Math Olympiad: MOEMS Division E (Grades 4-6)

272 core topics + 29 prerequisite topics taught as needed · approximately 96 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 Olympiad: MOEMS & AMC 8 · 7 topics

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.
Permutations [M]	Ordered arrangements.
The Pigeonhole Principle [M]	Guaranteeing a repeat.
Telescoping Sums [H]	A sum that collapses to its endpoints.
Sums, Parity & Invariants [M]	Elegant closed forms and even/odd reasoning.

MOEMS Depth: Divisions E & M · 55 topics

Divisibility Rules [H]	Test divisibility without doing the division.
Remainders [H]	The amount left over after fair sharing.
Digit Sums [H]	Add up the digits of a number.
Reversing Two-Digit Numbers [H]	The gap between a number and its reversal is a multiple of 9.
Counting Divisors [H]	Read the divisor count off the prime factorization.
GCF and LCM [H]	Common factors and common multiples.
Units Digit of a Power [H]	Last digits of powers repeat in a short cycle.
Sums of Consecutive Integers [H]	Pair the ends to add a run quickly.
Factors and Factor Pairs [H]	Factors come in pairs that multiply to the number.
Common Multiples in a Range [H]	Numbers divisible by both a and b are multiples of their LCM.
Gauss Sums [H]	Add 1 to n by pairing the ends.
Arithmetic Sequence Sums [H]	Average the first and last term, then multiply by the count.
Finding the nth Term [H]	Count the steps from the start.
Averages and a Missing Value [H]	The total is the average times the count.
A Fraction of a Quantity [H]	Divide into equal parts, then take some of them.
Successive Fractions (What's Left) [H]	Each fraction acts on the amount remaining, not the original.
Sharing in a Ratio [H]	Split into equal 'shares', then hand them out.
Proportions and Scaling [H]	Scale both quantities by the same factor.
Unit Rates [H]	How much for exactly one.
Distance, Speed, and Time [H]	Distance equals speed times time.
Perimeter of Rectangles [H]	Add up the distance around the outside.
Area of Rectangles [H]	Rows times columns of unit squares.
Area of Triangles [H]	Half of the base times the height.
Area of Composite Shapes [H]	Break the figure into rectangles, then add or subtract.
Angles in a Triangle [H]	The three angles always add to 180 degrees.
Angles on a Straight Line [H]	Angles along a straight line add to 180 degrees.
Counting Squares in a Grid [H]	Count squares of each size separately, then add.
The Painted Cube [H]	Where a small cube sits decides how many faces are painted.
Volume of a Box [H]	Stack layers of unit cubes.
The Multiplication Principle [H]	Multiply the number of independent choices.
Arrangements in a Row [H]	Multiply the shrinking number of choices for each spot.
Handshakes and Choosing Pairs [H]	Each handshake is a chosen pair.
Simple Probability [H]	Favorable outcomes over total outcomes.
Counting Grid Paths [H]	Every shortest path is a sequence of the same moves.
Two-Set Venn Counting [H]	Do not double-count the overlap.
The Pigeonhole Principle [H]	Plan for the worst case, then add one.
Age Problems [H]	Both people age by the same amount.
Chickens and Cows [H]	Assume all of one kind, then fix the leg count.
Coin Problems [H]	Assume the cheaper coin, then account for the extra value.
Work Rate (Inverse Proportion) [H]	More workers means proportionally less time.
Working Backwards [H]	Undo each step in reverse order.
Fence Posts (Off-by-One) [H]	A straight row of posts has one more post than gaps.
Days of the Week [H]	The weekday pattern repeats every 7 days.
Clock Angles [H]	Each hour mark is 30 degrees apart.
Magic Squares [H]	The magic sum is the total divided by the number of rows.
Even and Odd (Parity) [H]	Even/odd follows simple rules under addition and multiplication.
Remainders that Cycle [H]	Adding to a number cycles its remainder.
Perfect Squares [H]	Perfect squares come from squaring whole numbers.
Counting Multiples [H]	Divide to count multiples up to a limit.
Multiplying Patterns [H]	Find the constant ratio, then apply it again.
Total Value Problems [H]	Multiply each value by its count, then add.
Differences of Squares [H]	Consecutive squares differ by an odd number.
Grouping and Leftovers [H]	The leftover is the remainder after grouping.
Average Speed of a Round Trip [H]	Average speed is total distance over total time, not the mean of speeds.
Factorials [H]	Multiply every whole number down to 1.

