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 river-crossing puzzle below is remarkably old: it appears in Propositiones ad Acuendos Juvenes (“Problems to Sharpen the Young”), a collection of puzzles attributed to the English scholar Alcuin of York, written around 800 CE at the request of the Emperor Charlemagne. It is the oldest known collection of recreational mathematics puzzles in Europe, and the wolf-goat-cabbage crossing problem (told with a wolf, a goat, and a bundle of cabbages rather than a fox, a chicken, and grain) is one of its most famous entries – meaning people have been solving essentially the same logic puzzle for over 1 200 years.

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

A farmer must cross a river with a fox, a chicken, and a bag of grain. The boat holds only the farmer and one item. Left alone, the fox eats the chicken; the chicken eats the grain. How does the farmer get everything across safely?

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.

Complete the 3×3 magic square: the centre must be 5, corners must be even, edges must be odd. Numbers 1–9, magic constant 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.

Solve: SEND + MORE = MONEY.

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).

Solve: TWO + TWO = FOUR.

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

You have 9 balls. One is heavier. Using a balance scale, find the heavy ball in just 2 weighings.

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

At a party, every person shakes hands with every other person exactly once. There are 28 handshakes in total. How many people are at the party?

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: Classic Puzzle Types

Puzzle Type 5: Handshake Problems