Math Kangaroo: Benjamin (Grades 5-6)

306 core topics + 9 prerequisite topics taught as needed · approximately 97 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 · 7 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.

Math Kangaroo: Benjamin Depth (Grades 5-6) · 30 topics

Order of Operations [M]	Multiply and divide before you add and subtract.
Greatest Common Divisor [M]	The biggest number dividing both.
Least Common Multiple [M]	The first moment two cycles line up again.
Divisible by This or That [M]	Add the two counts, then remove the double count.
Units-Digit Cycles [M]	Last digits of powers repeat on a short loop.
Remainders with Shortcuts [M]	Digit sums reveal remainders by 3 and 9.
Fraction of a Quantity [M]	Divide into equal parts, take some.
Percent of a Number [M]	Percent means out of a hundred.
Sharing in a Ratio [M]	Count the shares, size one share, scale up.
Working Back from an Average [M]	The average fixes the total.
Distance, Speed, Time [M]	Distance is speed times time.
The 3×3 Magic Square [M]	Use a full line that already has two numbers.
Handshakes & Pairs [M]	Count pairs, not people.
Diagonals of a Polygon [M]	Each vertex connects to all but three others.
Counting Arrangements [M]	Fill each place in turn.
Sum of a Run of Numbers [M]	Pair the ends; each pair is the same.
Triangular Numbers [M]	Rows 1, 2, 3, … stack into a triangle.
Surface Area of a Cube [M]	Six equal square faces.
Volume of a Box [M]	Volume multiplies the three dimensions.
Clock Angles at Any Time [M]	The minute hand moves the hour hand too.
Age Problems [M]	Write both ages, then compare.
Counting Primes [M]	A prime has exactly two divisors.
Counting Factors [M]	Read the count off the prime powers.
Growing Figures [M]	Find the start and the step.
Simple Probability [M]	Favorable outcomes over all outcomes.
Multi-Step Money [M]	Total the cost, then subtract from what you paid.
The nth Term [M]	Jump straight to a far term.
Comparing Fractions [M]	Common denominators (or cross-multiplying) settle it.
Estimating by Rounding [M]	Round first, then compute.
Perimeter and Area [M]	Perimeter is the border; area is the fill.