MOEMS - Deep (Divisions E & M) · 86 topics

LCM of Three Numbers [H]	Build up the least common multiple two numbers at a time.
GCF with Equal Remainders [H]	Subtract the remainder, then take a common factor.
Sum of Divisors [H]	Add up every factor, from 1 up to the number itself.
Counting Even Divisors [H]	An even divisor must keep at least one factor of 2.
Counting Primes [H]	A prime has exactly two divisors: 1 and itself.
Digit Products [H]	Multiply the digits together instead of adding them.
Reversing Three-Digit Numbers [H]	The gap is always a multiple of 99.
Counting Palindromes [H]	A palindrome reads the same in both directions.
Converting to Another Base [H]	Repeatedly divide, reading remainders from bottom to top.
Two Remainders at Once [H]	Step through one list of remainders until the other matches.
Summing Multiples [H]	Factor out the common multiple, then use a Gauss sum.
Counting Non-Multiples [H]	Count the total, then remove the multiples.
Divisible by One or the Other [H]	Add the two counts, then subtract the overlap once.
Counting Perfect Cubes [H]	Perfect cubes come from cubing whole numbers.
Highest Power that Divides [H]	Keep dividing by the prime until it no longer goes evenly.
Recovering a Number from GCF and LCM [H]	The product of two numbers equals GCF times LCM.
The Second Largest Divisor [H]	Divide by the smallest prime factor.
Digital Roots [H]	Adding digits repeatedly lands on the remainder mod 9.
Counting Subsets [H]	Each element is either in or out - two choices each.
Seating Around a Circle [H]	Fix one person to remove the identical rotations.
Arranging Letters with Repeats [H]	Divide out the reorderings of identical letters.
Choosing with a Required Member [H]	Seat the required person first, then fill the rest.
Sharing Identical Objects [H]	Place dividers in the gaps between the objects.
Diagonals of a Polygon [H]	Each vertex connects to all but itself and its two neighbors.
Triangles from Points [H]	Every choice of 3 points makes one triangle.
Coloring with Adjacency Rules [H]	The first region is free; each next avoids its neighbor.
Two-Digit Numbers by Digit Sum [H]	List the tens digit and read off the units digit.
How Many n-Digit Numbers [H]	The first digit can't be zero; the rest are free.
Choosing from Two Groups [H]	Multiply the independent choices from each group.
Spinner Probability by Angle [H]	A region's chance is its angle over the full 360°.
Probability with a Deck of Cards [H]	Count the favorable cards out of 52.
Probability of Both Events [H]	Multiply the probabilities of independent events.
Probability an Event Does Not Happen [H]	Subtract the event's probability from 1.
Expected Value [H]	Average the outcomes, weighted by how likely each is.
Geometric Probability [H]	Compare the target area to the total area.
Numbers from Their Sum and Difference [H]	Half the sum plus half the difference gives the larger number.
Finding Consecutive Even Numbers [H]	Write each even number in terms of the first.
Pattern to Formula (Toothpicks) [H]	Find how much each new figure adds, then build a rule.
Balance Puzzles [H]	Replace one object with its equal in the other.
The Median [H]	Sort the numbers and take the middle one.
The Mode [H]	The mode is the value that appears most often.
The Range [H]	Subtract the smallest value from the largest.
Geometric Series Sums [H]	Use the doubling shortcut instead of adding term by term.
Inserting Arithmetic Means [H]	Count the equal gaps the inserted numbers create.
Iterating a Rule [H]	Apply the same operation repeatedly, tracking the result.
Average of Consecutive Integers [H]	The average is the midpoint of the run.
Area of a Parallelogram [H]	Base times height, using the straight-across height.
Area of a Trapezoid [H]	Average the parallel sides, then multiply by the height.
Area of a Square from its Diagonal [H]	A square's area is half the square of its diagonal.
Perimeter of an L-Shape [H]	Cutting a corner notch leaves the perimeter unchanged.
Rectangle from Area and Perimeter [H]	Half the perimeter is the sum of length and width.
Angles with Parallel Lines [H]	Same-side interior angles add to 180°.
Exterior Angles of a Polygon [H]	The exterior angles always add to 360°.
Complements and Supplements [H]	Complements make 90°; supplements make 180°.
Angles Around a Point [H]	Angles around a single point add to 360°.
Base Angles of an Isosceles Triangle [H]	The two equal sides sit opposite two equal angles.
Distance Between Points [H]	Make a right triangle from the horizontal and vertical gaps.
Midpoint of a Segment [H]	The midpoint averages the endpoints' coordinates.
Area from Coordinates [H]	The side lengths are the coordinate differences.
Cube Edge from Surface Area [H]	A cube has six equal square faces.
Cube Edge from Volume [H]	The volume of a cube is the edge cubed.
Euler's Formula for Solids [H]	Vertices minus edges plus faces always equals 2.
Scale Drawings [H]	Multiply the map distance by the scale factor.
Area of a Ring (Annulus) [H]	Subtract the inner circle's area from the outer circle's.
The Triangle Inequality [H]	The third side lies strictly between the sum and the difference.
Classifying Angles [H]	Compare the angle to 90° and 180°.
Approaching Objects [H]	Add the speeds to get the closing rate.
Catching Up [H]	Subtract the speeds to get the closing rate.
Boats and Currents [H]	The current helps one way and hinders the other.
Simple Interest [H]	Interest is principal times rate times time.
Mixture Prices [H]	The blended price is the total cost over the total weight.
Unit Conversion [H]	Multiply by how many small units fill one big unit.
Average Speed over Timed Legs [H]	Total distance divided by total time.
Filling Against a Drain [H]	Subtract the draining rate from the filling rate.
Elapsed Time [H]	Convert both clock times to minutes, then subtract.
Counting Days Inclusively [H]	Subtract the dates, then add one for both endpoints.
A Clock that Runs Fast [H]	The error grows by the same amount each hour.
Percent More Than [H]	Compare the increase to the original amount.
Fractions to Percents [H]	A percent is the fraction scaled to a denominator of 100.
From a Part to the Whole [H]	Find the value of one share, then count all the shares.
Shadows and Similar Triangles [H]	Height and shadow keep the same ratio for everything.
Meshing Gears [H]	The gear with fewer teeth spins faster.
Replacing One Member of an Average [H]	One swap shifts the total by n times the change in average.
A Percent of a Percent [H]	Take the percents one after the other.
Remainders Add [H]	The remainder of a sum is the sum of the remainders.
Sum of the First n Odd Numbers [H]	The running total of odd numbers is always a perfect square.

