AMC 8 (Middle School)

266 core topics + 112 prerequisite topics taught as needed · approximately 119 hours of instruction including spaced review · premium unlock $24.99

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

Geometry · 14 topics

Angle Relationships [E]	Vertical, complementary, and supplementary angle pairs.
Parallel Lines & Transversals [E]	Angle pairs formed when a transversal crosses parallel lines.
Triangle Angle Sum [E]	The three angles of a triangle always add to 180°.
The Pythagorean Theorem [M]	In a right triangle, a² + b² = c².
Distance & Midpoint [M]	Measuring segments in the coordinate plane.
Similar Triangles [M]	Same shape, different size: corresponding sides are proportional.
Perimeter & Area [E]	Measuring around and inside basic shapes.
Circles: Area & Circumference [M]	C = 2πr and A = πr².
Composite Areas [H]	Adding and subtracting simple shapes to measure a complicated one.
Volume: Prisms & Cylinders [M]	Volume = base area × height.
Volume: Cones, Pyramids & Spheres [M]	Pointed solids hold one third of the matching prism; spheres use 4/3 πr³.
Surface Area [M]	The total area of all the faces of a solid.
Special Right Triangles [H]	The 45-45-90 and 30-60-90 side ratios.
Arc Length & Sector Area [H]	A central angle takes the same fraction of the circumference and the area.

Math Olympiad: MOEMS & AMC 8 · 14 topics

Modular Arithmetic [M]	Working with remainders directly.
Counting Divisors [M]	Divisor count from the prime factorization.
GCD & LCM Relationships [M]	The identity gcd(a,b) · lcm(a,b) = a · b.
Digit Problems [M]	Reasoning about the digits of a number.
Units Digit of Powers [M]	Combine last-digit cycles, then add.
Permutations [M]	Ordered arrangements.
Combinations [M]	Unordered selections: n choose k.
The Pigeonhole Principle [M]	Guaranteeing a repeat.
Complementary Counting [M]	Count the opposite, then subtract.
Counting & Probability [H]	Favorable outcomes over total outcomes.
Vieta's Formulas [H]	Relating roots to coefficients without solving.
Telescoping Sums [H]	A sum that collapses to its endpoints.
Sums, Parity & Invariants [M]	Elegant closed forms and even/odd reasoning.
Recursion & Sequences [M]	Each term built from earlier ones.

AMC 8: Advanced Practice · 45 topics

Missing-Digit Divisibility [H]	Use the digit-sum rule to solve for an unknown digit.
Remainders of Powers [H]	Powers cycle through remainders; find the pattern.
LCM in Cycles Problems [H]	Events that repeat coincide at the least common multiple.
GCF in Grouping Problems [H]	The largest equal grouping is the greatest common factor.
Sum of Prime Factors [H]	Factor into primes, then add the distinct ones.
Counting Divisors [H]	Add one to each prime exponent and multiply.
Units Digit of a Sum of Powers [H]	Find each tail digit from its cycle, then add.
