Problem-Solving Strategies

What is Problem Solving?

Problem solving: Process of finding solutions to challenging questions

Not just calculation: Requires thinking, strategy, creativity

George Pólya's 4 steps:

1. Understand the problem
2. Devise a plan
3. Carry out the plan
4. Look back and reflect

Key insight: Having strategies makes hard problems easier

Strategy 1: Look for a Pattern

Find repeating relationships or sequences

Steps:

1. Calculate first few cases
2. Organize data (table, list)
3. Identify pattern
4. Extend or generalize

Example 1: Handshakes

Problem: 20 people shake hands once with each other. How many handshakes?

Start small:

• 2 people: 1 handshake
• 3 people: 3 handshakes
• 4 people: 6 handshakes
• 5 people: 10 handshakes

Pattern: 1, 3, 6, 10... (triangular numbers)

Formula: n(n-1)/2

Answer: 20(19)/2 = 190 handshakes

Example 2: Sum of Odd Numbers

Problem: 1 + 3 + 5 + 7 + ... + 99 = ?

Pattern:

• First 1 odd: 1 = 1²
• First 2 odds: 1 + 3 = 4 = 2²
• First 3 odds: 1 + 3 + 5 = 9 = 3²
• First 4 odds: 1 + 3 + 5 + 7 = 16 = 4²

Pattern: Sum of first n odds = n²

How many odds from 1 to 99? 50 numbers

Answer: 50² = 2500

Strategy 2: Work Backwards

Start from desired result, reverse the steps

Use when: Final state given, need to find initial state

Example 1: Number Puzzle

Problem: I think of a number, add 5, multiply by 3, subtract 12. Result is 18. What's my number?

Work backwards:

• End: 18
• Before subtracting 12: 18 + 12 = 30
• Before multiplying by 3: 30 ÷ 3 = 10
• Before adding 5: 10 - 5 = 5

Answer: Original number is 5

Check: (5 + 5) × 3 - 12 = 10 × 3 - 12 = 18 ✓

Example 2: Water Jug

Problem: Have 5L and 3L jugs. How to measure exactly 4L?

Work backwards from 4L:

• Need 4L in 5L jug
• That's 5L minus 1L
• Can get 1L by filling 3L from 5L twice

Solution:

1. Fill 5L jug
2. Pour into 3L jug (2L left in 5L jug)
3. Empty 3L jug
4. Pour 2L from 5L into 3L
5. Fill 5L jug again
6. Pour from 5L into 3L until full (1L added, 4L remains in 5L) ✓

Strategy 3: Draw a Diagram

Visualize the problem

Use when: Spatial relationships, geometry, logic puzzles

Example 1: Fence Posts

Problem: Fence is 100 meters long. Posts every 10 meters. How many posts?

Draw diagram:

|---|---|---|---|---|---|---|---|---|---|
0  10  20  30  40  50  60  70  80  90 100

Count: 11 posts (not 10!)

Pattern: n intervals need n+1 posts

Example 2: Path Counting

Problem: How many paths from A to B moving only right or up?

Draw grid:

    B
    |
    |
A---+

3×2 grid:

  4--10--B
  |   |   |
  2---6---+
  |   |   |
  A---2---+

Answer: 10 paths

Pattern: Pascal's triangle, or combinations

Strategy 4: Make a Table or List

Organize information systematically

Use when: Multiple variables, need to track combinations

Example: Pizza Toppings

Problem: Pizza shop has 3 sizes (S, M, L) and 4 toppings. How many different pizzas?

Make table:

Size | Toppings | Count
-----|----------|------
S    | 4 types  | 4
M    | 4 types  | 4
L    | 4 types  | 4

Total: 3 × 4 = 12 different pizzas

Or with multiple toppings: Use systematic list or tree diagram

Strategy 5: Guess and Check

Make educated guess, test it, refine

Use when: Direct solution difficult, answer among limited options

Important: Make systematic, logical guesses

Example: Age Problem

Problem: Father is 3 times son's age. In 12 years, father will be twice son's age. What are their ages now?

