Math Kangaroo: Cadet (Grades 7-8)

302 core topics + 100 prerequisite topics taught as needed · approximately 124 hours of instruction including spaced review

How the course runs

An adaptive diagnostic (up to 40 questions) places the student on the course's knowledge graph - topics already known are credited, and instruction begins exactly at the learning frontier. Every topic is taught with a worked-example lesson and auto-graded practice; a topic is mastered at 75%+ and then maintained through spaced reviews on an expanding schedule. Mixed checks follow every 6 lessons; each unit ends with a 12-item quiz, and course-wide assessments appear at 25%, 50%, 75%, and 100% mastery. Prerequisite gaps below the course are detected and taught rather than skipped, so completion certifies the whole tower, not just the top.

Core curriculum

Math Kangaroo: Benjamin & Cadet · 10 topics

Systematic Counting [E]	Counting without missing or double-counting.
Number Patterns [E]	Finding the rule behind a sequence.
Remainders & Divisibility [E]	Quick remainder tricks.
Units-Digit Patterns [M]	The last digit of a power repeats in a cycle.
Logic & Deduction [E]	Narrowing many possibilities to one.
Age & Word Problems [M]	Turning a story into an equation.
Coins & Making Change [M]	Counting the ways to reach a total.
Clock Angles [E]	The clock face as a circle of 360°.
Handshakes & Pairings [E]	Counting pairs among n people.
Magic Squares & Grids [E]	Every line sums to the same magic constant.

Math Kangaroo: Cadet Depth (Grades 7-8) · 30 topics

Word Equations [M]	Undo the operations in reverse.
Sum-and-Difference Systems [M]	Add the two facts to double the larger.
Proportions [M]	Equal ratios scale by the same factor.
Percent Change (Both Directions) [M]	A p% rise multiplies by (1 + p/100).
Averages and Totals [M]	Average times count gives the required total.
Average Speed (Equal Times) [M]	Average speed is total distance over total time.
Choosing a Group [M]	Order doesn't matter, so divide out the rearrangements.
Ordered Arrangements [M]	Multiply the choices place by place.
The Pigeonhole Principle [M]	Plan for the worst case, then add one.
Consecutive Integers [M]	The middle number is the average.
GCD, LCM and Products [M]	GCD times LCM equals the product of the numbers.
Remainders of Powers [M]	Remainders of powers repeat in a cycle.
Fraction Arithmetic [M]	Common denominators to add; straight across to multiply.
Angles in a Triangle [M]	The three angles always add to 180°.
The Pythagorean Theorem [M]	Legs squared add up to the hypotenuse squared.
Area of a Triangle [M]	Half of base times height.
Areas of Composite Shapes [M]	Add or subtract simple rectangles.
Circle Area and Circumference [M]	Area is πr²; circumference is 2πr.
Volume of a Prism [M]	Base area times height.
Summing an Arithmetic Sequence [M]	Average the first and last, times the count.
Geometric Sequences [M]	Each term multiplies by a fixed ratio.
Probability with Two Dice [M]	Count favorable pairs out of 36.
Counting Grid Paths [M]	Choose which steps go up.
Two-Set Counting [M]	Add the sets, subtract the overlap, subtract from the whole.
Age Relationships [M]	Set the future ages equal to the condition.
Mixture Ratios [M]	Find the scale factor from the part you know.
Working Together [M]	Add the rates, not the times.
Powers [M]	A power multiplies the base by itself.
Clock-Hand Angles [M]	Track both hands from 12.
Counting Coin Combinations [M]	Fix the big coins, then count the rest.

