Introduction to Problem Solving | Math Learning Guide

📋 Prerequisites

Competition math isn't just about knowing formulas — it's about knowing how to attack a problem. The best problem solvers have a toolkit of strategies. When you get stuck, try one of these four techniques before you give up.

📚 Key Concepts

Strategy 1: Find a Pattern (Small Cases)

When a problem asks about a large number (like "the 100th term" or "100 people"), start small:

Compute the first 2-3 cases manually
Look for a pattern or formula
Apply the pattern to the large case

This works for sequences, tiling problems, handshake problems, and more.

Strategy 2: Work Backwards

When you know the final result and need the starting value, reverse every operation:

Addition ↔ Subtraction
Multiplication ↔ Division
"What's left" problems are classic backwards problems

Key: reverse the order too — the last operation forward becomes the first operation backward.

Strategy 3: Draw a Picture / Diagram

If a problem feels abstract, draw it out:

Geometry problems: always draw a figure (even if described)
Word problems: bar models, tape diagrams, number lines
Logic / set problems: Venn diagrams
Rate / distance problems: timeline diagrams

Drawing turns words into something your visual brain can process. It's not "wasting time" — it saves time.

Strategy 4: Estimate & Check (Plug in Answers)

On multiple-choice tests (like AMC 8/10), you can often work from the answers:

Estimate roughly what the answer should be
Pick an answer choice and test it
If it's too big, try a smaller one; if too small, try bigger

This is called "backsolving" and it's a legitimate competition strategy. Use it when setting up the equation is hard.

⚠️ Common Mistake: Don't just stare at a problem you don't know how to solve. Try something — compute a small case, draw a picture, test an answer choice. Action leads to insight far more often than thinking does.

💡 Key Insight: Most competition problems are designed to have an elegant solution. If your approach involves 10 minutes of messy arithmetic, you're probably missing a pattern or trick. Step back, look for structure, and try a different strategy.

✏️ Example Problems

📝 Example 1 (Find a Pattern)

What is the 50th positive odd number?

Strategy: Find a pattern from small cases.

Step 1: List the first few odd numbers with their positions:

1st: 1, 2nd: 3, 3rd: 5, 4th: 7, 5th: 9, ...

Step 2: Find the pattern.

Each time, we add 2. The nth odd number = $ 2n - 1 $

$(Check: n=1 \\to 1, n=2 \\to 3, n=3 \\to 5.$ ✓)

Step 3: Apply to n = 50:

\[ 2(50) - 1 = 100 - 1 = 99 \]

Answer: 99

📝 Example 2 (Work Backwards)

I think of a number. I add 5, multiply by 2, subtract 3, and get 11. What number did I start with?

Strategy: Work backwards — reverse every operation, in reverse order.

$Forward steps: start \\to add 5 \\to multiply by 2 \\to subtract 3 \\to 11$

$Reverse steps: 11 \\to add 3 \\to divide by 2 \\to subtract 5 \\to answer$

Step 1: Reverse "subtract 3" $\\to add 3 to 11$

$ 11 + 3 = 14 $

Step 2: Reverse "multiply by 2" $\\to divide by 2$

$ 14 \div 2 = 7 $

Step 3: Reverse "add 5" $\\to subtract 5$

$ 7 - 5 = 2 $

Check forward: $Start with 2. Add 5 \\to 7. Multiply by 2 \\to 14. Subtract 3 \\to 11.$ ✓

Answer: 2

📝 Example 3 (Handshake Problem)

6 people are in a room. Each person shakes hands with every other person exactly once. How many handshakes happen in total?

Strategy: Draw it out / find the pattern.

Method 1 (Drawing): Draw 6 points (people). Connect every pair with a line (handshake). Count the lines.

Method 2 (Pattern / Formula):

$Person 1 shakes hands with 5 others \\to 5 handshakes$

Person 2 already shook with Person 1, so 4 new handshakes

Person 3: 3 new, Person 4: 2, Person 5: 1, Person 6: 0

\[ 5 + 4 + 3 + 2 + 1 = 15 \]

General formula: For $n$ people, handshakes = $ \frac{n(n-1)}{2} $

Check: $ \frac{6 \times 5}{2} = 15 $. ✓

Answer: 15