Guess 1: Son = 10

• Father = 30
• In 12 years: Son = 22, Father = 42
• Is 42 = 2 × 22? No (too large)

Guess 2: Son = 12

• Father = 36
• In 12 years: Son = 24, Father = 48
• Is 48 = 2 × 24? Yes! ✓

Answer: Son is 12, Father is 36

Strategy 6: Solve a Simpler Problem

Reduce complexity, find insight, apply to original

Use when: Problem too complex to tackle directly

Example: Chessboard Squares

Problem: How many squares on 8×8 chessboard (all sizes)?

Simpler problem: Start with smaller boards

1×1 board: 1 square

2×2 board:

• 1×1 squares: 4
• 2×2 squares: 1
• Total: 5

3×3 board:

• 1×1: 9
• 2×2: 4
• 3×3: 1
• Total: 14

Pattern: 1² + 2² + 3² + ... + n²

Formula: n(n+1)(2n+1)/6

8×8 board: 8(9)(17)/6 = 204 squares

Strategy 7: Use Logic and Elimination

Rule out impossible cases systematically

Use when: Logic puzzles, constraints given

Example: Three Boxes

Problem: Three boxes labeled "Apples," "Oranges," "Both." All labels wrong. Pick one fruit from one box. Which box to identify all?

Logic:

• All labels wrong
• Must pick from "Both" box

If get apple:

• "Both" box is actually "Apples"
• "Oranges" box can't be Oranges (wrong label) → must be "Both"
• "Apples" box must be "Oranges"

Answer: Pick from box labeled "Both"

Strategy 8: Use Symmetry

Exploit symmetrical properties to simplify

Example: Digit Sum

Problem: What's 1 + 2 + 3 + ... + 98 + 99 + 100?

Use symmetry:

• Pair: (1 + 100) = 101
• Pair: (2 + 99) = 101
• Pair: (3 + 98) = 101
• ...
• 50 pairs total

Answer: 50 × 101 = 5050

Gauss's method!

Strategy 9: Consider Extreme Cases

Test boundaries and special cases

Helps understand problem structure

Example: Function Behavior

Problem: For what x is x² > 2x?

Test extremes:

• x = 0: 0² vs 0 → equal
• x = 1: 1 vs 2 → less
• x = 2: 4 vs 4 → equal
• x = 3: 9 vs 6 → greater
• x = -1: 1 vs -2 → greater

Answer: x < 0 or x > 2

Strategy 10: Change Your Perspective

Reframe problem, look from different angle

Example: Lily Pad

Problem: Lily pad doubles in size daily. Takes 30 days to cover pond. How long to cover half?

Wrong thinking: Half time = 15 days

Correct thinking: If doubles daily, one day before full is half

Answer: 29 days

Common Problem Types

Counting problems: Use combinations, permutations

Optimization: Find maximum or minimum

Number puzzles: Use algebra, logic

Geometry: Draw diagrams, use formulas

Probability: Make tree diagrams, lists

Sequences: Look for patterns

Avoiding Common Mistakes

1. Rushing: Read carefully, understand before solving

2. No plan: Choose strategy before calculating

3. Not checking: Verify answer makes sense

4. Giving up: Try different strategy if stuck

5. Arithmetic errors: Calculate carefully

Practice Problem-Solving Process

Example: Locker Problem

Problem: 100 lockers, all closed. 100 students. Student n toggles every nth locker. Which lockers are open after all students?

Understand: Each locker toggled by divisors of its number

Plan: Locker open if odd number of toggles (odd number of divisors)

Insight: Only perfect squares have odd divisors

Solution: Lockers 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 (10 total)

Reflect: Perfect squares because one divisor (square root) pairs with itself

Building Problem-Solving Skills

Practice regularly: Solve variety of problems

Learn from mistakes: Analyze what went wrong

Study solutions: Learn new techniques

Collaborate: Discuss strategies with others

Be persistent: Don't give up quickly

Enjoy the process: Problem solving is creative!

Practice