Math Kangaroo: Cadet - Deep (Grades 7-8) · 124 topics

Solving Two-Step Equations [H]	Subtract the constant, then divide by the coefficient.
Linear Inequalities [H]	Solve like an equation, then take the largest integer that fits.
Combining Like Terms [H]	Add the coefficients of matching variable terms.
Evaluating Two-Variable Expressions [H]	Substitute each value, then follow the order of operations.
Systems of Two Equations [H]	Eliminate one variable to solve for the other.
Systems: Combined Values [H]	Solve the system, then form the requested combination.
Difference of Two Squares [H]	Factor a² − b² into (a − b)(a + b) for a fast computation.
Absolute-Value Equations [H]	Two solutions sit symmetrically around the center.
Consecutive Even Integers [H]	The middle term is the sum divided by three.
Sum, Product and Squares [H]	x² + y² = (x + y)² − 2xy.
Chained Substitution [H]	Work from the known value up through the chain.
Rearranging a Formula [H]	Solve the perimeter formula for the unknown side.
Equations with a Fraction [H]	Clear the denominator first by multiplying both sides.
Finding the Percent [H]	Divide the part by the whole and multiply by 100.
Reversing a Discount [H]	Divide the sale price by (1 − p/100) to recover the original.
Successive Percent Increases [H]	Percents do not simply add - combine the multipliers.
Combining Two Discounts [H]	Two discounts stack to less than their sum.
Profit Percent [H]	Profit percent is profit divided by cost, times 100.
Inverse Proportion [H]	More workers means proportionally less time - the product stays fixed.
Unit Rates [H]	Find the price for one, then scale to any amount.
Map Scales [H]	Multiply the map distance by the scale factor.
Ratios and a Known Difference [H]	The difference in ratio parts equals the real difference.
Mixing Solutions [H]	The mixture concentration is a weighted average by volume.
Sharing in a Three-Part Ratio [H]	Divide the total by the sum of the ratio parts.
Counting Divisors [H]	Add one to each prime exponent and multiply.
Sum of Divisors [H]	List the divisors in pairs and add.
Largest Prime Factor [H]	Strip out small primes until only the largest remains.
Divisibility by 9 [H]	A number is divisible by 9 exactly when its digit sum is.
When Cycles Coincide [H]	The next shared moment is the least common multiple.
Greatest Common Divisor in Action [H]	Equal groups with no leftovers means the GCD.
Remainders [H]	The remainder is what is left after taking out full groups.
Simultaneous Remainders [H]	Search the numbers with the first remainder for the second condition.
Reversing a Two-Digit Number [H]	The difference is always 9 times the digit gap.
Reading Other Bases [H]	Each digit is weighted by a power of the base.
Converting to Another Base [H]	Divide repeatedly by the base and read remainders upward.
Counting Perfect Squares [H]	Count the whole numbers whose square lands in the range.
The Largest Impossible Amount [H]	For coprime a and b, the answer is ab − a − b.
Trailing Zeros of a Factorial [H]	Count the factors of 5 in the factorial.
Digit Sums [H]	Add the individual digits of the number.
Counting Multiples in a Range [H]	Subtract counts of multiples up to each endpoint.
Factor Pairs [H]	Pair each divisor with its partner.
Power of a Power [H]	Multiply the exponents.
Product of Powers [H]	Add the exponents when the base is the same.
Negative Exponents [H]	A negative exponent means a reciprocal.
Scientific Notation [H]	The exponent counts places the decimal point moves.
Multiplying in Scientific Notation [H]	Multiply the fronts, add the exponents, then renormalize.
Simplifying Radicals [H]	Pull out the largest perfect-square factor.
Cube Roots [H]	Find the number that cubes to the given value.
Estimating Square Roots [H]	Find the largest square that does not exceed the number.
Comparing Powers [H]	Estimate each power's size to rank them.
Solving Exponential Equations [H]	Rewrite the right side as a power of the same base.
Arithmetic nth Term [H]	Add the common difference n − 1 times to the first term.
Counting Terms in a Sequence [H]	Divide the total gap by the step, then add one.
Triangular Numbers [H]	The nth triangular number is n(n + 1)/2.
Fibonacci Numbers [H]	Each term is the sum of the two before it.
Summing a Geometric Sequence [H]	Use a(rⁿ − 1)/(r − 1).
Summing Multiples [H]	Factor out the common multiple, then sum 1 to n.
Sum of Odd Numbers [H]	The first n odd numbers add to n².
Sum of Even Numbers [H]	The first n even numbers add to n(n + 1).
Sum of Squares [H]	Use n(n + 1)(2n + 1)/6.
Filling In a Sequence [H]	Find the constant step, then apply it.