Math Kangaroo: Benjamin - Deep (Grades 5-6) · 131 topics

Adding Unlike Fractions [M]	Rewrite over a common denominator, then add.
Subtracting Unlike Fractions [M]	Common denominator first, then subtract.
Multiplying Fractions [M]	Multiply straight across, then reduce.
Dividing Fractions [M]	Multiply by the reciprocal of the divisor.
Fraction of a Fraction [M]	Take a part of a part by multiplying.
Mixed to Improper Fractions [M]	Whole times denominator, plus the numerator.
Multiplying Decimals [M]	Multiply as whole numbers, then place the point.
Dividing Decimals [M]	Scale both numbers so the divisor is whole.
Decimals to Fractions [M]	Read place value, write over a power of ten, reduce.
Fractions to Decimals [M]	A fraction is a division problem.
Percents to Fractions [M]	Percent means out of one hundred.
Finding the Percent [M]	Compare the part to the whole, times 100.
Finding the Whole from a Percent [M]	Undo the percent to recover the whole.
Percent Increase [M]	Add the percent of the original amount.
Percent Decrease [M]	Subtract the percent of the original amount.
Comparing Decimals [M]	Compare place by place from the left.
Rounding Decimals [M]	Look at the next digit: 5 or more rounds up.
Fraction Halfway Between [M]	The midpoint is the average of the two fractions.
Simplifying Fractions [M]	Divide top and bottom by their common factor.
Solving Proportions [M]	Equal ratios scale by the same factor.
Unit Rate [M]	Divide to find the amount per one.
Comparing Deals (Best Buy) [M]	Compare price per item, not total price.
Scaling a Recipe [M]	Multiply every amount by the same factor.
Map Scale [M]	Multiply map distance by the scale.
Three-Part Ratios [M]	Total the parts, size one share, scale each.
Equivalent Ratios [M]	Multiply both parts by the same number.
Average Speed [M]	Average speed is total distance over total time.
Combined Work Rate [M]	Add the per-hour rates, then invert.
Applying a Rate [M]	Multiply the rate by the amount of time.
Divisibility Rules [M]	Digit tests decide divisibility quickly.
Sum of Divisors [M]	List every factor, then add them up.
Counting Prime Factors [M]	Break a number down to primes.
GCF in Word Problems [M]	The biggest equal-groups number is the GCF.
LCM in Word Problems [M]	Cycles realign at the least common multiple.
Perfect Squares [M]	Perfect squares come from whole numbers squared.
Counting Multiples [M]	Divide the range end by the step.
Missing Digit for Divisibility [M]	Adjust the digit sum to a multiple of 3.
Even and Odd Reasoning [M]	Parity follows simple rules.
Adding and Subtracting Integers [M]	Move along the number line by the sign.
Multiplying Integers [M]	Same signs positive, different signs negative.
Absolute Value [M]	Absolute value is distance from zero.
Distance on the Number Line [M]	Distance is the absolute difference.
Temperature Change [M]	A rise adds, a fall subtracts.
Evaluating Powers [M]	A power is repeated multiplication.
Powers of Two [M]	Match the value to its exponent.
Square Roots [M]	The square root undoes squaring.
Comparing Powers [M]	Compute each power, then compare.
Order of Operations with Powers [M]	Exponents before multiply, multiply before add.
Working Inside Brackets [M]	Do the parentheses before anything else.
Geometric Sequence Term [M]	Multiply by the ratio, term by term.
Fibonacci-Style Sequences [M]	Each term is the sum of the previous two.
Square Number Sequence [M]	The nth square number is n × n.
Sum of Odd Numbers [M]	The first n odd numbers add to n².
Sum of Even Numbers [M]	The first n even numbers add to n(n+1).
Summing an Arithmetic Sequence [M]	Average the ends, times the count.
Second-Difference Patterns [M]	When first differences grow evenly, look one level deeper.
Missing Term in a Pattern [M]	Find the step, then fill the gap.
Digit Reversal Difference [M]	The difference is always a multiple of 9.
Digit Sums [M]	Add up the individual digits.
Digit Products [M]	Multiply the digits together.
Next Palindrome [M]	A palindrome reads the same both ways.
Consecutive Number Sums [M]	The middle of three consecutives is the average.
Working Backward [M]	Undo each step in reverse order.
Number Pyramids [M]	Add pairs going up, level by level.
Number Frames [M]	Subtract the known parts from the target total.
Area of a Triangle [M]	Half of base times height.
Area of a Parallelogram [M]	Base times height, like a rectangle.
Area of a Trapezoid [M]	Average the parallel sides, times height.
Area of Composite Shapes [M]	Add or subtract simple rectangles.
Perimeter of Composite Shapes [M]	Trace the outer boundary all the way round.
Area of a Circle (as kπ) [M]	Area is π times the radius squared.
Circumference of a Circle (as kπ) [M]	Circumference is π times the diameter.
Complementary Angles [M]	Complementary angles add to 90°.
Supplementary Angles [M]	Supplementary angles add to 180°.
Angles Around a Point [M]	Angles around a point add to 360°.
Vertical and Adjacent Angles [M]	Opposite angles are equal; neighbours make 180°.
Triangle Angle Sum [M]	A triangle's angles add to 180°.
Isosceles Triangle Angles [M]	Equal sides give equal base angles.
Exterior Angle of a Triangle [M]	An exterior angle equals the two far interior angles.
Polygon Angle Sum [M]	Split the polygon into triangles.
Interior Angle of a Regular Polygon [M]	Share the total angle sum equally.
Exterior Angle of a Regular Polygon [M]	The exterior angles always total 360°.
Right Triangle Sides [M]	The squares of the legs add to the hypotenuse squared.
Missing Side from Area [M]	Divide the area by the known side.
Square Side from Area or Perimeter [M]	Undo squaring for area, divide by 4 for perimeter.
Scaling Similar Figures [M]	Lengths scale by k, areas by k².
Surface Area of a Box [M]	Add the areas of all six faces.
Cube Surface from Volume [M]	Recover the edge, then find the faces.
The Painted Cube [M]	Sort the small cubes by position.
Opposite Faces of a Die [M]	Opposite faces of a die sum to seven.
Faces, Edges, and Vertices [M]	Count the flat faces, straight edges, and corners.
Counting Stacked Cubes [M]	Add the cubes layer by layer.
Volume as Unit Cubes [M]	Volume counts the unit cubes inside.
Lines of Symmetry [M]	A line of symmetry folds a shape onto itself.
Reflecting a Point [M]	Reflection flips the sign of one coordinate.
Distance on the Coordinate Plane [M]	Points sharing a coordinate differ only along one axis.
Area from Coordinates [M]	Side lengths are differences of coordinates.
Quadrants of the Plane [M]	Signs of the coordinates pick the quadrant.
The Multiplication Principle [M]	Multiply the choices at each stage.
Arranging in a Row [M]	Count the choices for each position.
Counting Grid Paths [M]	Every path uses the same total of right and up moves.
Outcomes of Coin Flips [M]	Two outcomes per flip multiply together.
Probability of the Complement [M]	The chance of not-happening is 1 minus the chance.
Probability with Two Dice [M]	Count the favourable pairs out of 36.
Spinner Probability [M]	Favourable sections over total sections.
Two-Set Venn Diagrams [M]	Add the groups but count the overlap once.
Venn Diagrams: Neither [M]	Subtract those in at least one from the total.
The Pigeonhole Principle [M]	Plan for the worst case, then add one.
Counting Rectangles in a Grid [M]	Choose two vertical and two horizontal lines.
Counting Squares on a Grid [M]	Add squares of each possible size.
Counting True Statements [M]	Judge each statement on its own.
Knights and Knaves [M]	Test each possibility against every statement.
Ordering by Deduction [M]	Chain the comparisons into one line.
Coin Problems [M]	Use the count and the total value together.
Mixture Problems [M]	Total the cost, divide by the total amount.
Finding the Mean [M]	Add the values, divide by how many.
Finding the Median [M]	Sort, then take the middle value.
Finding the Mode [M]	The mode is the value that appears most.
Finding the Range [M]	Subtract the smallest from the largest.
Combining Averages [M]	Weight each average by its group size.
Meeting in the Middle [M]	Add the speeds to close the gap.
Catching Up [M]	The gap closes at the speed difference.
Ages in the Future [M]	Everyone ages by the same number of years.
The Fence-Post Problem [M]	A straight line has one more post than gaps.
Cuts and Pieces [M]	Making k pieces needs k−1 cuts.
Climbing Floors [M]	Floors 1 to n have n−1 flights between them.
Elapsed Time [M]	Count the minutes from start to finish.
Minutes Between Two Times [M]	Convert both times to minutes, then subtract.
Day-of-Week Problems [M]	Days repeat every 7, so use the remainder.
Reading a Bar Graph [M]	Read the values, then combine them.
Frequency Tables [M]	Multiply each value by how often it occurs.

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.

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