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For teachers

Maths club sessions

Ready-made 50-minute sessions for a maths club, a stretch group or an enrichment lesson. Each session is built around one problem-solving strategy: a warm-up, three main problems, a challenge for the fastest, and a discussion question taken from the ‘where it leads’ notes.

The session plans and problems are free to use from this page. The printable packs (problems, teacher notes and full solutions on one handout) are included with every A Level, IB, IGCSE and CBSE plan, including school licences.

PartTime
Warm-up5 min
Main problem 110 min
Main problem 210 min
Main problem 310 min
Challenge (for anyone who finishes)10 min
Discussion5 min

24 sessions

Junior · Intermediate · Senior · Olympiad-style

Junior sessions (ages 11 to 13)

Session 1: Organised cases

Split a problem into cases that cannot overlap and together cover everything, then deal with each small case. Strategy guide.

  1. Warm-up: Problem J08 (Junior, combinatorics)
  2. Main 1: Problem J01 (Junior, number theory)
  3. Main 2: Problem J03 (Junior, number theory)
  4. Main 3: Problem J06 (Junior, number theory)
  5. Challenge: Problem I01 (Intermediate, number theory)
  6. Discussion: Counting numbers with a given digit sum is a ‘stars and bars’ problem in disguise; the restriction that digits are at most 9 is what makes larger cases interesting.

Session 2: Working backwards

Start from the end state or the answer you want and undo each step. Strategy guide.

  1. Warm-up: Problem J15 (Junior, geometry)
  2. Main 1: Problem J02 (Junior, number theory)
  3. Main 2: Problem J05 (Junior, number theory)
  4. Main 3: Problem J14 (Junior, combinatorics)
  5. Challenge: Problem I03 (Intermediate, number theory)
  6. Discussion: When the remainders do not line up so neatly, the Chinese remainder theorem still guarantees a solution as long as the divisors share no common factor.

Session 3: Symmetry

Use a reflection, rotation or swap that leaves the problem unchanged to halve the work or pin down the answer. Strategy guide.

  1. Warm-up: Problem J34 (Junior, probability)
  2. Main 1: Problem J23 (Junior, algebra)
  3. Main 2: Problem J29 (Junior, probability)
  4. Main 3: Problem J31 (Junior, probability)
  5. Challenge: Problem I21 (Intermediate, geometry)
  6. Discussion: Adding and subtracting equations (elimination) generalises to solving any system of linear equations.

Session 4: Spot the pattern and generalise

Try small cases, spot the pattern, then explain why it must continue. Strategy guide.

  1. Warm-up: Problem J32 (Junior, probability)
  2. Main 1: Problem J04 (Junior, number theory)
  3. Main 2: Problem J07 (Junior, number theory)
  4. Main 3: Problem J16 (Junior, geometry)
  5. Challenge: Problem I02 (Intermediate, number theory)
  6. Discussion: Cycles of last digits are modular arithmetic; Euler’s theorem says the cycle length always divides 4 for powers of numbers ending in 1, 3, 7 or 9.

Session 5: Count the opposite

When the cases you want are messy, count or find the ones you don't want and subtract. Strategy guide.

  1. Warm-up: Problem J43 (Junior, number theory)
  2. Main 1: Problem J10 (Junior, combinatorics)
  3. Main 2: Problem J19 (Junior, geometry)
  4. Main 3: Problem J33 (Junior, probability)
  5. Challenge: Problem I08 (Intermediate, combinatorics)
  6. Discussion: Counting lattice paths with obstacles by subtracting is inclusion–exclusion; with many obstacles it becomes a dynamic-programming table.

Session 6: Parity and remainders

Odd and even, or remainders on dividing by a small number, rule out whole families of answers at once. Strategy guide.

  1. Warm-up: Problem J51 (Junior, number theory)
  2. Main 1: Problem J37 (Junior, logic)
  3. Main 2: Problem J42 (Junior, number theory)
  4. Main 3: Problem J49 (Junior, number theory)
  5. Challenge: Problem I30 (Intermediate, probability)
  6. Discussion: Calendar questions are arithmetic modulo 7; Zeller’s congruence turns any date into a weekday by formula.

Intermediate sessions (ages 13 to 16)

Session 1: Organised cases

Split a problem into cases that cannot overlap and together cover everything, then deal with each small case. Strategy guide.

  1. Warm-up: Problem J01 (Junior, number theory)
  2. Main 1: Problem I01 (Intermediate, number theory)
  3. Main 2: Problem I05 (Intermediate, number theory)
  4. Main 3: Problem I07 (Intermediate, number theory)
  5. Challenge: Problem S03 (Senior, number theory)
  6. Discussion: Solving n2 ≡ c modulo powers of 10 digit by digit is Hensel lifting, a key tool in number theory.

Session 2: Spot the pattern and generalise

Try small cases, spot the pattern, then explain why it must continue. Strategy guide.