Angles in a Quadrilateral [H]	The four angles add to 360°.
Polygon Angle Sum [H]	Split the polygon into (n − 2) triangles.
Each Angle of a Regular Polygon [H]	Divide the total angle sum by the number of angles.
Exterior Angles of Regular Polygons [H]	The exterior angles always total 360°.
Angles with Parallel Lines [H]	Same-side interior angles are supplementary.
Isosceles Triangle Angles [H]	The two base angles are equal and share the leftover.
The Exterior Angle Theorem [H]	An exterior angle equals the two remote interior angles.
Area from Perimeter and a Side [H]	Recover the other side, then multiply.
Square Area from Perimeter [H]	The side is a quarter of the perimeter.
Similar Triangles [H]	Corresponding sides share one scale factor.
Ratio of Areas [H]	Area scales with the square of the side ratio.
Area of a Sector [H]	Take the angle's fraction of the whole circle's area.
Arc Length [H]	Take the angle's fraction of the circumference.
Volume of a Cylinder [H]	Volume is the circular base area times the height.
Surface Area of a Cylinder [H]	Add the two circular ends to the curved side.
Surface Area of a Cube [H]	Six identical square faces.
Surface Area of a Box [H]	Three pairs of matching rectangular faces.
Volume of a Sphere [H]	Volume is four-thirds π r cubed.
Volume of a Cone [H]	A cone is one-third of the matching cylinder.
Distance Between Points [H]	Use the horizontal and vertical gaps with the Pythagorean theorem.
Midpoint of a Segment [H]	Average the endpoints' coordinates.
Slope of a Line [H]	Slope is rise over run.
Reflecting a Point [H]	Reflecting over the y-axis negates the x-coordinate.
Translating a Point [H]	Add the horizontal shift to the x-coordinate.
Rotating a Point 90° [H]	A 90° counterclockwise turn sends (x, y) to (−y, x).
Area of a Trapezoid [H]	Average the parallel sides, then multiply by the height.
Area of a Parallelogram [H]	Base times perpendicular height.
Area of a Rhombus [H]	Half the product of the diagonals.
Area of a Ring [H]	Subtract the inner circle's area from the outer.
Diagonal of a Rectangle [H]	The diagonal is the hypotenuse of the two sides.
Space Diagonal of a Box [H]	Combine all three edge lengths under one square root.
Ladder Against a Wall [H]	The wall, ground, and ladder form a right triangle.
The Multiplication Principle [H]	Multiply the choices at each independent stage.
Arranging Distinct Objects [H]	The number of orderings is n factorial.
Arrangements with Repeated Letters [H]	Divide n! by the factorials of each repeat count.
Circular Arrangements [H]	Fix one seat to remove equivalent rotations.
Complementary Probability [H]	Subtract the unwanted probability from 1.
Basic Probability [H]	Favorable outcomes over total outcomes.
Drawing Without Replacement [H]	Multiply the probabilities, updating after the first draw.
Expected Value [H]	Average the outcomes weighted by their probabilities.
Geometric Probability [H]	Probability is the ratio of favorable area to total area.
Inclusion-Exclusion Counting [H]	Add the two counts, then subtract the overlap.
Diagonals of a Polygon [H]	Each vertex connects to all but its two neighbors and itself.
Counting Subsets [H]	Each element is either in or out - that is 2ⁿ.
Even Sums and Parity [H]	A sum is even only from two evens or two odds.
Speed from Distance and Time [H]	Speed is distance divided by time.
Trains Approaching [H]	Add the speeds to get the closing rate.
Catching Up [H]	The gap closes at the difference of the speeds.
Average Speed (Equal Distances) [H]	With equal distances, use the harmonic mean of the speeds.
Combined Filling Rates [H]	Add the individual rates per hour.
Boat and Current [H]	Still-water speed is the average of downstream and upstream.
Weighted Averages [H]	Total everything, then divide by the total count.
A Changing Average [H]	Rebuild the total, add the new value, divide by the new count.
Finding the Median [H]	Sort the values and take the middle one.
The Range of Data [H]	Subtract the smallest value from the largest.
Mean from a Frequency Table [H]	Weight each value by how often it occurs.
Average of Consecutive Integers [H]	The mean is the average of the first and last.
Pie-Chart Angles [H]	A percent of the whole is that percent of 360°.
Days of the Week [H]	Weekdays cycle every 7 days - use the remainder.
Elapsed Time [H]	Convert both times to minutes and subtract.
24-Hour Clock Arithmetic [H]	Hours wrap around every 24 - take the remainder.
Working Backward [H]	Undo each operation with its inverse, in reverse order.
Maximizing a Product [H]	For a fixed sum, the product is largest when the numbers are closest.

