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There are **55** NRICH Mathematical resources connected to **Combinatorics**, you may find related items under Decision mathematics and combinatorics.

Problem
Primary curriculum
Secondary curriculum
### In a Box

Chris and Jo put two red and four blue ribbons in a box. They each pick a ribbon from the box without looking. Jo wins if the two ribbons are the same colour. Is the game fair?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Snooker Frames

It is believed that weaker snooker players have a better chance of winning matches over eleven frames (i.e. first to win 6 frames) than they do over fifteen frames. Is this true?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Stage 5 Cipher Challenge

Can you crack these very difficult challenge ciphers? How might you systematise the cracking of unknown ciphers?

Age 16 to 18

Challenge Level

Article
Primary curriculum
Secondary curriculum
### An Introduction to Computer Programming and Mathematics

This article explains the concepts involved in scientific mathematical computing. It will be very useful and interesting to anyone interested in computer programming or mathematics.

Age 16 to 18

Problem
Primary curriculum
Secondary curriculum
### Molecular Sequencer

Investigate the molecular masses in this sequence of molecules and deduce which molecule has been analysed in the mass spectrometer.

Age 14 to 18

Challenge Level

Article
Primary curriculum
Secondary curriculum
### Symmetric Tangles

The tangles created by the twists and turns of the Conway rope trick are surprisingly symmetrical. Here's why!

Age 14 to 16

Article
Primary curriculum
Secondary curriculum
### Tangles

A personal investigation of Conway's Rational Tangles. What were the interesting questions that needed to be asked, and where did they lead?

Age 11 to 16

Problem
Primary curriculum
Secondary curriculum
### Cube Net

How many tours visit each vertex of a cube once and only once? How many return to the starting point?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Lost in Space

How many ways are there to count 1 - 2 - 3 in the array of triangular numbers? What happens with larger arrays? Can you predict for any size array?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Bell Ringing

Suppose you are a bellringer. Can you find the changes so that, starting and ending with a round, all the 24 possible permutations are rung once each and only once?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Ordered Sums

Let a(n) be the number of ways of expressing the integer n as an ordered sum of 1's and 2's. Let b(n) be the number of ways of expressing n as an ordered sum of integers greater than 1. (i) Calculate a(n) and b(n) for n<8. What do you notice about these sequences? (ii) Find a relation between a(p) and b(q). (iii) Prove your conjectures.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Postage

The country Sixtania prints postage stamps with only three values 6 lucres, 10 lucres and 15 lucres (where the currency is in lucres).Which values cannot be made up with combinations of these postage stamps? Prove that all other values can be made up.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Counting Binary Ops

How many ways can the terms in an ordered list be combined by repeating a single binary operation. Show that for 4 terms there are 5 cases and find the number of cases for 5 terms and 6 terms.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Magic W Wrap Up

Prove that you cannot form a Magic W with a total of 12 or less or with a with a total of 18 or more.

Age 16 to 18

Challenge Level

Article
Primary curriculum
Secondary curriculum
### Ways of Summing Odd Numbers

Sanjay Joshi, age 17, The Perse Boys School, Cambridge followed up the Madrass College class 2YP article with more thoughts on the problem of the number of ways of expressing an integer as the sum of odd numbers.

Age 11 to 14

Article
Primary curriculum
Secondary curriculum
### An Investigation Based on Score

Class 2YP from Madras College was inspired by the problem in NRICH to work out in how many ways the number 1999 could be expressed as the sum of 3 odd numbers, and this is their solution.

Age 11 to 14

Article
Primary curriculum
Secondary curriculum
### The Eternity Puzzle

A big prize was offered for solving The Eternity Puzzle, a jigsaw with no picture and every piece is the same on both sides. The finished result forms a regular dodecagon (12 sided polygon).

Age 16 to 18

Article
Primary curriculum
Secondary curriculum
### Transitivity

Suppose A always beats B and B always beats C, then would you expect A to beat C? Not always! What seems obvious is not always true. Results always need to be proved in mathematics.

Age 16 to 18

Article
Primary curriculum
Secondary curriculum
### Links and Knots

Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots, prime knots, crossing numbers and knot arithmetic.

