Tolling Bells
Four church bells ring at different intervals. Students find when all bells will ring together again — a hands-on introduction to the lowest common multiple through sound and pattern.
LCM · MultiplesY7–Y8
Problems worth chewing on.
These are open-ended, unfamiliar problems — not worksheets. They are designed to teach students how to attack something they have never seen before: to try a small case, spot a pattern, make a conjecture, and then prove or disprove it. Some take twenty minutes. Some take a week. The point is the process.
Number
The 1089 Trick
Take any 3-digit number, reverse it, subtract the smaller from the larger, then reverse and add. The answer is always 1089. Students investigate why this algebraic magic works.
Place value · AlgebraY7–Y9
Magic Squares
Arrange integers in a grid so every row, column, and diagonal adds to the same total. Students discover the constraint relationships that force the magic constant and explore 4×4 extensions.
Number properties · ReasoningY6–Y8
Chickens and Rabbits
A farmer has some chickens and rabbits. There are 20 heads and 56 legs. How many of each? Students move from trial-and-improvement to forming a system of equations.
Simultaneous equations · Trial & improvementY6–Y8
Fibonacci & Nature
Sunflower seed spirals, pinecone rows, and petal counts almost always follow the Fibonacci sequence. Students generate the sequence, notice its ratio converges to φ, and search for it in the natural world.
Sequences · Golden ratioY7–Y9
Algebra
Frogs (Slides & Jumps)
Coloured frogs on lily pads must swap sides using a defined set of moves. Students count moves for 1, 2, 3 … frogs per side, spot the quadratic pattern, and write a general rule.
Sequences · GeneralisationY9–Y10
Pascal's Triangle
Each entry is the sum of the two entries above it. Students colour multiples, spot triangular and Fibonacci numbers hidden in the rows, and connect the triangle to binomial expansions.
Patterns · BinomialsY8–Y10
Tower of Hanoi
Move a stack of discs from peg A to peg C, never placing a larger disc on a smaller one. The minimum number of moves follows a doubling pattern — a concrete gateway to exponential growth and recursion.
Exponential growth · RecursionY8–Y10
Geometry
100 Locker (Prison Cell) Problem
Student 1 opens every locker; student 2 closes every second locker; student 3 toggles every third; and so on. Which lockers are open at the end? The surprising answer reveals a deep connection to square numbers.
Factors · Square numbersY7–Y9
Painted Cube
A large 3×3×3 cube is painted on all six faces and then cut into 27 small cubes. How many small cubes have paint on exactly 3 faces? Exactly 2? Exactly 1? None? Students generalise to an n×n×n cube.
Volume · Surface area · AlgebraY8–Y10
Polygon Diagonals
How many diagonals can you draw in a pentagon? A hexagon? A 20-gon? Students collect data, spot the pattern, and derive the formula n(n−3)/2, linking geometry to combinatoric reasoning.
Algebra · CombinatoricsY8–Y9
Probability & Combinatorics
Handshake Problem
If everyone in a room shakes hands with everyone else exactly once, how many handshakes take place? Students count small cases, build a table, and arrive at the formula n(n−1)/2 — the same as triangle numbers.
Combinatorics · Triangle numbersY8–Y10
Locker Combinations
How many ways can 4 students stand in a line? What if there are 10 students and 3 chairs? Students build up from small cases to discover factorial counting and the ideas behind permutations.
Factorials · PermutationsY8–Y10
Investigation
Königsberg Bridges
Can you walk through the city of Königsberg crossing each of its seven bridges exactly once? Euler proved it is impossible — and in doing so invented graph theory. Students replicate his reasoning on modern networks.
Graph theory · ProofY9–Y11
Don Steward Problem Collections
Don Steward's Median blog is one of the richest free repositories of mathematical problems in existence, covering every topic from fractions to calculus with unusual depth and variety.
All topicsY7–Y13
Mathematical Jigsaws (Tarsia)
Tarsia jigsaws are triangular or domino-style puzzles where edges must match question to answer. They're self-checking, so students know immediately when a piece is wrong. Useful for consolidation of any topic.
Any topic · Self-checkingY7–Y11