Base Conversion [H]	Each place is a power of the base.
Trailing Zeros of a Factorial [H]	Count the factors of 5 in the product.
Two-Digit Number Puzzles [H]	Turn the clues into equations in the two digits.
Multiplication Principle with Restrictions [H]	Count choices one position at a time as options shrink.
Permutations [H]	Ordered selections multiply the shrinking choices.
Combinations [H]	Unordered choices divide out the reorderings.
Probability with Two Dice [H]	Count favorable ordered pairs out of 36.
Probability Without Replacement [H]	The second draw sees one fewer marble.
Complementary Probability [H]	P(at least one) = 1 − P(none).
Arrangements with a Restriction [H]	Glue the pair together, then account for their internal order.
Counting Lattice Paths [H]	Choose which steps go right among all the steps.
Venn Diagrams from Word Problems [H]	Use the counts of each region to solve for the overlap.
Percent of a Number [H]	Turn the percent into hundredths, then multiply.
Working Back from a Percent [H]	A discount multiplies the price by (1 − rate); undo it by dividing.
Successive Percent Changes [H]	Apply each change to the running amount, not the original.
Weighted Averages [H]	Weight each average by how many it represents.
Changing an Average [H]	Compare the old total with the new required total.
Three-Part Ratios [H]	Count the total shares, then value one share.
Average Speed [H]	Average speed is total distance over total time.
Combined Work Rates [H]	Add the per-hour rates, then invert.
Arithmetic Sequences [H]	Add the common difference n − 1 times.
Geometric Sequences [H]	Multiply by the common ratio n − 1 times.
Finding Consecutive Integers [H]	The middle of an odd run is the sum divided by the count.
Composite Areas [H]	Add and subtract rectangles.
Shaded Fractions of a Grid [H]	Compare shaded squares to the total squares.
The Pythagorean Theorem [H]	In a right triangle, leg² + leg² = hypotenuse².
Angles in Polygons [H]	The interior angles of an n-gon sum to 180(n − 2).
Circle Area and Circumference [H]	Area is πr²; circumference is 2πr.
Similar Figures and Area Ratios [H]	Areas scale as the square of the side ratio.
Surface Area of a Box [H]	Add the areas of the three pairs of faces.
Volume of a Cylinder [H]	Volume is the base area times the height.
Counting Rectangles in a Grid [H]	Choose two vertical lines and two horizontal lines.
Age Word Problems [H]	Sum and difference pin down two unknown ages.
Coin Value Problems [H]	Assume all the cheaper coin, then account for the extra value.
Clock Hand Angles [H]	Track how far each hand has moved from 12.
Fibonacci-Style Recursion [H]	Each term is the sum of the two previous.
Pigeonhole Guarantees [H]	Fill every category to k − 1, then add one more.
Parity of a Sum [H]	The parity of a total depends only on how many odd numbers it has.