  1. Warm-up: Problem J04 (Junior, number theory)
  2. Main 1: Problem I02 (Intermediate, number theory)
  3. Main 2: Problem I04 (Intermediate, number theory)
  4. Main 3: Problem I09 (Intermediate, combinatorics)
  5. Challenge: Problem S02 (Senior, number theory)
  6. Discussion: Repunits (numbers made of 1s) factor according to the divisors of their length; Rn can only be prime when n is prime.

Session 3: Working backwards

Start from the end state or the answer you want and undo each step. Strategy guide.

  1. Warm-up: Problem J02 (Junior, number theory)
  2. Main 1: Problem I03 (Intermediate, number theory)
  3. Main 2: Problem I16 (Intermediate, geometry)
  4. Main 3: Problem I19 (Intermediate, geometry)
  5. Challenge: Problem S01 (Senior, number theory)
  6. Discussion: Adding a constant to complete a product (Simon’s favourite factoring trick) turns many equations into divisor counts.

Session 4: Symmetry

Use a reflection, rotation or swap that leaves the problem unchanged to halve the work or pin down the answer. Strategy guide.

  1. Warm-up: Problem J23 (Junior, algebra)
  2. Main 1: Problem I21 (Intermediate, geometry)
  3. Main 2: Problem I31 (Intermediate, probability)
  4. Main 3: Problem I64 (Intermediate, combinatorics)
  5. Challenge: Problem S10 (Senior, combinatorics)
  6. Discussion: 65 = 5 × 13 is a product of two primes of the form 4k + 1, which is why it has two different representations as a sum of two squares.

Session 5: Count the opposite

When the cases you want are messy, count or find the ones you don't want and subtract. Strategy guide.

  1. Warm-up: Problem J06 (Junior, number theory)
  2. Main 1: Problem I08 (Intermediate, combinatorics)
  3. Main 2: Problem I12 (Intermediate, combinatorics)
  4. Main 3: Problem I13 (Intermediate, combinatorics)
  5. Challenge: Problem S12 (Senior, combinatorics)
  6. Discussion: ‘Not together’ is easiest by complement; the ‘gaps method’ (place the other letters, then drop the Es into gaps) is an alternative.

Session 6: Proof techniques

Contradiction, contrapositive, induction and construction: ways to be certain, not just convinced. Strategy guide.

  1. Warm-up: Problem J36 (Junior, logic)
  2. Main 1: Problem I35 (Intermediate, logic)
  3. Main 2: Problem I41 (Intermediate, number theory)
  4. Main 3: Problem I45 (Intermediate, number theory)
  5. Challenge: Problem S35 (Senior, logic)
  6. Discussion: Encoding ‘same type / opposite type’ links turns knight–knave puzzles into colouring a graph with two colours.

Senior sessions (ages 16 to 18)

Session 1: Organised cases

Split a problem into cases that cannot overlap and together cover everything, then deal with each small case. Strategy guide.

  1. Warm-up: Problem I01 (Intermediate, number theory)
  2. Main 1: Problem S03 (Senior, number theory)
  3. Main 2: Problem S04 (Senior, number theory)
  4. Main 3: Problem S05 (Senior, number theory)
  5. Challenge: Problem O04 (Olympiad-style, number theory, plan)
  6. Discussion: n2 + n + 1 divides n3 − 1, so the solutions are the n with n3 ≡ 1, n ≠ 1: cube roots of unity mod 7.

Session 2: Spot the pattern and generalise

Try small cases, spot the pattern, then explain why it must continue. Strategy guide.

  1. Warm-up: Problem I02 (Intermediate, number theory)
  2. Main 1: Problem S02 (Senior, number theory)
  3. Main 2: Problem S09 (Senior, combinatorics)
  4. Main 3: Problem S11 (Senior, combinatorics)
  5. Challenge: Problem O23 (Olympiad-style, number theory, plan)
  6. Discussion: The cycle length 20 divides φ(100) = 40, as Euler’s theorem says it must.

Session 3: Symmetry

Use a reflection, rotation or swap that leaves the problem unchanged to halve the work or pin down the answer. Strategy guide.

  1. Warm-up: Problem I21 (Intermediate, geometry)
  2. Main 1: Problem S10 (Senior, combinatorics)
  3. Main 2: Problem S15 (Senior, geometry)
  4. Main 3: Problem S16 (Senior, geometry)
  5. Challenge: Problem O22 (Olympiad-style, number theory, plan)
  6. Discussion: As the number of sides grows, three random vertices form an obtuse triangle with probability approaching 3/4.

Session 4: Working backwards

Start from the end state or the answer you want and undo each step. Strategy guide.

  1. Warm-up: Problem I03 (Intermediate, number theory)
  2. Main 1: Problem S01 (Senior, number theory)
  3. Main 2: Problem S08 (Senior, combinatorics)
  4. Main 3: Problem S14 (Senior, combinatorics)
  5. Challenge: Problem O05 (Olympiad-style, number theory, plan)
  6. Discussion: Making a number a perfect power by multiplying is done prime by prime: raise each exponent to the next multiple of the power.

Session 5: Extremal principle

Look at the largest, smallest, first or last object: it often has to behave in a special way. Strategy guide.