Math Kangaroo (Grades 5-6) - Number Puzzles and Arithmetic Strategy · 46 topics

Sharing With Nothing Left Over [H]	Decide which candidate total can be shared equally by testing divisibility instead of dividing.
The Digit Under the Blot [H]	Choose a hidden digit so that the digit sum or the last digit satisfies a divisibility test.
Crossing Out One Digit [H]	Remove the single digit that leaves behind a digit sum with the required divisibility.
Divisibility Hidden in a Product [H]	Decide whether a product is divisible by a number by inspecting its factors instead of multiplying.
The Total That Cannot Be Made [H]	Test which totals can be built from whole packets of two given sizes.
Digits With a Given Sum [H]	Find the largest or smallest number whose digits add to a stated total.
Clues About a Reversed Number [H]	Recover a two-digit number from clues about the number with its digits swapped.
Hidden Digits in an Addition [H]	Recover blotted digits from a written column addition, carrying where needed.
Digits With a Given Product [H]	Find a number from the product of its digits, sometimes with a second clue.
Digits Tied Together by a Rule [H]	Use a stated relation between the digits to list the candidates and pick the one that fits.
Counting Numbers With a Digit Property [H]	Count how many numbers in a range satisfy a condition on their digits.
The Bead at a Far Position [H]	Locate a position inside a repeating pattern by dividing and reading the remainder.
How Many Beads of One Color [H]	Count how often an item occurs in a repeating pattern using whole periods plus the leftover part.
Finding a Number From Its Remainders [H]	Identify a number from the remainders it leaves and the range it lies in.
Remainders After Adding or Multiplying [H]	Work out the remainder of a sum or product from the remainders of its parts.
A Fraction of What Is Left [H]	Take a fraction of the remainder rather than of the original whole.
The Whole From a Fractional Part [H]	Recover the whole quantity when a fraction of it, or the part left over, is known.
Which Fraction Is Nearest [H]	Compare fractions by measuring their distance from a benchmark such as 1/2 or 1.
The Shaded Fraction of a Figure [H]	Read a fraction off a tiled figure and use it to answer a further question.
How Many Fit the Fractions [H]	Use the denominators of a fraction description to pin down the possible size of the whole.
Comparing Percent Amounts [H]	Work out several percent amounts of different bases and compare them.
Two Percent Changes in a Row [H]	Apply a second percent change to the amount produced by the first, not to the original.
Percent More and Percent Fewer [H]	Compare two groups when one is a percent more or a percent fewer than the other.
Percents Against Fractions [H]	Order amounts written as fractions, percents and decimals by rewriting them in one form.
Sharing When the Difference Is Known [H]	Size one share from the difference between two parts of a ratio, then answer the question asked.
Linking Two Ratios Together [H]	Pass through a shared quantity to connect two ratios into one.
From a Ratio to a Fraction [H]	Convert between a ratio of parts and the fraction each part is of the whole.
Replacing One Value in an Average [H]	Track how the average moves when a single value is replaced by another.
The Score Needed Next [H]	Use totals to find the extra value that moves an average to a target.
Removing a Value From an Average [H]	Compare the totals before and after a value is removed to identify it.
Averages of Numbers in a Run [H]	Use the symmetry of a run of consecutive numbers to link its average, ends and total.
Two Rules Taking Turns [H]	Discover a rule in which two different steps alternate, then continue the row.
Running a Rule Backwards [H]	Undo a repeated rule step by step, reversing each operation in turn.
Add the Digit Sum Each Time [H]	Follow a rule whose step depends on the digits of the current number.
Where a Number Sits in a Row [H]	Turn a step rule around to find a position, or the first term past a limit.
Rows Where Every Three Add to the Same Total [H]	Use a constant sum of neighbouring cells to show that the row repeats every three places.
Putting In Plus and Minus Signs [H]	Decide which totals can be reached by choosing plus or minus in front of each number.
Where the Brackets Go [H]	Compare the values an expression takes under every possible single bracketing.
Arranging Digit Cards for the Best Result [H]	Place digits into two numbers so that their sum, product or difference is extreme.
The Order of Two Machines [H]	Compare the results of applying two operations in either order, and undo them.
Finding the Hidden Operations [H]	Choose the operations that make a number sentence true, respecting the order of operations.
Counting Heads and Legs [H]	Split a group of two kinds of creature using the head count and the leg count.
Two or Three Numbers From Their Sums [H]	Find individual amounts from totals and differences by combining the given equations.
Two Kinds of Item, One Bill [H]	Compare two shopping bills so that one item cancels and the other is revealed.
Passing Some Over [H]	Track how a transfer changes two amounts, remembering that one loses exactly what the other gains.
A Journey in Stages [H]	Choose the smallest stage as the unknown and write every other stage in terms of it.

