Puzzle Solving - Classic Problems and How to Crack Them
Puzzle solving sits at the heart of mathematical thinking. A good puzzle is one that seems impossible at first, then suddenly obvious once you find the right approach. Working through puzzles builds persistence, lateral thinking, and the ability to see problems from new angles – skills that transfer directly to advanced mathematics and real-world problem solving.
The Puzzle-Solving Mindset
The most important habit when solving puzzles:
1. Read the puzzle carefully – restate it in your own words.
2. Try small or simple cases first.
3. Look for hidden constraints you may have missed.
4. If stuck, change your point of view entirely.
5. Check your solution against every condition in the puzzle.
Puzzle Type 1: River Crossing
1. Take the chicken across. Return alone.
2. Take the fox across. Bring the chicken back.
3. Take the grain across. Return alone.
4. Take the chicken across.
All four are safely on the far side in 7 trips.
Puzzle Type 2: Magic Squares
A magic square is a grid of distinct numbers where every row, column, and main diagonal sums to the same value (the magic constant).
For an n × n magic square using the numbers 1 to n²: Magic constant = n(n²+1)/2.
For a 3 × 3 square: 3(10)/2 = 15.
The unique solution (up to rotation and reflection):
2 7 6
9 5 1
4 3 8
Check rows: 15, 15, 15. Columns: 15, 15, 15. Diagonals: 15, 15. ✓
Puzzle Type 3: Cryptarithmetic
Each letter represents a distinct digit (0–9). Find the digits that make the equation true. No number begins with a zero.
This is the most famous cryptarithmetic puzzle. The unique solution is:
S=9, E=5, N=6, D=7, M=1, O=0, R=8, Y=2.
9 567 + 1 085 = 10 652 ✓
Key deductions: M must be 1 (the only carry possible). S must be 9 (to produce a carry into M). O must be 0 (since M=1 and there is only a carry of 1).
TWO is a 3-digit number, FOUR is 4 digits. So TWO ≥ 500 (since 500+500=1000).
One solution: T=7, W=3, O=4. 734 + 734 = 1 468. F=1, O=4, U=6, R=8. ✓
Puzzle Type 4: Weighing Problems
Weighing 1: Put 3 balls on each side, leave 3 aside.
• If balanced: the heavy ball is in the 3 left aside.
• If unbalanced: the heavy ball is in the heavier group of 3.
Weighing 2: From the group of 3 suspects, put 1 on each side, leave 1 aside.
• If balanced: the left-aside ball is heavy.
• If unbalanced: the heavy side holds it.
2 weighings suffice to find 1 heavy ball among 9.
Puzzle Type 5: Handshake and Pairing Problems
Number of handshakes = n(n−1)/2 = 28.
n(n−1) = 56. Try n = 8: 8 × 7 = 56. ✓
8 people are at the party.
Puzzle Type 6: The Pigeonhole Principle
If you have more pigeons than pigeonholes, at least one pigeonhole must contain more than one pigeon. This simple idea proves powerful results with no calculation required.
Example: In any group of 13 people, at least two were born in the same month. (12 months = 12 pigeonholes; 13 people = 13 pigeons.)
Example: In any group of 367 people, at least two share a birthday. (366 possible birthdays including Feb 29.)
Key Takeaways
- River crossing: identify which item causes conflict and manage it directly.
- Magic squares: for a 3×3 grid the magic constant is 15; the centre must be 5.
- Cryptarithmetic: start with the leading digits and carries, then work inwards.
- Weighing problems: divide into thirds at each step to minimise weighings.
- Pigeonhole principle: n+1 items in n categories guarantees at least one category has 2 or more.
Practice Questions
- A zookeeper must cross a river with a wolf, a goat, and a cabbage. Same rules as the farmer puzzle. Find the sequence of crossings.
- Create a 3×3 magic square using 3, 5, 7, 9, 11, 13, 15, 17, 19. What is the magic constant?
- Solve: BASE + BALL = GAMES (each letter is a unique digit).
- You have 12 balls, one is either heavier or lighter (unknown). Find the odd ball and whether it is heavy or light in exactly 3 weighings.
- How many people must be in a room to guarantee that at least two of them were born on the same day of the week?