AMC 8 - Deep · 50 topics

Last Two Digits of a Power [H]	Work modulo 100; the last two digits cycle.
Prime Powers in a Factorial [H]	Count multiples of p, then p², then p³ inside n!.
Three Simultaneous Remainders [H]	Satisfy two conditions, then step by their product for the third.
Solving a Linear Congruence [H]	Test x = 1, 2, 3, … until the remainder matches.
Adding in Another Base [H]	Carry whenever a column reaches the base.
Digits of a Repeating Decimal [H]	Long division cycles; find where the digit lands.
Counting Numbers Containing a Digit [H]	Count those without the digit, then subtract.
Largest Square Factor [H]	Pair up the primes; each pair forms a square.
Counting Square Divisors [H]	A square divisor uses each prime an even number of times.
Sum of Proper Divisors [H]	Add every divisor except the number itself.
Inclusion-Exclusion with Three Sets [H]	Add singles, subtract pairs, add back the triple.
Paths Through a Required Point [H]	Multiply the paths of the two legs.
Paths Avoiding a Point [H]	Subtract the paths through the point from all paths.
Derangements (No Fixed Points) [H]	Count arrangements where nothing lands in its own place.
Circular Seating with a Pair Together [H]	Glue the pair, seat the block in a circle, then order the pair.
At Least One, by Complement [H]	Subtract the all-men committees from all committees.
Distributing with Empties Allowed [H]	Arrange the objects and the dividers together.
Conditional Probability [H]	Update the counts to reflect what already happened.
At Least k Successes [H]	Add the ways for k, k+1, …, n successes.
Expected Value with Unequal Chances [H]	Weight each payoff by its probability, then add.
Who Wins Going First [H]	Sum the chances of winning on each of your turns.
Three-Digit Numbers by Digit Sum [H]	Count digit choices with a fixed total, minding the ranges.
Systems of Two Equations [H]	Assume all cheapest, then account for the extra cost.
Consecutive Integers from a Product [H]	Estimate with a square root, then check neighbors.
Sum of Consecutive Squares [H]	Use the closed formula n(n+1)(2n+1)/6.
Sum of Consecutive Cubes [H]	It equals the square of the triangular number.
Finding the Number of Terms [H]	The sum grows predictably; solve for the count.
Geometric Mean [H]	The geometric mean is the square root of the product.
Sum and Product of Roots [H]	Read the root sum and product off the coefficients.
Quadratic Sequences [H]	Constant second differences extend the pattern.
Undoing Repeated Operations [H]	Reverse each step: add, then divide, from the end back.
Alternating Sums [H]	Pair consecutive terms to collapse the sum.
Median from a Frequency Table [H]	Add up the counts to locate the middle value.
Ratio from a Difference [H]	One 'share' is the difference divided by the ratio gap.
Mixing Concentrations [H]	Total the pure acid, then divide by total volume.
Area from Coordinates (Shoelace) [H]	Cross-multiply the coordinates in a loop.
Space Diagonal of a Box [H]	Apply the Pythagorean theorem in three dimensions.
Missing Length in Similar Figures [H]	Corresponding sides share one scale factor.
Area of a Sector [H]	A sector is a fraction of the whole circle.
Volume of a Cone [H]	A cone is one third of the matching cylinder.
Volume of a Sphere [H]	Use the four-thirds pi r cubed formula.
The Exterior Angle Theorem [H]	An exterior angle equals the two far interior angles added.
Reflecting a Point [H]	Reflecting flips the sign of one coordinate.
Rotating a Point 90° [H]	A quarter turn sends (x, y) to (−y, x).
Heron's Formula [H]	Use the semiperimeter to find area from three sides.
Overlap of Two Rectangles [H]	Overlap in x times overlap in y.
Classifying Triangles by Sides [H]	Compare the longest side squared to the other two squared.
Three Workers Together [H]	Add the three per-hour rates, then invert.
A Train Passing a Platform [H]	The train must cover its own length plus the platform.
When Clock Hands Overlap [H]	The hands meet every 720/11 minutes.

