rhombuses
Polygons and Angles
Equilateral triangle inside a regular pentagon
Polygons and Angles
Why a parallelogram's opposite angles are equal
Polygons and Angles
Folding a rectangle makes an isosceles triangle
Transformations and Angles
Angles where overlapped figures share a vertex
Transformations and Angles
Angle where a rotated figure meets the original
Transformations and Angles
Simplify calculations by regrouping
Mixed Operations with Whole Numbers
Perimeter of joined and cut rectangles
Perimeter and Area
Area of overlapping figures
Area of Figures
Area of pieces that split a rectangle
Area of Figures
Move a figure to make its area easy
Equal-Area Transformation (1)
Equal-area triangles between parallel sides
Equal-Area Transformation (2)
Straighten a border without changing areas
Equal-Area Transformation (2)
Use odd and even to rule out totals
Making a Target Number
Lockers left open by divisor-count parity
Applications of Prime Factorization
The Euclidean algorithm for the GCD
Greatest Common Divisor and Least Common Multiple
Product equals GCD times LCM
Greatest Common Divisor and Least Common Multiple
Angles between segments drawn on a cube
Exploring the Cube
Cut and fold edges of a net
Rectangular Prisms
Two-coloring decides domino tiling
Parity
Decide odd or even without computing
Parity
Flip count parity pins down hidden faces
Parity
Move the mean by adding a value
Averages
Solve using the sum of one line
Foundations of Magic Squares
Move a triangle onto its congruent twin
Congruent Figures
Figures turned through 180 degrees
Line-Symmetric and Point-Symmetric Figures
Slide a face to an equivalent net
Nets of a Patterned Cube
Rolling a die along a path
Dice
Simplify complex fractions
Fraction and Decimal Calculations
Simplify messy fraction-decimal expressions
Fraction and Decimal Calculations
Surface area of a composite solid
Surface Area and Volume of Solids
Read the time on a numberless clock
Clocks and Angles
Euler's formula for polyhedra
Prisms and Pyramids
Edges of a solid from its net
Using Nets
Dual polyhedron from face centers
Dual and Semiregular Polyhedra
Descartes' angle-defect theorem
Dual and Semiregular Polyhedra
Concentration of a salt solution
Ratio and Rate (1)
Unbracket, reorder, and factor out
Mixed Operations with Fractions
Collapse a repeated number or expression
Mixed Operations with Fractions
Central angle of a cone net
Shortest Distance and Surface Area
Parity bounds the fewest three-coin flips
Swapping and Flipping
Simplify messy decimal computations
Mixed Fraction and Decimal Calculations
Surface area unchanged by removing a cube
Surface Area of Cube Stacks
Area of the shaded part (2)
Area of the Shaded Region