Success in the Math Kangaroo competition is not about knowing more advanced mathematics — it is about thinking more creatively. The most accomplished Math Kangaroo students are not necessarily those who have memorized the most formulas, but those who have built a rich toolkit of problem-solving strategies and know when to deploy each one. The good news is that these strategies are not innate talents — they can be learned, practiced, and mastered by any motivated student. This guide presents the ten most powerful problem-solving techniques used by Math Kangaroo champions around the world. For each technique, you will learn what it is, when to use it, how to practice it, and how it appears in real competition problems. By the end of this article, your child will have a mental toolkit that transforms Math Kangaroo problems from intimidating obstacles into inviting puzzles.
I. Why Strategies Matter More Than Knowledge
Math Kangaroo problems are carefully designed to reward creative thinking over rote memorization. You will rarely encounter a problem that can be solved simply by applying a formula you learned in class. Instead, you will face puzzles that require you to see the problem from a new angle, to reorganize information in a clever way, or to discover a hidden pattern. This is why problem-solving strategies are so valuable — they are the "keys" that unlock the doors to solutions.
| Common Myth | Reality |
| "I need to know advanced math to do well." | Math Kangaroo problems are solvable with grade-level mathematics. What matters is how you think, not how much you know. |
| "Some people are just naturally good at math." | Problem-solving is a skill, not a gift. With deliberate practice, anyone can improve dramatically. |
| "I either see the solution immediately or I never will." | Most problems are solved through systematic exploration — trying approaches, refining ideas, and gradually converging on the answer. |
| "Memorizing solutions to past problems is the best preparation." | Understanding the strategies behind solutions is far more valuable than memorizing specific answers, because new problems require new combinations of strategies. |
With this mindset, let us explore the ten techniques that will transform your child's approach to Math Kangaroo.
II. Technique #1: Draw a Picture or Diagram
The single most useful technique for Math Kangaroo — and perhaps for all of mathematics — is to draw a picture. When a problem seems confusing, a diagram transforms abstract words into concrete visual information that your brain can process more naturally.
When to Use It
Geometry problems (obviously!)
Word problems involving distance, arrangements, or physical situations
Any problem where you are confused about what is happening
Problems involving folding, cutting, rotating, or assembling shapes
How to Practice
For every past Math Kangaroo problem, challenge yourself to draw a picture even if the problem does not seem to require one. You will be surprised how often a diagram reveals insights that words alone do not.
Pro Tips
Draw larger than you think you need to — tiny diagrams hide important details.
Use different colors for different elements (if possible).
Label everything: lengths, angles, names, quantities.
Draw multiple versions if the first one does not reveal the answer.
III. Technique #2: Make an Organized List or Table
When a problem has multiple cases, possibilities, or numerical relationships, making an organized list or table brings order to chaos. This technique is especially powerful for counting problems, scheduling problems, and "find all possible..." problems.
When to Use It
Counting problems ("How many numbers between 1 and 100 have...")
Problems asking for "all possible" arrangements or combinations
Scheduling or sequencing problems
Any problem where you feel you might miss some cases
Pro Tips
Organize systematically (alphabetically, by size, by starting point) to ensure you do not miss cases.
Look for patterns in your list — often the answer can be computed directly without listing everything.
Count your cases twice — once forward, once backward — to verify.
IV. Technique #3: Look for Patterns
The human brain is a pattern-recognition machine — and Math Kangaroo problems are designed to reward this ability. Many problems that seem impossible at first become obvious once you spot the underlying pattern.
When to Use It
Sequence problems (2, 5, 10, 17, 28, ...)
Repeating processes or operations
Geometric patterns (tessellations, fractals, growing shapes)
Problems involving large numbers or many steps
Pro Tips
When a problem involves "the 100th term" or "after 1000 steps," start by computing the first few terms by hand. The pattern almost always emerges within the first 3–5 cases.
Look for repeating cycles — many processes return to a starting state after a fixed number of steps.
If you spot a pattern, verify it on one more case before committing to an answer.
V. Technique #4: Work Backwards
Sometimes the path from start to finish is hard to see, but the path from finish to start is obvious. When this happens, work backwards: start with the desired outcome and trace your steps back to the beginning.