Olympiad - Number Sense and Algebraic Reasoning · 48 topics

Digits Forced by a Divisor [H]	Split the divisor into coprime parts and let each part pin down a digit.
Divisors That Are Always There [H]	Among k consecutive integers, every residue class modulo k appears exactly once.
GCD of Two Linear Forms [H]	A common divisor of two linear forms also divides any integer combination of them.
The 1001 Trick [H]	Because 1001 = 7 x 11 x 13, a six-digit number with a repeated block is divisible by all three.
Prime Powers Inside a Factorial [H]	Count multiples of p, then of p squared, then of p cubed, and add.
Which Claim Holds for Every n [H]	A statement about all integers needs a residue argument, while one counterexample kills it.
Forcing a Divisor to Be Constant [H]	Reduce the dividend modulo the divisor until only a constant remains, then list its divisors.
A Parity That Survives Every Move [H]	Replacing two numbers by their difference leaves the parity of the total unchanged.
Coloring a Board [H]	Color the board like a chessboard: every domino covers one square of each color.
Flipping a Fixed Number of Switches [H]	Each move changes the number of ON switches by an amount with the same parity as the move size.
Choosing the Argument That Settles It [H]	An impossibility proof needs a quantity that no move can change, not an arithmetic coincidence.
The Handshake Sum Must Be Even [H]	Adding everyone's handshake count double counts each handshake, so that total is even.
Reachable Positions Form One Progression [H]	Swapping one jump for the other always changes the finish by the same fixed amount.
A Total That Drifts by a Fixed Amount [H]	Track how much the total changes at each move, not which numbers were chosen.
Adding One Turns a Sum Rule into a Product Rule [H]	Since ab + a + b + 1 = (a+1)(b+1), the product of all numbers increased by one is preserved.
Color Counts Modulo Three [H]	Each meeting changes two color counts by one and the third by two, so differences modulo three are preserved.
Every Amount Is a Combination of the Two Jugs [H]	Any amount reachable with two jugs is a multiple of the greatest common divisor of their sizes.
A Score That Does Not Depend on Your Choices [H]	Each split scores exactly the pairs of stones it separates, so the total counts all pairs once.
Undoing a Repeated Rule [H]	Reverse each step in turn: undoing halve-then-subtract means add back, then double.
Three Rounds Run in Reverse [H]	Undo a doubling round by halving the receivers and returning what was given.
Shortest Route to a Target Number [H]	Work backwards from the target: undo a multiplication when possible, otherwise undo an addition.
Losing Positions Found Backwards [H]	A position is losing exactly when every move from it leads to a winning position.
Make the Others as Small as Possible [H]	To maximise one member of a set with a fixed total, minimise everything else.
Pushing the Middle Value Up [H]	To maximise the median, make the values below it minimal and those above it as tight as allowed.
Pair Up the Forbidden Partners [H]	Split the numbers into pairs that add to the forbidden total and take one from each pair.
How Many Can Clear the Bar [H]	Count how many members can meet a threshold by giving everyone the least the rules allow.
Squeezing the Largest Member Down [H]	If the biggest number is M, the whole set fits inside M, M-1, ..., M-k+1, which caps the total.
Remainders as Pigeonholes [H]	Two numbers differ by a multiple of m exactly when they leave the same remainder on division by m.
Pigeonholes of Different Sizes [H]	A color with fewer than k socks caps the worst case at its own supply.
None Divides Another [H]	Writing each number as an odd number times a power of two sorts them into chains.
Something Beats the Average [H]	Some box holds at least the average, rounded up, and an even spread shows no more is guaranteed.
Factoring an Equation into a Product [H]	Adding the right constant turns xy + ax + by into a product of two brackets.
Counting Differences of Squares [H]	Every way of writing N as a difference of squares comes from a factor pair of N whose two factors have the same parity.
Building Symmetric Expressions [H]	Any symmetric expression in x and y can be rebuilt from their sum and product alone.
Powers of x Plus Its Reciprocal [H]	Multiplying x^n + 1/x^n by x + 1/x produces the next power and the previous one.
A Sum of Fourth Powers That Factors [H]	The identity a^4 + 4b^4 = (a^2 + 2ab + 2b^2)(a^2 - 2ab + 2b^2) factors a sum of fourth powers.
Alternating Squares Collapse [H]	Pair the terms so each pair is a difference of squares equal to the sum of its two bases.
Splitting a Power Minus One [H]	For every divisor d of e, the number m^d - 1 divides m^e - 1.
Spotting the Hidden Factorisation [H]	A number close to a square or to a round power usually hides an algebraic factorisation.
Telescoping with a Gap [H]	Split 1/(k(k+d)) as (1/d)(1/k - 1/(k+d)) so terms cancel d places apart.
A Product That Telescopes [H]	Factor each term as (k-1)(k+1)/k^2 so neighbouring numerators and denominators cancel.
Rationalising Makes It Telescope [H]	Multiplying by the conjugate turns 1/(root k + root (k+1)) into a difference of square roots.
Three Factors in the Denominator [H]	Half the difference of two neighbouring products of pairs gives 1/(k(k+1)(k+2)).
Adding Rates, Not Times [H]	Let the unknown be the tank per hour rate of each pipe; the three pairwise sums then add to twice the total rate.
One Worker Leaves Partway [H]	Measure the fraction of the job finished before the change, then divide what remains by the surviving rate.
The Escalator Steps You Never Tread [H]	The escalator supplies the steps you do not walk, so steps walked plus steps carried is a constant.
Meeting, Then Finishing [H]	Let the meeting time be the unknown; each rider covers the other's first leg in the time stated.
Grass That Grows While the Cows Eat [H]	Take one cow's daily ration as the unit and treat the starting grass and the daily growth as two unknowns.

