22 original competition-style problems: counting arrangements, paths, subsets and tilings. 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.
In how many ways can 30 be written as a sum of three positive whole numbers, if the order of the three numbers does not matter?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Organised cases
How many subsets of {1, 2, 3, …, 10} (including the empty set) have a sum divisible by 5?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Symmetry, Spot the pattern and generalise
A staircase has 10 steps. In how many ways can you climb it if each stride covers 1, 2 or 3 steps?
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
A graph has 10 vertices and contains no triangle (no three vertices all joined to each other). What is the largest possible number of edges? Prove it.
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Extremal principle, Proof techniques
What is the largest number of whole numbers you can choose from 1 to 100 so that no chosen number is exactly double another chosen number?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Organised cases, Extremal principle
How many different triangles (counting triangles with the same three side lengths once) have whole-number side lengths and perimeter 2026?
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
The four cells of a 2 by 2 grid are each coloured with one of 3 colours. Two colourings count as the same if one can be rotated (by 90°, 180° or 270°) into the other. How many different colourings are there?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Symmetry
What is the largest number of subsets of {1, 2, 3, 4, 5, 6} that can be chosen so that no chosen subset is contained in another chosen subset?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Extremal principle, Proof techniques
What is the smallest n such that every sequence of n different real numbers contains an increasing subsequence of length 4 or a decreasing subsequence of length 4?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Pigeonhole principle, Extremal principle
How many arrangements of 1, 2, 3, 4, 5, 6 in a row have the property that every number (after the first) is either larger than all the numbers before it or smaller than all the numbers before it?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Working backwards
Four of the twelve corners of a regular 12-sided polygon are chosen. In how many ways can they be chosen so that they are the corners of a rectangle?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Symmetry, Proof techniques
A path goes from (0, 0) to (6, 6) using steps right (1, 0), up (0, 1) or diagonal (1, 1). How many such paths are there?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Organised cases
Five towns are to be connected by roads so that every town can reach every other, using exactly 4 roads (each road joins two towns directly). How many different road networks are possible?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Organised cases
What is the largest number of pieces into which 5 flat plane cuts can divide space (a block of cheese that is large enough)?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Spot the pattern and generalise, Working backwards
A chess knight starts in one corner of an 8 by 8 board. What is the smallest number of moves it needs to reach the diagonally opposite corner?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Parity and remainders, Extremal principle
How many strings of six 1s and six 0s have the property that, reading from the left, the number of 1s is never less than the number of 0s?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Count the opposite, Symmetry
What is the smallest n such that among any n points in the plane with whole-number coordinates, there are always two whose midpoint also has whole-number coordinates?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Pigeonhole principle, Parity and remainders
In how many ways can a 3 by 4 rectangle be tiled with 1 by 2 dominoes (placed either way)?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Organised cases, Spot the pattern and generalise
How many subsets of {1, 2, …, 15} (including the empty set) contain no two numbers that differ by exactly 3?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Organised cases
Six people sit around a round table. Arrangements that differ only by a rotation count as the same. In how many arrangements are Asha and Bilal not next to each other?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Symmetry, Count the opposite
How many sets {x, y, z} of three different positive whole numbers have x + y + z = 20?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Organised cases, Working backwards
A convex polyhedron has only pentagonal and hexagonal faces, and exactly three faces meet at every vertex. It has 20 hexagonal faces. How many vertices does it have?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Invariants, Proof techniques
More Olympiad-style problems: Number theory · Geometry · Algebra · Probability · Logic · Calculus and functions
Combinatorics 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
IGCSE Math Revision