When to Use It
Problems involving a sequence of operations (e.g., "I doubled it, then added 5, then...")
Maze or path-finding problems
Problems where you know the final state and need to find the initial state
Game theory problems where you know the winning position
Pro Tips
When working backwards, each step is the inverse of the forward step (addition becomes subtraction, multiplication becomes division, etc.).
Draw a flow diagram to keep track of the steps.
Verify your answer by working forwards from your found starting point.
VI. Technique #5: Use Logical Reasoning and Elimination
Math Kangaroo is a multiple-choice competition — and this means you can often solve problems by eliminating wrong answers even when you cannot find the right answer directly. Logical reasoning is your most powerful ally in this context.
When to Use It
When you do not know how to solve a problem directly
When you can easily identify 1–2 answers that are clearly wrong
When you can narrow the answer down to 2–3 possibilities
When time is running out and you need to make a strategic guess
Pro Tips
If the answer must be even, cross out all odd options.
If the answer must be between 10 and 20, cross out everything outside that range.
If you can test one answer choice quickly and it does not work, cross it out and try another.
Never leave a question blank — Math Kangaroo has no penalty for wrong answers. Even an educated guess is better than no answer.
VII. Technique #6: Try Simpler Versions of the Problem
When a problem seems overwhelming because the numbers are large or the situation is complex, try a simpler version first. Solve the same problem with smaller numbers, fewer objects, or fewer steps. Often, the solution method becomes clear — and you can apply it to the original problem.
When to Use It
Problems with very large numbers (e.g., "2026" or "1,000,000")
Problems with many objects or steps
Problems involving complicated rules or conditions
Any problem where you feel completely stuck
Pro Tips
Choose representative simple cases — they should preserve the essential structure of the problem.
After solving 2–3 simpler versions, ask yourself: "Can I see a formula or pattern?"
If the simple versions give different answers, make sure you understand why — this usually reveals the key insight.
VIII. Technique #7: Use Symmetry
Many Math Kangaroo problems have hidden symmetry — aspects of the problem that remain unchanged when you transform it. Recognizing and exploiting symmetry can reduce a seemingly complex problem to a trivial one.
When to Use It
Geometry problems involving regular shapes, reflections, or rotations
Counting problems where many cases are equivalent
Problems with multiple equivalent paths or arrangements
Any problem where many elements seem interchangeable
Pro Tips
In geometry, if a problem has a line of symmetry, draw it. It often splits the problem in half.
If a counting problem has "equivalent" cases, count one and multiply by the number of equivalents.
Symmetry is often the key to the most elegant solutions — if your solution feels long and ugly, you might be missing a symmetry.
IX. Technique #8: Consider Extreme or Special Cases
When a problem asks about a general situation, consider what happens in extreme or special cases. What happens when one variable is zero? When it is as large as possible? When the shape is a circle, a square, or a single point? These special cases often reveal the answer or at least narrow down the possibilities.
When to Use It
Problems asking for the maximum or minimum of something
Problems involving "all possible" configurations
When you want to test whether an answer is plausible
When multiple choice options seem close and you need to distinguish them
Pro Tips
For geometry, consider degenerate cases: a triangle with zero area, a circle with zero radius, a rectangle that is actually a line.
For counting, consider what happens with 0 objects, 1 object, or all objects.
Extreme cases are excellent for eliminating answer choices — if an option does not work in an extreme case, it cannot be the general answer.
X. Technique #9: Count in Two Different Ways
Some of the most elegant solutions in mathematics come from counting the same thing in two different ways. When both counts must give the same answer, you get an equation that can be solved. This technique, also called "double counting," is a powerful tool for combinatorial problems.
When to Use It
Counting problems where direct counting is difficult
Problems involving graphs, networks, or relationships
Problems where you need to prove that two quantities are equal
Advanced combinatorics problems
Pro Tips
Look for pairings: can you match each item you want to count with something else?
Count from the perspective of different elements: count edges from vertices, count handshakes from people, count games from teams.
This technique is especially useful for proving that seemingly different quantities are actually equal.