Math Kangaroo (Grades 5-6) - Geometry and Spatial Reasoning · 46 topics

Same Fence, Different Field [H]	Compare the areas of rectangles that share one perimeter, and find the extreme cases.
Shortest Fence for a Given Area [H]	List the whole-number side pairs that give an area and pick the extreme perimeter.
Cutting a Rectangle into Equal Strips [H]	Find the perimeter of one strip, or of all of them, after a rectangle is cut into equal strips.
The Path Around the Pond [H]	Find the area of a border of constant width by subtracting the inner rectangle from the outer one.
Rectangles Built from Identical Tiles [H]	Work out the size of a rectangle assembled from identical tiles, and the fewest tiles that make a square.
The Perimeter of a Staircase [H]	Slide the steps of a staircase outwards to see that its perimeter equals that of the surrounding rectangle.
The Missing Side of an L-Shape [H]	Use the fact that opposite sides of an L-shape add up in pairs to find a missing length or the perimeter.
Two Rugs That Overlap [H]	Add the two areas and subtract the overlap once, because the overlap has been counted twice.
The Perimeter of a Square Tile Shape [H]	Count the exposed unit edges of a shape built from squares instead of adding side lengths.
A Rectangle with a Hole [H]	Subtract the hole for area, but add the hole's edge for the total length of edge.
A Triangle Inside a Rectangle [H]	Use base times height divided by two, noticing that the tip may sit anywhere on the opposite side.
What Fraction Is Shaded [H]	Express a shaded part of a figure as a fraction of the whole in lowest terms.
The Area of a Tilted Shape [H]	Find the area of a tilted lattice figure by subtracting the corner triangles from the surrounding rectangle.
Shaded Against White [H]	Compare a shaded region with the rest of a figure by counting or by subtracting from the whole.
Three Angles on a Straight Line [H]	Use the fact that angles sitting side by side on a straight line total 180 degrees.
Angles Filling a Whole Turn [H]	Share 360 degrees among angles that meet at a point, including equal shares and shares in a ratio.
A Line Drawn Across a Triangle [H]	Chase angles through a triangle cut by a line from one vertex to the opposite side.
Turning Round and Round [H]	Track a direction through repeated turns, remembering that a full turn is 360 degrees.
Slices of a Round Cake [H]	Convert between the center angle of a slice and the fraction of the circle it takes up.
Angles Made by Folding Paper [H]	Use the fact that a fold reflects an angle, creating two equal angles at the fold line.
Which Net Folds into a Cube [H]	Decide whether six joined squares fold into a closed cube by checking that no two squares land on the same face.
Opposite Faces on a Net [H]	Identify which squares of a net become opposite faces once the net is folded up.
From the Flat Net to the Box [H]	Recover a box's dimensions from the rectangles of its net, then find its volume or its card area.
Where the Sixth Square Can Go [H]	Test each free position for a sixth square by checking whether the completed shape folds into a cube.
Numbers on the Net of a Die [H]	Combine the opposite-face rule with the folding of a net to find a missing or extreme face total.
Counting the Squares in a View [H]	Count the squares in a view of a cube stack by taking the tallest pile in each line of sight.
Which View Shows the Most [H]	Work out all three views of a cube stack and compare how many squares each one shows.
What the Stack Looks Like from There [H]	Read off the bar heights of a view, in the correct left-to-right order for the direction of viewing.
Fewest and Most Cubes for Two Views [H]	Find the least or the greatest number of cubes that can produce a given front view and side view.
The Cubes You Cannot See [H]	Count the cubes of a block that have no face on any visible surface, remembering that the floor hides one layer.
Filling the Box with Cubes [H]	Subtract the cubes already placed from the capacity of a box, or from the smallest cube that contains a block.
Reading a Plan of Piles [H]	Use a top view labeled with pile heights to count the cubes, the visible tops, or the cubes still missing.
Glued Faces and Surface [H]	Relate the surface of a solid made of unit cubes to the number of glued face-to-face joins.
Turning a Pattern onto Itself [H]	Measure the rotational symmetry of a pattern by counting the turns that leave it unchanged.
Shading to Make It Symmetric [H]	Add the fewest squares needed for a pattern to gain a mirror line or half-turn symmetry.
How Many Mirror Lines [H]	Test each candidate line of a grid pattern by checking whether every shaded square has a shaded partner.
The Shape After a Turn [H]	Predict how a shape drawn on a grid looks after a quarter turn, a half turn, or a reflection.
Covering a Floor with Tiles [H]	Count the tiles that cover a rectangle exactly, and decide when an exact covering is possible at all.
Covering a Board with Dominoes [H]	Use area and the black-and-white coloring of a board to decide whether dominoes can cover it.
Square Tiles That Fit Exactly [H]	Cover a rectangle with equal square tiles by using the greatest common divisor of its sides.
The Shortest Walk Through the Streets [H]	Find the length of a shortest route along grid streets, including detours forced by closed crossings.
Counting Routes Past a Closed Crossing [H]	Count shortest grid routes when some crossings are closed or one crossing must be used.
An Ant Walking on a Box [H]	Measure routes along the edges of a box, from a single crossing to a tour of every corner.
Drawing a Figure in One Stroke [H]	Count the corners where an odd number of lines meet to decide whether a figure can be drawn in one stroke.
Cutting into Identical Pieces [H]	Divide a figure's area by the number of identical pieces, and scale lengths when the pieces are small copies.
Cut Up and Put Together Again [H]	Use the fact that cutting and rearranging changes the shape but never the total area.