Olympiad - Geometry and Combinatorial Reasoning · 47 topics

Areas Along a Divided Side [H]	Triangles with the same apex and bases on one line have areas in the ratio of those bases.
Shrinking a Triangle at One Corner [H]	Cutting both sides at a vertex multiplies the area by the product of the two ratios.
A Chain of Midpoints [H]	Each median drawn in a triangle halves the area of the triangle it is drawn in.
A Point Inside a Rectangle [H]	Opposite triangles from an interior point of a rectangle have areas summing to half the rectangle.
The Midpoint Quadrilateral [H]	Joining the midpoints of a convex quadrilateral produces a parallelogram of exactly half the area.
Pick's Theorem on a Lattice [H]	A lattice polygon has area I + B/2 - 1, where I and B count interior and boundary lattice points.
Which Fact Forces Equal Areas [H]	Two triangles have equal areas exactly when equal bases are paired with equal heights.
Rectangles Inside a Grid [H]	A rectangle in a grid is fixed by choosing two horizontal lines and two vertical lines.
Tilted Squares in a Point Array [H]	Every square in a lattice sits inside a unique upright square that circumscribes it.
Counting Inside a Fan [H]	In a fan of cevians every triangle is fixed by choosing two of the rays from the apex.
Parallelograms From Two Line Families [H]	Two lines from each of two parallel families bound exactly one parallelogram.
Triangles With Collinear Points Removed [H]	Count all vertex triples, then subtract the triples that are collinear and so degenerate.
Intersections of a Family of Lines [H]	Each pair of lines meets once unless the pair is parallel, so subtract the parallel pairs.
Rectangles in an L-Shape [H]	Count rectangles in the full grid and subtract exactly those that reach into the removed block.
A Board With Two Squares Removed [H]	A domino always covers one square of each color, so a color imbalance blocks any covering.
Counting Tilings of a Strip [H]	Tilings of a 2 by n strip satisfy a recurrence found by looking at the last column only.
Deficient Boards and L-Trominoes [H]	A tromino covering of a deficient board uses exactly one third of the remaining unit squares.
Covering a Board With Long Pieces [H]	An m by n board takes 1 by k pieces exactly when k divides m or k divides n.
Fewest Tiles on a Floor [H]	Fewest tiles means most 2 by 2 tiles, and an even-coordinate invariant caps how many fit.
Which Covering Argument Is Sound [H]	An impossibility needs an invariant that every piece respects; a possibility needs an actual covering.
Degrees Add Up to Twice the Edges [H]	The friendship counts of all members add to twice the number of friendly pairs.
When Everyone Has the Same Degree [H]	Equal degrees d for n people are achievable exactly when d is at most n - 1 and n times d is even.
Walking Every Path Exactly Once [H]	A connected network can be walked in one route exactly when it has no odd corner or exactly two.
How Many Strokes to Draw a Figure [H]	A connected figure with 2k odd points needs exactly k strokes, and k is never smaller.
Counting Routes Through a Network [H]	Routes to a town total the routes to each town with a road into it.
Roads You Can Close [H]	A connected network on n towns keeps only n - 1 roads at minimum, so the rest are removable.
Can These Handshake Counts Happen [H]	A list of handshake counts needs an even total and no entry above the number of other people.
Regions Cut by Straight Lines [H]	Each new line adds one region for every region it crosses, that is one more than its intersections.
Regions Inside a Circle of Chords [H]	Each interior crossing comes from a unique set of four marked points, which drives the region count.
Pentagons Forced by Euler's Formula [H]	Counting each edge and vertex through the faces turns Euler's formula into a fixed pentagon count.
Regions Cut by Circles [H]	A new circle meeting the earlier ones in 2(k - 1) points gains exactly that many regions.
Integer Triangles of a Given Perimeter [H]	List sides in increasing order and let the triangle inequality bound the largest side.
Counting Coin Combinations by Cases [H]	Fix the number of the largest coin first; the rest of each case is then a short count.
Splitting a Number into Three Parts [H]	Order the three parts to count each split once, then sweep the smallest part.
Numbers With Ordered Digits [H]	A strictly increasing digit string is just a choice of digits, since the order is then forced.
Choosing an Exhaustive Case Split [H]	A case split must cover every possibility, overlap nowhere, and be fine enough to settle the claim.
Largest Product With a Fixed Sum [H]	Parts that differ by more than one can be evened out to increase the product, so the best split is level.
Kings That Never Touch [H]	Cutting the board into 2 by 2 blocks caps the kings, and an odd-row odd-column placement attains the cap.
Bishops on Their Diagonals [H]	No two bishops may share a diagonal, and two of the corner diagonals can never both be used.
A Set With No Fixed Difference [H]	Splitting the numbers into chains that step by d turns the problem into alternating along each chain.
Shading Without a Full 2 by 2 Block [H]	Disjoint 2 by 2 blocks each need one blank square, and blanking the even-even squares attains that.
Does the Construction Attain the Bound [H]	An extremal answer needs both an upper bound and one arrangement that reaches it and obeys every rule.
Bracelets Up to Rotation [H]	Averaging the colorings left unchanged by each rotation corrects the overcount from turning the ring.
Painting the Faces of a Cube [H]	The 24 rotations of a cube each fix their own colorings, and the answer is the average of those counts.
Shadings Up to Rotation and Reflection [H]	Group shadings into families joined by the eight symmetries of the square and count one per family.
Rook Placements That Mirror Themselves [H]	A self-mirroring rook placement pairs up its columns, so it is built from fixed points and swapped pairs.
Using Symmetry Without Overcounting [H]	Dividing by the number of symmetries is valid only when every object is repeated that many times.

