8 original competition-style problems: truth-tellers, calendars, games and reasoning puzzles. Try each one before opening the hints; the second hint gives more away, and the full solution explains why the method works and where the idea leads.
Every question is free to read. The hints, answer checking and full solutions are included with every A Level, IB, IGCSE and CBSE plan. See plans.
Answer a problem to track what you have solved.
Press Project on any problem to show it full screen with a timer, the hints, the answer and the worked solution one step at a time (arrow keys move between problems; Space reveals the next step; F full screen; Esc closes). Switch on the ‘Add to worksheet’ buttons, pick problems, then print them from the worksheet builder or set them as homework for a class, with the full solutions as the mark scheme. Free problems are free for every class; problems marked ‘With a plan’ can be set by teachers with a plan or school licence. Ready-made sessions: maths club packs.
A strip of 2026 squares is empty. Two players take turns placing a domino on two neighbouring empty squares. A player who cannot move loses. With perfect play, who wins?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Symmetry, Invariants
Ten coins lie heads up. A move turns over exactly four of the coins (any four). What is the smallest number of moves that makes all ten coins tails up?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Parity and remainders, Extremal principle
Ten people stand in a row, numbered 1 to 10. Each is a truth-teller or a liar. Person k says: ‘At least k of us ten are liars.’ How many truth-tellers are there?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Organised cases, Parity and remainders
The numbers 1, 2, 3 are written on a board. A move replaces two of the numbers a and b by (a + b)/√2 and (a − b)/√2. Can the board ever show 1, √2 and 1 + √2?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Invariants, Proof techniques
Some squares of a 4 by 4 board are to be shaded so that every 2 by 2 block of squares (there are 9 of them, overlapping) contains at least one shaded square. What is the smallest number of shaded squares needed?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Extremal principle, Pigeonhole principle
What is the largest number of whole numbers you can choose from 1 to 20 so that no chosen number is the sum of two chosen numbers (the two may be equal)?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Extremal principle, Pigeonhole principle
A counter starts at 0. Each move either adds 1 to it or doubles it. What is the smallest number of moves needed to reach 100?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Working backwards, Spot the pattern and generalise
Each of 6 people at a party knows some of the others (knowing is mutual). Nobody knows everyone else, and no two people know the same number of others. Is this possible? If so, how many people does the best-connected person know; if not, answer 0.
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Pigeonhole principle, Proof techniques
More Olympiad-style problems: Number theory · Combinatorics · Geometry · Algebra · Probability · Calculus and functions
Logic at other levels: Junior (ages 11 to 13) · Intermediate (ages 13 to 16) · Senior (ages 16 to 18)
Problem-solving strategies · Where next · Extension & competition maths
A Level Math Revision