Math Kangaroo (Grades 5-6) - Logic, Counting and Combinations · 46 topics

Lining Up When One Place Is Fixed [H]	Count the orders of a line when one person is tied to an end place, by ordering only the free places.
Neighbours Who Must, or Must Not, Be Together [H]	Treat a group that must stay together as one block, and subtract the together count to get the apart count.
Seats Around a Round Table [H]	Count circular seatings by holding one place fixed, then halve the count when mirror images also match.
Coloring Stripes with No Two Neighbours Alike [H]	Color cells one at a time, counting the choices left for each cell once its neighbours are fixed.
Choosing a Pair When Order Does Not Matter [H]	Count unordered pairs as n(n-1)/2, then adjust for a banned pair or for a pair that must include one girl.
Choosing Three from a Small Group [H]	Count unordered triples as n(n-1)(n-2)/6, then adjust for a member who is fixed or a pair that is banned.
Numbers Built from Given Digits [H]	Count numbers made from a digit set by filling the most restricted place first, such as the last digit or the leading digit.
Choices with One Combination Banned [H]	Multiply the free choices, then subtract the combinations a rule forbids, adding back any removed twice.
Rearranging Letters When Some Repeat [H]	Divide the count of all orders by the orders of each repeated letter, since swapping identical letters changes nothing.
How Many Different Selections [H]	Count selections from n kinds as 2 to the power n, one yes-or-no decision per kind, then remove the selections a rule forbids.
Routes When a Crossing Is Closed [H]	Count shortest routes on a street grid by writing on each crossing the number of routes that reach it, entering 0 at a closed crossing.
Following the One-Way Roads [H]	Count journeys through a one-way network by labeling each town with the number of ways to reach it from the start.
Climbing Steps One or Two at a Time [H]	Count the ways to reach each step as the sum of the ways to reach the steps a single hop below it.
Drawing a Figure in One Stroke [H]	Count the points where an odd number of lines meet: half that number is the smallest number of strokes needed.
Routes That Must Pass a Given Crossing [H]	Split a route at a compulsory crossing and multiply the routes of the two halves, adding the two cases when either of two crossings will do.
Games in an All-Play-All Tournament [H]	Count the games of an all-play-all tournament as the number of pairs of teams, doubling it when every pair meets twice.
Finding the Number of Teams from the Games [H]	Work backwards from the number of games to the number of teams by finding which consecutive pair of numbers has the right product.
Matches in a Knockout Tournament [H]	Count knockout games by counting the losses needed, since every game produces exactly one loss and every player but the champion is knocked out.
Reading a Points Table [H]	Turn wins, draws and losses into points with 3 and 1, and work backwards from a points total to the number of wins or draws.
Games Still to Be Played [H]	Halve the total of the games-played column, because each game is counted once by each of the two teams in it, then subtract from the full fixture list.
When Exactly One Statement Is True [H]	Test each offered number against every statement and keep the one whose number of true statements matches the rule.
How Many of Them Are Lying [H]	Use the fact that all truthful speakers must have said the same number, so the count of speakers claiming a value has to match that value.
Who Broke the Window [H]	Suppose each suspect in turn is guilty, count how many statements that makes true, and keep the suspect whose count fits.
Knights and Liars Talking About Each Other [H]	Pass the known type along the chain: a speaker calling someone a liar is of the opposite type when the claim is true.
Finding the Light Coin on a Balance [H]	Split the coins into three equal piles so that one weighing cuts the search to a third, giving three to the power of the weighings.
Matching People from a Table of Clues [H]	Cross out the impossible squares of a matching grid until each row and each column has exactly one square left.
Houses in a Row [H]	Place the most restricted clue first, such as a pair that must be side by side, and slide it along the row until every clue fits.
Ordering People from Comparisons [H]	Build one line from the comparisons by inserting each person to the correct side of those already placed.
The Finishing Order of a Race [H]	Turn each clue into a gap between finishing places, then fit the fixed blocks into the five places.
Balance Scales and Swapping Shapes [H]	Reduce each balance to the value of one shape, then swap shapes for their equals until only the wanted shape is left.
Who Sits Where at the Round Table [H]	Fix one person to remove the turning, then walk round the table placing each neighbour clue in the seats that remain.
Socks in the Dark [H]	Find the largest unlucky handful that still fails, then add one sock to force the wanted outcome.
Making Sure of Several of One Color [H]	Add up how many of each color can be taken while staying one short of the target, then take one more.
Making Sure of One of Every Kind [H]	Take everything except the scarcest kind for the worst case, then add one so that kind must appear.
How Many People Before Two Must Share [H]	Treat each category as a box: one more person than there are boxes forces two into one box.
The Fullest Box and the Emptiest Box [H]	Share the objects out as evenly as the rules allow, since the fullest box can never be below the average and the emptiest can never be above it.
Worst Case with Gloves and Tickets [H]	Describe the largest handful that still fails, choosing the worse side of every pairing, and then add one.
Probability After Some Are Taken Away [H]	Recount the favourable beads and the total beads after the change, then read the probability as one count over the other.
Two Beads Drawn Without Replacing [H]	Multiply the chance of the first draw by the chance of the second when one bead has already gone, and add the separate color cases.
Two Spins of a Spinner [H]	List the ordered pairs of results as the total count, then count the pairs that satisfy the condition.
Two Dice and a Condition [H]	Work over the thirty-six ordered outcomes of two dice and count the ones that meet the condition.
At Least One, Counted the Short Way [H]	Find the probability of at least one success by taking one minus the probability that every trial fails.
The Chance That a Random Line-Up Works [H]	Divide the number of arrangements that satisfy the condition by the total number of arrangements.
Counting the Ones Left Out [H]	Add the group totals, subtract each overlap once and add the triple overlap back, then subtract from the whole class.
Choosing Seats with No Two Together [H]	Seat the chosen children first and then slot the empty chairs into the gaps, which turns the problem into a plain selection.
How Many Numbers Contain a Given Digit [H]	Count the numbers whose tens digit matches and those whose units digit matches, then subtract the ones counted twice.