  1. Warm-up: Problem I05 (Intermediate, number theory)
  2. Main 1: Problem S07 (Senior, number theory)
  3. Main 2: Problem S24 (Senior, algebra)
  4. Main 3: Problem S90 (Senior, algebra)
  5. Challenge: Problem O07 (Olympiad-style, number theory, plan)
  6. Discussion: The largest prime factor usually decides how big n must be for n! to be divisible by a given number.

Session 6: Count the opposite

When the cases you want are messy, count or find the ones you don't want and subtract. Strategy guide.

  1. Warm-up: Problem I08 (Intermediate, combinatorics)
  2. Main 1: Problem S12 (Senior, combinatorics)
  3. Main 2: Problem S13 (Senior, combinatorics)
  4. Main 3: Problem S17 (Senior, geometry)
  5. Challenge: Problem O09 (Olympiad-style, number theory, plan)
  6. Discussion: This is 3! × S(6, 3), where S is a Stirling number of the second kind: surjections onto k students.

Olympiad-style sessions (ages 15 to 18)

Session 1: Symmetry

Use a reflection, rotation or swap that leaves the problem unchanged to halve the work or pin down the answer. Strategy guide.

  1. Warm-up: Problem S10 (Senior, combinatorics)
  2. Main 1: Problem O22 (Olympiad-style, number theory, plan)
  3. Main 2: Problem O26 (Olympiad-style, combinatorics, plan)
  4. Main 3: Problem O31 (Olympiad-style, combinatorics, plan)
  5. Challenge: Problem O35 (Olympiad-style, combinatorics, plan)
  6. Discussion: where the main idea leads (in the printable pack, included with every A Level, IB, IGCSE and CBSE plan)

Session 2: Working backwards

Start from the end state or the answer you want and undo each step. Strategy guide.

  1. Warm-up: Problem S01 (Senior, number theory)
  2. Main 1: Problem O05 (Olympiad-style, number theory, plan)
  3. Main 2: Problem O11 (Olympiad-style, number theory, plan)
  4. Main 3: Problem O13 (Olympiad-style, number theory, plan)
  5. Challenge: Problem O27 (Olympiad-style, combinatorics, plan)
  6. Discussion: where the main idea leads (in the printable pack, included with every A Level, IB, IGCSE and CBSE plan)

Session 3: Extremal principle

Look at the largest, smallest, first or last object: it often has to behave in a special way. Strategy guide.

  1. Warm-up: Problem S07 (Senior, number theory)
  2. Main 1: Problem O07 (Olympiad-style, number theory, plan)
  3. Main 2: Problem O08 (Olympiad-style, number theory, plan)
  4. Main 3: Problem O10 (Olympiad-style, number theory, plan)
  5. Challenge: Problem O14 (Olympiad-style, number theory, plan)
  6. Discussion: where the main idea leads (in the printable pack, included with every A Level, IB, IGCSE and CBSE plan)

Session 4: Spot the pattern and generalise

Try small cases, spot the pattern, then explain why it must continue. Strategy guide.

  1. Warm-up: Problem S02 (Senior, number theory)
  2. Main 1: Problem O23 (Olympiad-style, number theory, plan)
  3. Main 2: Problem O24 (Olympiad-style, number theory, plan)
  4. Main 3: Problem O30 (Olympiad-style, combinatorics, plan)
  5. Challenge: Problem O38 (Olympiad-style, combinatorics, plan)
  6. Discussion: where the main idea leads (in the printable pack, included with every A Level, IB, IGCSE and CBSE plan)

Session 5: Proof techniques

Contradiction, contrapositive, induction and construction: ways to be certain, not just convinced. Strategy guide.

  1. Warm-up: Problem S35 (Senior, logic)
  2. Main 1: Problem O01 (Olympiad-style, number theory, plan)
  3. Main 2: Problem O02 (Olympiad-style, number theory, plan)
  4. Main 3: Problem O03 (Olympiad-style, number theory, plan)
  5. Challenge: Problem O06 (Olympiad-style, number theory, plan)
  6. Discussion: where the main idea leads (in the printable pack, included with every A Level, IB, IGCSE and CBSE plan)

Session 6: Organised cases

Split a problem into cases that cannot overlap and together cover everything, then deal with each small case. Strategy guide.

  1. Warm-up: Problem S03 (Senior, number theory)
  2. Main 1: Problem O04 (Olympiad-style, number theory, plan)
  3. Main 2: Problem O12 (Olympiad-style, number theory, plan)
  4. Main 3: Problem O15 (Olympiad-style, number theory, plan)
  5. Challenge: Problem O16 (Olympiad-style, number theory, plan)
  6. Discussion: where the main idea leads (in the printable pack, included with every A Level, IB, IGCSE and CBSE plan)

Set problems as homework

Open any problem page, switch on ‘Add to worksheet’ in the teachers’ drawer and pick problems: they go into your worksheet basket, ready to print or to set as homework for a class, with the solutions as the mark scheme. Start with Intermediate · Strategy guides · Extension & competition maths