Olympiad (Elementary) - Reasoning Puzzles and Strategy · 48 topics

Smallest Number with a Given Digit Sum [H]	Use as few digits as possible, then make the leading digit as small as possible.
Greatest Number with a Given Digit Product [H]	Make the hundreds digit as large as possible, then the tens digit.
Hidden Digits and Divisibility [H]	Split the divisor into coprime parts and apply one divisibility rule at a time.
Swapping Digits and Place Value [H]	Swapping the digits changes a two-digit number by 9 times the digit gap.
Counting One Digit in a Range [H]	Count the digit once for the units place and once for the tens place.
Digits Used in Page Numbers [H]	Peel off the one-digit and two-digit pages, then share out what is left.
A Number That Is a Multiple of Its Digit Sum [H]	Write the number as 10t + u and compare it with k(t + u).
Logic Grids [H]	Cross out every impossible pairing, then read off the row that is forced.
Putting People in Order from Clues [H]	Build a single line from the comparisons, using the end positions first.
Truth Tellers and Liars [H]	Test each suspect in turn and count how many statements come out true.
Finding a Number from Clues [H]	Apply the clue that leaves fewest candidates first, then filter what remains.
Which Clue Is Enough [H]	A clue decides the answer only when exactly one candidate survives it.
An Invariant on the Board [H]	Track a quantity the move changes in a fixed way, so the order of moves cannot matter.
Signs, Parity and Possible Totals [H]	Flipping one sign changes the total by an even amount, so the parity never changes.
Why a Cut Board Cannot Be Covered [H]	Color the board and compare the color counts a covering would need.
Counting Squares of One Color [H]	Colors alternate, so an odd total leaves the corner color one square ahead.
Counting Handshakes Twice [H]	Adding everybody's handshake count counts each handshake twice.
The Lockers Problem [H]	A locker ends open exactly when its number has an odd number of divisors.
Patterns with Growing Gaps [H]	When the gaps grow by a fixed amount, add up the gaps instead of guessing.
Justifying the Rule of a Pattern [H]	A rule is right only if it reproduces every term shown, not just the first jump.
Working Back Along an Adding Sequence [H]	Each term equals the one after it minus the one before it, so the sequence can be run backwards.
Alternating Sums of a Long List [H]	Group the terms in pairs, each pair contributing the same amount.
Counting Inside a Repeating Pattern [H]	Count whole repeats first, then handle the part-repeat at the end.
Half and a Bit More, Undone [H]	Undo the last day first: add back what was eaten extra, then double.
Doubling, Read Backwards [H]	Doubling forwards means halving backwards, one day per halving.
Passing Counters Until All Are Equal [H]	Undo each turn in reverse: halve the two who received, and give the total back.
Undoing a Discount and a Coupon [H]	Undo the steps in reverse order: add the coupon back before undoing the percent.
Ages from a Sum and a Future Ratio [H]	Write the future condition in terms of the younger age now, then use the sum.
Ages with Reversed Digits [H]	Write both ages with the same two digits, then test the few digit pairs that fit.
The Fewest Coins That Make an Amount [H]	Take as many of the largest coin as possible, then repeat on what is left.
Three Kinds of Coin, One Unknown [H]	Use the linking condition to write every count in terms of one unknown.
The Largest Total You Cannot Buy [H]	List the reachable totals by remainder class, then look at where each class starts.
Arrangements with Two People Together [H]	Glue the pair into one block, arrange the blocks, then swap inside the pair.
Arrangements with Two People Apart [H]	Count all the orders and subtract the ones where the pair is together.
Keeping One Group Apart in a Row [H]	Seat the other group first, then drop the restricted children into the gaps.
Grid Routes Past a Closed Corner [H]	Count every route, then subtract the routes that use the closed corner.
Counting All the Rectangles in a Grid [H]	A rectangle is fixed by choosing two of the horizontal lines and two of the vertical lines.
Too Many and Too Few [H]	The gap between the leftover and the shortfall is the extra given to each child.
Going Round a Circle in Steps [H]	The marks repeat after the least common multiple of the step and the circle size.
Splitting into Unordered Piles [H]	List the piles from smallest to largest so each share is counted once.
Divisors That Leave a Given Remainder [H]	Subtract the remainder, then count the divisors that are bigger than it.
Fair Shares That Must All Differ [H]	Count the unordered splits with different sizes, then multiply by the orderings.
Chained Balance Puzzles [H]	Convert step by step through the middle item, scaling both sides to match.
Fewest Weighings on a Balance [H]	Each weighing has three outcomes, so it can cut the suspects to a third.
Measuring with Two Jugs [H]	Every amount you can reach is a whole number of jugfuls added and removed.
Distances on a Marked Ruler [H]	Every pair of marks gives a distance, but repeated gaps must be counted once.
Choosing the First Step [H]	Identify the quantity every later step depends on, and round it the way the situation demands.
Working Back Through Fractions of What Is Left [H]	Each fraction acts on the amount remaining, so undo the steps from the end.