Prerequisite material - taught automatically when the diagnostic finds gaps

Arithmetic Foundations · 7 topics

Adding & Subtracting Whole Numbers	Multi-digit addition and subtraction.
Multiplication	Multiplying whole numbers.
Division	Dividing whole numbers.
Order of Operations	Parentheses first, then multiplication/division, then addition/subtraction.
Negative Numbers: Adding & Subtracting	Working with numbers below zero on the number line.
Negative Numbers: Multiplying & Dividing	Sign rules for products and quotients.
Exponents	Repeated multiplication in shorthand.

Expressions & Equations · 2 topics

Evaluating Expressions	Substituting a value for a variable.
One-Step Equations	Undoing a single operation.

Math Kangaroo: Benjamin Depth (Grades 5-6) · 19 topics

Order of Operations	Multiply and divide before you add and subtract.
Greatest Common Divisor	The biggest number dividing both.
Units-Digit Cycles	Last digits of powers repeat on a short loop.
Remainders with Shortcuts	Digit sums reveal remainders by 3 and 9.
Fraction of a Quantity	Divide into equal parts, take some.
Percent of a Number	Percent means out of a hundred.
Sharing in a Ratio	Count the shares, size one share, scale up.
Working Back from an Average	The average fixes the total.
Distance, Speed, Time	Distance is speed times time.
Handshakes & Pairs	Count pairs, not people.
Counting Arrangements	Fill each place in turn.
Surface Area of a Cube	Six equal square faces.
Volume of a Box	Volume multiplies the three dimensions.
Age Problems	Write both ages, then compare.
Simple Probability	Favorable outcomes over all outcomes.
Multi-Step Money	Total the cost, then subtract from what you paid.
The nth Term	Jump straight to a far term.
Comparing Fractions	Common denominators (or cross-multiplying) settle it.
Perimeter and Area	Perimeter is the border; area is the fill.