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.

Olympiad (Elementary) - Number & Operation Puzzles · 38 topics

A Missing Digit in a Column Addition [H]	Read one column of an addition at a time and let the carry decide what the hidden digit must be.
A Missing Digit in a Subtraction [H]	Undo a subtraction column by column and watch for the column that had to borrow.
A Missing Digit in a Multiplication [H]	Recover a hidden digit of a product by dividing, or by testing the ten digits that could fill the box.
Letters Standing for Digits [H]	Turn each letter into its place value, then use the arithmetic to pin the letters down.
A Box Forced by a Divisibility Rule [H]	Use the digit-sum test and the last-digit tests to decide which digits can fill a box.
A Digit Found from the Ones Digit Alone [H]	The ones digit of a product depends only on the ones digits, so it can identify a hidden digit by itself.
One Less Than a Common Multiple [H]	A number that is always one short of dividing evenly is one less than a common multiple.
The Remainder of a Product [H]	Replace each factor by its own remainder before multiplying, then take the remainder again.
Counting the Numbers with a Given Remainder [H]	Numbers with the same remainder form an evenly spaced list, so counting them is counting steps.
The Remainder After a Rule Is Applied [H]	Carry the remainder alone through the arithmetic instead of carrying the whole number.
The Next Multiple Up or Down [H]	Divide, keep the whole-number part, and step one multiple in the direction the question asks.
How Many Common Factors [H]	Every common factor of two numbers is a factor of their greatest common factor, so list the factors of that one number.
A Factor Pair from a Product and a Clue [H]	List the factor pairs of the product and pick out the pair that also matches the second clue.
Making a Perfect Square [H]	Pair up the prime factors, and multiply or divide by exactly the primes that are left unpaired.
The Greatest Number of Equal Bags [H]	Splitting two piles into identical bags with nothing left over means dividing by a common factor, and the most bags come from the greatest one.
Adding Up the Prime Factors [H]	Break a number all the way down to primes, then decide whether repeats are counted once or every time.
Jumps Along a Number Line [H]	Equal jumps mean repeated addition, so multiply by the number of jumps and add the starting point.
Reading Equally Spaced Marks [H]	Find the gap between two labeled marks, divide by the number of spaces, and step from there.
Two Counts That Land Together [H]	Two skip counts meet on the common multiples of their step sizes, shifted by wherever they started.
How Many Numbers Are in the List [H]	Count the steps between the first and last number, then add one for the number you started on.
Undoing a Chain of Operations [H]	Run the chain in reverse from the answer, turning every operation into its opposite.
Undoing a Run of Gains and Losses [H]	Total the gains and losses once, then step backwards from the final score to whatever the question wants.
Working Back Through a Sharing [H]	Give the shared amounts back, take away anything that was found, and the starting pile reappears.
Passing Counters to Even Things Up [H]	A transfer moves the gap by twice the amount passed, while the total never changes.
The Score Needed for a Target Average [H]	Turn both averages into totals; the score needed is the difference between the two totals.
Averaging Two Groups of Different Sizes [H]	Add the two totals and divide by the two counts; never average the averages.
Averages of Pairs [H]	Each pair average gives a pair total, and the three pair totals together count every number twice.
The Value That Was Removed [H]	Compare the total before with the total after: their difference is exactly what left the set.
When the Leftovers Need One More [H]	A leftover group still needs a whole container, so divide and then round up unless the division comes out even.
How Many More for an Equal Share [H]	The number still needed is the divisor minus the remainder, and it is zero when the division already comes out even.
Full Boxes and the Partly Filled One [H]	The quotient counts the full boxes and the remainder fills the last one, so read whichever the question wants.
Finding the Group Size from the Leftover [H]	Take the leftover away first: the number of children must divide what is left, and must be bigger than any leftover.
Counting Numbers with a Given Digit Sum [H]	Fix the digits you can and list the possibilities for the rest, in order, so none is missed.
All the Digits Used in a Run of Numbers [H]	Group the run into tens: within one ten the ones digits always contribute the same total.
A Number Minus Its Own Digit Sum [H]	Write the number in place value and subtract: the ones digit cancels, leaving a multiple of nine.
Arranging Digits for an Extreme Value [H]	Put the digits that matter most in the places that are worth most, then check the few close rivals.
Regrouping an Expanded Form [H]	A place can hold more than nine of its own units, and the extra ones simply carry into the next place.
What a Digit Is Worth Where It Stands [H]	A digit's value is the digit multiplied by the value of its place, so compare values, not digits.