XI. Technique #10: Use Invariants and Monovariants
An invariant is something that does not change throughout a process. A monovariant is something that changes in only one direction (always increasing or always decreasing). Identifying invariants and monovariants can solve problems that seem impossible by other methods.
When to Use It
Problems involving repeated operations or transformations
"Is it possible to reach..." type problems
Game theory problems
Problems asking whether a certain final state is achievable
Pro Tips
Parity (odd/even) is the simplest invariant — always check it first.
If a problem asks "Can we reach state X from state Y?", look for an invariant that has different values in X and Y. If such an invariant exists, the answer is impossible.
For games, a monovariant often reveals the winning strategy: the player who always moves in the direction of the monovariant wins.
XII. Putting It All Together: The Problem-Solving Process
Knowing these ten techniques is only half the battle. The other half is knowing how to combine them during the actual competition. Here is a proven process that top Math Kangaroo students follow for every problem:
| Step | Action | Time Budget |
| 1. Read carefully | Read the problem twice. Identify what is given and what is asked. Underline key information. | 30 seconds |
| 2. Classify | What type of problem is this? (Counting, geometry, logic, pattern, etc.) Which techniques might apply? | 15 seconds |
| 3. Try the easiest approach first | Start with the simplest technique that might work. Often, drawing a picture or making a list is enough. | 1–2 minutes |
| 4. If stuck, switch techniques | If your first approach is not working after 2 minutes, try a different technique. Often, a new perspective unlocks the solution. | 1–2 minutes |
| 5. Verify | Once you have an answer, verify it. Does it make sense? Can you solve the problem a different way to check? | 30 seconds |
| 6. Move on if necessary | If you have spent 5+ minutes on a single problem, mark your best guess and move on. Come back later if time permits. | Ongoing decision |
XIII. How to Practice These Techniques Over Time
Mastering these ten techniques is not a one-time event — it is a journey of continuous improvement. Here is a recommended practice plan:
| Phase | Duration | Focus |
| Phase 1: Learn | 1–2 weeks | Read this article carefully. For each of the 10 techniques, find 2–3 past Math Kangaroo problems that illustrate it. Work through them, identifying which technique applies. |
| Phase 2: Practice | 2–4 weeks | Work through 1–2 past Math Kangaroo exams per week. For each problem, explicitly identify which technique(s) you used. After solving, check the official written solution to see if a more elegant approach was available. |
| Phase 3: Reflect | Ongoing | Keep a "technique journal" — for each problem, write down which technique(s) you used and which you wish you had used. Over time, you will develop an intuitive sense of which technique to reach for in each situation. |
| Phase 4: Master | Months before competition | Do timed mock exams. Focus not just on solving problems correctly, but on solving them efficiently using the right technique. This is what separates good students from great ones. |
XIV. Final Thoughts: The True Goal Is Not to Win — It Is to Think
As you and your child embark on this journey of mastering problem-solving strategies, remember the deeper purpose. These ten techniques are not just tools for winning Math Kangaroo — they are ways of thinking that will serve your child for life. The ability to draw a picture to clarify confusion. The discipline to make an organized list. The curiosity to look for patterns. The creativity to try a simpler version. The wisdom to work backwards. These are the habits of mind that define not just great mathematicians, but great thinkers in any field.
Math Kangaroo, at its heart, is not about trophies or rankings. It is about the joy of discovery — that magical moment when a confusing problem suddenly becomes clear, when a clever insight reveals an elegant solution, when your child realizes they can think their way through something they did not know how to approach. That joy is the true reward. The awards, the scores, the rankings — these are just pleasant side effects of a much deeper transformation.
So encourage your child to embrace these techniques not as a chore, but as a collection of powerful tools for exploration. Each one opens a new door. Each one makes the world of mathematics a little more accessible, a little more beautiful, a little more fun. And who knows? The student who masters these ten techniques today might become the scientist, engineer, artist, or entrepreneur who changes the world tomorrow — not because they memorized formulas, but because they learned to think clearly, creatively, and fearlessly.
Ready to start mastering these techniques? Visit mathkangaroo.org/mks/practice for free past exams, written solutions, and the interactive Play and Learn platform. Each problem is an opportunity to practice one of these powerful strategies. Happy problem-solving!