Age 14 to 18

Problem
Primary curriculum
Secondary curriculum
### Master Minding

Your partner chooses two beads and places them side by side behind a screen. What is the minimum number of guesses you would need to be sure of guessing the two beads and their positions?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Domino Tetrads

Is it possible to use all 28 dominoes arranging them in squares of four? What patterns can you see in the solution(s)?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Plate Spotting

I was in my car when I noticed a line of four cars on the lane next to me with number plates starting and ending with J, K, L and M. What order were they in?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Deep Roots

Find integer solutions to: $\sqrt{a+b\sqrt{x}} + \sqrt{c+d.\sqrt{x}}=1$

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Paving Paths

How many different ways can I lay 10 paving slabs, each 2 foot by 1 foot, to make a path 2 foot wide and 10 foot long from my back door into my garden, without cutting any of the paving slabs?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Euromaths

How many ways can you write the word EUROMATHS by starting at the top left hand corner and taking the next letter by stepping one step down or one step to the right in a 5x5 array?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Walkabout

A walk is made up of diagonal steps from left to right, starting at the origin and ending on the x-axis. How many paths are there for 4 steps, for 6 steps, for 8 steps?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### One Basket or Group Photo

Libby Jared helped to set up NRICH and this is one of her favourite problems. It's a problem suitable for a wide age range and best tackled practically.

Age 7 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### How Many Dice?

A standard die has the numbers 1, 2 and 3 are opposite 6, 5 and 4 respectively so that opposite faces add to 7? If you make standard dice by writing 1, 2, 3, 4, 5, 6 on blank cubes you will find there are 2 and only 2 different standard dice. Can you prove this ?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Flagging

How many tricolour flags are possible with 5 available colours such that two adjacent stripes must NOT be the same colour. What about 256 colours?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Greetings

From a group of any 4 students in a class of 30, each has exchanged Christmas cards with the other three. Show that some students have exchanged cards with all the other students in the class. How many such students are there?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Snowman

All the words in the Snowman language consist of exactly seven letters formed from the letters {s, no, wm, an). How many words are there in the Snowman language?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Cube Paths

Given a 2 by 2 by 2 skeletal cube with one route `down' the cube. How many routes are there from A to B?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Penta Colour

In how many different ways can I colour the five edges of a pentagon so that no two adjacent edges are the same colour?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Scratch Cards

To win on a scratch card you have to uncover three numbers that add up to more than fifteen. What is the probability of winning a prize?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Tri-colour

Six points are arranged in space so that no three are collinear. How many line segments can be formed by joining the points in pairs?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Painting Cubes

Imagine you have six different colours of paint. You paint a cube using a different colour for each of the six faces. How many different cubes can be painted using the same set of six colours?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### W Mates

Show there are exactly 12 magic labellings of the Magic W using the numbers 1 to 9. Prove that for every labelling with a magic total T there is a corresponding labelling with a magic total 30-T.

Age 16 to 18

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Problem
Primary curriculum
Secondary curriculum
### Knight Defeated

The knight's move on a chess board is 2 steps in one direction and one step in the other direction. Prove that a knight cannot visit every square on the board once and only (a tour) on a 2 by n board for any value of n. How many ways can a knight do this on a 3 by 4 board?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Magic W

Find all the ways of placing the numbers 1 to 9 on a W shape, with 3 numbers on each leg, so that each set of 3 numbers has the same total.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Olympic Magic

in how many ways can you place the numbers 1, 2, 3 … 9 in the nine regions of the Olympic Emblem (5 overlapping circles) so that the amount in each ring is the same?

Age 14 to 16

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Problem
Primary curriculum
Secondary curriculum
### Magic Caterpillars

Label the joints and legs of these graph theory caterpillars so that the vertex sums are all equal.

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Euler's Officers

How many different ways can you arrange the officers in a square?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### N000ughty Thoughts

How many noughts are at the end of these giant numbers?

Age 14 to 16

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Problem
Primary curriculum
Secondary curriculum
### Snooker

A player has probability 0.4 of winning a single game. What is his probability of winning a 'best of 15 games' tournament?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Russian Cubes

I want some cubes painted with three blue faces and three red faces. How many different cubes can be painted like that?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Doodles

Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?

Age 14 to 16

Challenge Level