Olympiad (Elementary) - Geometry, Counting & Logic · 38 topics

Squares of One Chosen Size in a Grid [H]	Slide a square of a fixed size over the grid and count the positions its top left corner can take.
Rectangles That Contain a Marked Square [H]	A rectangle is fixed by its two horizontal edges and its two vertical edges, so containing a cell is a condition on each choice separately.
Rectangles Sorted by Area or Perimeter [H]	List the shapes that meet the size condition first, then count where each shape can be placed.
Squares Inside an L or a Cross [H]	Count the squares of each size separately and reject every position that pokes out of the figure.
Finding the Grid From the Count [H]	Read the counting formula backwards: the count fixes the missing dimension because the formula never repeats a value.
Triangles in a Triangular Grid [H]	Sort the triangles by size and then by which way they point, and count each family with its own rule.
A Fan of Rays Cut by a Crossbar [H]	Every triangle here has its apex at P, so choosing two rays and one of the two parallel lines names it.
Triangles Made by Straight Lines [H]	Start from every set of three lines and subtract the sets that fail to close up into a triangle.
Perimeter After a Piece Is Cut Away [H]	Removing area does not automatically change perimeter: check which edges are lost and which new edges appear.
The Cross-Shaped Figure [H]	Split the cross into a middle square and four arms, and count its twelve boundary edges by length.
The Frame Around a Picture [H]	The frame is the big rectangle minus the small one, and the small one has both dimensions cut by twice the width.
Sheets That Overlap [H]	Area covered is the sum of the pieces with every doubly counted overlap taken back out once.
The Perimeter of a Block of Unit Squares [H]	For a rectangle of unit squares the perimeter depends only on the two side counts, so the squarest factor pair wins.
Area of a Shape Drawn on Grid Paper [H]	Box the shape in the smallest rectangle with sides on the grid lines and take away the right triangles left in the corners.
Cutting a Rectangle Into Equal Squares [H]	A square of side k tiles the rectangle exactly when k divides both sides, so the possible sizes are the common factors.
Cut Into Strips and Put Back Together [H]	Cutting and rejoining keeps the area fixed while the perimeter changes, so track the two side lengths of every arrangement.
Pieces Made by Straight Cuts [H]	Cuts one way make strips and cuts the other way make columns, so the pieces multiply and each direction contributes one more than its cut count.
Cutting Off the Largest Square Again and Again [H]	Stripping the largest square repeatedly is the subtraction method for the greatest common factor, so the last square has that side.
Routes That Visit a Checkpoint [H]	Split a route at the checkpoint and multiply the count of the first leg by the count of the second.
Routes That Use a Particular Street [H]	A route uses a block exactly when it reaches one end and leaves from the other, so count the two legs and multiply.
Routes Through a Map With Corners Missing [H]	Write the number of routes on each corner in turn: it is the sum of the numbers on the corner to the west and the corner to the south.
Spelling a Word Along a Path [H]	Each step offers a fixed number of choices, so multiply the choices, and subtract the paths that run through a forbidden letter.
Handshakes Between Two Groups [H]	Handshakes across two groups multiply, handshakes inside one group are pairs, and the two families never overlap.
When Every Pair Plays Twice [H]	Count the pairs first and then multiply by how many times each pair meets, keeping divisions separate.
Points in a League Table [H]	Every game hands out exactly two points whatever the result, so the total points is fixed by the number of games alone.
Rounds and Byes in a Knockout [H]	Every match removes exactly one player, and every round roughly halves the field, so count losers for matches and doublings for rounds.
Handshakes That Did Not Happen [H]	Count all the pairs first and then subtract the pairs that are excluded, taking care to count each excluded pair once.
Exactly One of the Two [H]	Split every pupil into four boxes - only the first, only the second, both, neither - and every question becomes addition.
Three Activities at Once [H]	Add the three totals, take back the three overlaps, then put the middle region back once.
Reading a Two-Way Table [H]	Every row and every column of a two-way table adds to its own total, which turns any missing entry into a subtraction.
The Smallest and Largest Possible Overlap [H]	Push the two sets as far apart as the room allows for the smallest overlap, and nest one inside the other for the largest.
At Least Two Out of Three [H]	A pairwise count already contains the children in all three, so decide how many times the middle region should be counted before adding.
Counting From Both Ends of a Line [H]	Two counts of the same child overlap on that child, so the line length is one less than their sum.
Times and Gaps in a Finishing Order [H]	Lay the runners out on a time line and let each clue fix one gap, then read any other gap straight off.
Pinning Down a Ranking From Clues [H]	Turn each clue into a picture of who sits above whom, then merge the pictures until only one full order survives.
Mirror Lines and Turns of a Pattern [H]	Test each candidate move one at a time and check that every shaded square lands on a shaded square.
Shading More Squares to Make It Symmetric [H]	Shade the partner of every shaded square, and then the partner of every square you just shaded, until nothing new appears.
Turns That Leave a Figure Unchanged [H]	The turns that fix a figure are exactly the multiples of one smallest turn, and that smallest turn must divide 360.