Math Kangaroo: Benjamin - Deep (Grades 5-6) · 72 topics

Adding Unlike Fractions	Rewrite over a common denominator, then add.
Subtracting Unlike Fractions	Common denominator first, then subtract.
Multiplying Fractions	Multiply straight across, then reduce.
Fraction of a Fraction	Take a part of a part by multiplying.
Percents to Fractions	Percent means out of one hundred.
Finding the Percent	Compare the part to the whole, times 100.
Finding the Whole from a Percent	Undo the percent to recover the whole.
Percent Increase	Add the percent of the original amount.
Percent Decrease	Subtract the percent of the original amount.
Simplifying Fractions	Divide top and bottom by their common factor.
Solving Proportions	Equal ratios scale by the same factor.
Three-Part Ratios	Total the parts, size one share, scale each.
Equivalent Ratios	Multiply both parts by the same number.
Divisibility Rules	Digit tests decide divisibility quickly.
Counting Prime Factors	Break a number down to primes.
GCF in Word Problems	The biggest equal-groups number is the GCF.
LCM in Word Problems	Cycles realign at the least common multiple.
Counting Multiples	Divide the range end by the step.
Missing Digit for Divisibility	Adjust the digit sum to a multiple of 3.
Even and Odd Reasoning	Parity follows simple rules.
Powers of Two	Match the value to its exponent.
Order of Operations with Powers	Exponents before multiply, multiply before add.
Working Inside Brackets	Do the parentheses before anything else.
Fibonacci-Style Sequences	Each term is the sum of the previous two.
Missing Term in a Pattern	Find the step, then fill the gap.
Digit Reversal Difference	The difference is always a multiple of 9.
Digit Sums	Add up the individual digits.
Digit Products	Multiply the digits together.
Consecutive Number Sums	The middle of three consecutives is the average.
Working Backward	Undo each step in reverse order.
Number Frames	Subtract the known parts from the target total.
Area of a Triangle	Half of base times height.
Area of Composite Shapes	Add or subtract simple rectangles.
Perimeter of Composite Shapes	Trace the outer boundary all the way round.
Supplementary Angles	Supplementary angles add to 180°.
Angles Around a Point	Angles around a point add to 360°.
Vertical and Adjacent Angles	Opposite angles are equal; neighbours make 180°.
Triangle Angle Sum	A triangle's angles add to 180°.
Isosceles Triangle Angles	Equal sides give equal base angles.
Exterior Angle of a Triangle	An exterior angle equals the two far interior angles.
Exterior Angle of a Regular Polygon	The exterior angles always total 360°.
Missing Side from Area	Divide the area by the known side.
Square Side from Area or Perimeter	Undo squaring for area, divide by 4 for perimeter.
Surface Area of a Box	Add the areas of all six faces.
The Painted Cube	Sort the small cubes by position.
Opposite Faces of a Die	Opposite faces of a die sum to seven.
Faces, Edges, and Vertices	Count the flat faces, straight edges, and corners.
Counting Stacked Cubes	Add the cubes layer by layer.
Volume as Unit Cubes	Volume counts the unit cubes inside.
Lines of Symmetry	A line of symmetry folds a shape onto itself.
Distance on the Coordinate Plane	Points sharing a coordinate differ only along one axis.
Area from Coordinates	Side lengths are differences of coordinates.
The Multiplication Principle	Multiply the choices at each stage.
Arranging in a Row	Count the choices for each position.
Counting Grid Paths	Every path uses the same total of right and up moves.
Probability of the Complement	The chance of not-happening is 1 minus the chance.
Probability with Two Dice	Count the favourable pairs out of 36.
Spinner Probability	Favourable sections over total sections.
Two-Set Venn Diagrams	Add the groups but count the overlap once.
Venn Diagrams: Neither	Subtract those in at least one from the total.
The Pigeonhole Principle	Plan for the worst case, then add one.
Counting Squares on a Grid	Add squares of each possible size.
Counting True Statements	Judge each statement on its own.
Knights and Knaves	Test each possibility against every statement.
Ordering by Deduction	Chain the comparisons into one line.
Mixture Problems	Total the cost, divide by the total amount.
Finding the Mean	Add the values, divide by how many.
Finding the Median	Sort, then take the middle value.
Combining Averages	Weight each average by its group size.
The Fence-Post Problem	A straight line has one more post than gaps.
Cuts and Pieces	Making k pieces needs k−1 cuts.
Frequency Tables	Multiply each value by how often it occurs.

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