Prerequisite material - taught automatically when the diagnostic finds gaps

Arithmetic Foundations · 8 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.
Square Roots	Undoing a square.

Fractions · 6 topics

Equivalent Fractions	Different fractions can name the same amount.
Simplifying Fractions	Reducing a fraction to lowest terms.
Adding Fractions (Like Denominators)	Same-denominator addition.
Adding Fractions (Unlike Denominators)	Rewrite over a common denominator first.
Multiplying Fractions	Multiply straight across.
Dividing Fractions	Multiply by the reciprocal.

Decimals, Percents & Ratios · 1 topics

Ratios & Proportions

Two quantities that scale together.

Expressions & Equations · 4 topics

Evaluating Expressions	Substituting a value for a variable.
Combining Like Terms	Adding the coefficients of matching variable parts.
The Distributive Property	Multiplying across a sum.
One-Step Equations	Undoing a single operation.

Linear Functions · 1 topics

The Coordinate Plane

Locating points with (x, y) pairs.

Quadratics & Polynomials · 8 topics

Adding & Subtracting Polynomials	Combining polynomials by collecting like terms.
Multiplying Binomials (FOIL)	Expanding products of binomials.
Factoring Out the GCF	Undoing the distributive property.
Factoring Trinomials	Reversing FOIL: finding two numbers that multiply to c and add to b.
Special Factoring Patterns	Difference of squares and perfect-square trinomials.
Solving x² = k	Taking square roots of both sides - remembering ±.
Completing the Square	Turning any quadratic into a perfect square plus a constant.
The Quadratic Formula	x = (−b ± √(b² − 4ac)) / 2a solves any quadratic.

Radicals & Exponentials · 1 topics

Simplifying Radicals

Pulling perfect-square factors out of a square root.

Math Kangaroo: Benjamin & Cadet · 2 topics

Systematic Counting	Counting without missing or double-counting.
Number Patterns	Finding the rule behind a sequence.