Prerequisite material - taught automatically when the diagnostic finds gaps

Arithmetic Foundations · 4 topics

Adding & Subtracting Whole Numbers	Multi-digit addition and subtraction.
Multiplication	Multiplying whole numbers.
Division	Dividing whole numbers.
Exponents	Repeated multiplication in shorthand.

Fractions · 4 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.

Math Kangaroo: Benjamin & Cadet · 2 topics

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

Math Kangaroo: Pre-Écolier & Écolier · 2 topics

Counting Pictures & Objects	Careful counting beats fast counting.
Sides & Corners	Shapes hide multiplication; cuts make one more piece.

Elementary Math Skills · 4 topics

Place Value & Comparing	Each digit's position multiplies its value by ten.
Multi-Digit Multiplication	Split by place value, multiply each part, add.
Long Division	Divide, multiply, subtract, bring down - repeat.
Multi-Step Word Problems	One sentence at a time: what do I know, what changes, what's asked?

Grade 4: Angles, Lines & Measurement · 4 topics

Angles as Turns	An angle measures how far one ray turns away from another.
Classifying Triangles	Sort triangles by their angles or by how many sides match.
Classifying Quadrilaterals	Name quadrilaterals by their parallel sides and right angles.
Lines of Symmetry	A line of symmetry folds a shape exactly onto itself.

Olympiad - Geometry and Combinatorial Reasoning · 9 topics

Pick's Theorem on a Lattice	A lattice polygon has area I + B/2 - 1, where I and B count interior and boundary lattice points.
Rectangles Inside a Grid	A rectangle in a grid is fixed by choosing two horizontal lines and two vertical lines.
Counting Inside a Fan	In a fan of cevians every triangle is fixed by choosing two of the rays from the apex.
Triangles With Collinear Points Removed	Count all vertex triples, then subtract the triples that are collinear and so degenerate.
Intersections of a Family of Lines	Each pair of lines meets once unless the pair is parallel, so subtract the parallel pairs.
Rectangles in an L-Shape	Count rectangles in the full grid and subtract exactly those that reach into the removed block.
Regions Cut by Straight Lines	Each new line adds one region for every region it crosses, that is one more than its intersections.
Bracelets Up to Rotation	Averaging the colorings left unchanged by each rotation corrects the overcount from turning the ring.
Shadings Up to Rotation and Reflection	Group shadings into families joined by the eight symmetries of the square and count one per family.

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