6 Best Mathcounts Problem Solving Techniques Most Students Overlook

You’ve seen it happen: your bright, capable child stares at a math problem, completely stuck. They know the formulas and have done all the homework, but this particular question feels like a brick wall. The frustration is real, but the solution often isn’t more knowledge—it’s a better toolkit for thinking.

The Core Mindset for Creative Problem Solving

Many students approach math as a series of recipes to be memorized. If a problem doesn’t match a recipe they know, they hit a wall. The first, and most important, shift is moving from a "what formula do I use?" mindset to a "what is the story of this problem?" mindset. This is the pivot from calculation to true problem-solving. It’s about fostering curiosity and persistence.

Encourage your child to see being "stuck" not as a failure, but as the starting line. This is where the real learning happens! The goal isn’t just to get the right answer, but to enjoy the process of unraveling the puzzle. When they learn to embrace a challenge with curiosity instead of anxiety, they unlock a level of creativity that rote memorization can never offer. This mental resilience is the foundation for every other technique.

Making a Sketch: Your First Problem-Solving Step

A child is faced with a complex problem about trains traveling at different speeds from different cities. The words swim together, and the numbers feel abstract and confusing. What’s the first move? Before reaching for a calculator, they should reach for a pencil and paper.

Drawing a simple diagram, a map, or a chart is the single most effective first step for a huge range of problems. It’s not about artistic talent; a simple line with points labeled for cities and arrows for trains is enough. This act translates abstract language into a concrete visual, instantly clarifying relationships between the moving parts. It helps organize information and often reveals a path to the solution that was hidden in the dense text.

The Reversal Method: Working from Solution to Start

Have you ever watched your child build a complex equation to solve a problem, only to get lost in the algebra? Sometimes, the most elegant path is the one that feels completely backward. The reversal method involves starting with the end result and working your way back to the beginning, undoing each step along the way.

This is especially powerful for problems that say something like, "After a series of transactions, John had $10 left. What did he start with?" Instead of setting up a variable x and performing all the operations, you simply start with the $10 and reverse the process. If he gave away $5, you add $5 back. If he doubled his money, you cut it in half. This technique can turn a multi-step algebraic mess into a simple sequence of arithmetic.

Simplify and Scale: Tackle a Smaller Problem First

Some problems are designed to intimidate with large numbers. "If 100 people are in a room and everyone shakes hands with everyone else exactly once, how many handshakes are there?" A student might try to imagine all 100 people and quickly become overwhelmed.

The trick is to ignore the big number and solve a simplified version of the exact same problem. What if there were only 2 people? 1 handshake. What about 3 people? 3 handshakes. 4 people? 6 handshakes. By solving these mini-puzzles, the student can focus on the process and find the underlying pattern. Once they see the logic in a small, manageable case, they can "scale up" that logic to solve for 100 people, or a million.

Finding the Pattern: The Key to Repetitive Logic

Many competition problems involve sequences or operations that repeat. A classic example might ask for the last digit of 3 raised to the power of 2023. No student is expected to calculate that number! The problem is not about computation; it’s about pattern recognition.

Help your child explore the first few steps and look for a cycle.

• 3¹ = 3
• 3² = 9
• 3³ = 27 (last digit is 7)
• 3⁴ = 81 (last digit is 1)
• 3⁵ = 243 (last digit is 3)

The pattern of the last digits—3, 9, 7, 1—repeats every four powers. Once they see that cycle, the massive number 2023 becomes a simple division problem to see where in the cycle the final answer falls. This technique transforms a seemingly impossible task into a simple, logical deduction.

Systematic Listing: Ensuring You Don’t Miss a Case

"How many different ways can you make 30 cents using quarters, dimes, and nickels?" A common mistake is to randomly guess combinations, almost certainly missing some or counting others twice. The solution is to work systematically, creating an organized list or table.

Start with the largest coin to create structure. First, how many quarters can you use? You can use one quarter or zero quarters. That’s it. This creates two main branches for your list.

1. Case 1: Use one quarter (25¢). You need 5¢ more. There’s only one way: one nickel. (1Q, 0D, 1N)
2. Case 2: Use zero quarters (0¢). You need 30¢. Now, move to the next largest coin: dimes. You could use three dimes, two, one, or zero. You list each of these sub-cases and figure out the nickels needed for each.

This methodical approach guarantees that every single possibility is accounted for, turning a chaotic guessing game into a calm, orderly process.

Using Logic and Elimination to Narrow Down Choices

Even when a problem isn’t multiple-choice, there’s often a finite universe of possible answers. This is where your child can play detective. Using the constraints and conditions given in the problem, they can eliminate possibilities and dramatically narrow the search field.

Consider a problem like: "I am a three-digit number. I am a multiple of 5. The sum of my digits is 13. My tens digit is double my hundreds digit." Instead of testing hundreds of numbers, you use the clues. "Multiple of 5" means the last digit must be 0 or 5. "Tens digit is double the hundreds digit" means the first two digits could be 12, 24, 36, or 48. Now you just combine these clues to find the very few numbers that fit all the criteria, and test which one has digits summing to 13.

Making These Techniques Second Nature for Your Child

As a parent, your role isn’t to be the answer key. It’s to be the coach who asks the right questions. When your child is stuck, resist the urge to show them the solution. Instead, empower them by prompting them to use their toolkit.

Ask questions that guide their thinking process:

• "What would happen if you tried to draw a picture of this problem?"
• "This seems complicated. Is there a simpler version with smaller numbers we could solve first?"
• "What if we started from the end? What would the step right before the last one have been?"
• "Have you tried making an organized list of the possibilities?"

By consistently prompting them with these strategies, you are helping to build the mental habits of a flexible, resilient problem-solver. You’re teaching them how to think, not what to think. This approach transforms homework frustration into an opportunity for them to build confidence and intellectual independence, one challenging problem at a time.

Ultimately, these techniques are about more than just scoring points in a math competition. They are fundamental strategies for logical reasoning, creativity, and persistence. By mastering these approaches, your child isn’t just becoming a better math student; they’re becoming a more resourceful and confident thinker, ready for any challenge that comes their way.