MOEMS Depth: Divisions E & M · 42 topics

Divisibility Rules	Test divisibility without doing the division.
Remainders	The amount left over after fair sharing.
Digit Sums	Add up the digits of a number.
Reversing Two-Digit Numbers	The gap between a number and its reversal is a multiple of 9.
Counting Divisors	Read the divisor count off the prime factorization.
GCF and LCM	Common factors and common multiples.
Sums of Consecutive Integers	Pair the ends to add a run quickly.
Factors and Factor Pairs	Factors come in pairs that multiply to the number.
Gauss Sums	Add 1 to n by pairing the ends.
Finding the nth Term	Count the steps from the start.
Averages and a Missing Value	The total is the average times the count.
A Fraction of a Quantity	Divide into equal parts, then take some of them.
Successive Fractions (What's Left)	Each fraction acts on the amount remaining, not the original.
Sharing in a Ratio	Split into equal 'shares', then hand them out.
Proportions and Scaling	Scale both quantities by the same factor.
Unit Rates	How much for exactly one.
Distance, Speed, and Time	Distance equals speed times time.
Area of Rectangles	Rows times columns of unit squares.
Area of Triangles	Half of the base times the height.
Counting Squares in a Grid	Count squares of each size separately, then add.
The Painted Cube	Where a small cube sits decides how many faces are painted.
The Multiplication Principle	Multiply the number of independent choices.
Arrangements in a Row	Multiply the shrinking number of choices for each spot.
Handshakes and Choosing Pairs	Each handshake is a chosen pair.
Counting Grid Paths	Every shortest path is a sequence of the same moves.
Two-Set Venn Counting	Do not double-count the overlap.
The Pigeonhole Principle	Plan for the worst case, then add one.
Age Problems	Both people age by the same amount.
Chickens and Cows	Assume all of one kind, then fix the leg count.
Coin Problems	Assume the cheaper coin, then account for the extra value.
Work Rate (Inverse Proportion)	More workers means proportionally less time.
Working Backwards	Undo each step in reverse order.
Fence Posts (Off-by-One)	A straight row of posts has one more post than gaps.
Even and Odd (Parity)	Even/odd follows simple rules under addition and multiplication.
Remainders that Cycle	Adding to a number cycles its remainder.
Perfect Squares	Perfect squares come from squaring whole numbers.
Counting Multiples	Divide to count multiples up to a limit.
Multiplying Patterns	Find the constant ratio, then apply it again.
Total Value Problems	Multiply each value by its count, then add.
Differences of Squares	Consecutive squares differ by an odd number.
Grouping and Leftovers	The leftover is the remainder after grouping.
Factorials	Multiply every whole number down to 1.

MOEMS - Deep (Divisions E & M) · 39 topics

LCM of Three Numbers	Build up the least common multiple two numbers at a time.
GCF with Equal Remainders	Subtract the remainder, then take a common factor.
Counting Primes	A prime has exactly two divisors: 1 and itself.
Digit Products	Multiply the digits together instead of adding them.
Divisible by One or the Other	Add the two counts, then subtract the overlap once.
Counting Perfect Cubes	Perfect cubes come from cubing whole numbers.
Highest Power that Divides	Keep dividing by the prime until it no longer goes evenly.
The Second Largest Divisor	Divide by the smallest prime factor.
Counting Subsets	Each element is either in or out - two choices each.
Seating Around a Circle	Fix one person to remove the identical rotations.
Arranging Letters with Repeats	Divide out the reorderings of identical letters.
Choosing with a Required Member	Seat the required person first, then fill the rest.
Sharing Identical Objects	Place dividers in the gaps between the objects.
Diagonals of a Polygon	Each vertex connects to all but itself and its two neighbors.
Triangles from Points	Every choice of 3 points makes one triangle.
Coloring with Adjacency Rules	The first region is free; each next avoids its neighbor.
Two-Digit Numbers by Digit Sum	List the tens digit and read off the units digit.
How Many n-Digit Numbers	The first digit can't be zero; the rest are free.
Choosing from Two Groups	Multiply the independent choices from each group.
Numbers from Their Sum and Difference	Half the sum plus half the difference gives the larger number.
Pattern to Formula (Toothpicks)	Find how much each new figure adds, then build a rule.
Balance Puzzles	Replace one object with its equal in the other.
The Median	Sort the numbers and take the middle one.
Geometric Series Sums	Use the doubling shortcut instead of adding term by term.
Inserting Arithmetic Means	Count the equal gaps the inserted numbers create.
Iterating a Rule	Apply the same operation repeatedly, tracking the result.
Area of a Parallelogram	Base times height, using the straight-across height.
Area of a Trapezoid	Average the parallel sides, then multiply by the height.
Area of a Square from its Diagonal	A square's area is half the square of its diagonal.
Angles with Parallel Lines	Same-side interior angles add to 180°.
Area from Coordinates	The side lengths are the coordinate differences.
Euler's Formula for Solids	Vertices minus edges plus faces always equals 2.
Area of a Ring (Annulus)	Subtract the inner circle's area from the outer circle's.
The Triangle Inequality	The third side lies strictly between the sum and the difference.
Approaching Objects	Add the speeds to get the closing rate.
Catching Up	Subtract the speeds to get the closing rate.
Unit Conversion	Multiply by how many small units fill one big unit.
Filling Against a Drain	Subtract the draining rate from the filling rate.
Percent More Than	Compare the increase to the original amount.

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