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Broad Topics > Decision Mathematics and Combinatorics > Combinatorics

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In a Box

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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?

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Snooker Frames

Stage: 5 Challenge Level: Challenge Level:1

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?

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Ip?

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Seventh challenge cipher

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Jluuis or Even Asutguus?

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Sixth challenge cipher

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Up a Semitone?

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Fifth challenge cipher

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Semicircle

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Fourth challenge cipher

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Vital?

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Third challenge cipher

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A Fine Thing?

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Second challenge cipher

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Stage 5 Cipher Challenge

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

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An Introduction to Computer Programming and Mathematics

Stage: 5

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.

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Molecular Sequencer

Stage: 4 and 5 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Symmetric Tangles

Stage: 4

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

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Tangles

Stage: 3 and 4

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

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Shuffle Shriek

Stage: 3 Challenge Level: Challenge Level:1

Can you find all the 4-ball shuffles?

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Cube Net

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Lost in Space

Stage: 4 Challenge Level: Challenge Level:1

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?

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More Beads

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

With red and blue beads on a circular wire; 'put a red bead between any two of the same colour and a blue between different colours then remove the original beads'. Keep repeating this. What happens?

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Bell Ringing

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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?

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Ordered Sums

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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. . . .

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Factorial Fun

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

How many divisors does factorial n (n!) have?

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Postage

Stage: 4 Challenge Level: Challenge Level:1

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. . . .

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Counting Binary Ops

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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.

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Magic W Wrap Up

Stage: 5 Challenge Level: Challenge Level:1

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.

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Ways of Summing Odd Numbers

Stage: 3

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. . . .

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An Investigation Based on Score

Stage: 3

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.

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The Eternity Puzzle

Stage: 5

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).

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Transitivity

Stage: 5

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.

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Links and Knots

Stage: 4 and 5

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.

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Master Minding

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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?

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Domino Tetrads

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

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Plate Spotting

Stage: 2 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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?

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Deep Roots

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

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Small Change

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

In how many ways can a pound (value 100 pence) be changed into some combination of 1, 2, 5, 10, 20 and 50 pence coins?

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Paving Paths

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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?

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Euromaths

Stage: 3 Challenge Level: Challenge Level:1

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?

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Walkabout

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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?

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One Basket or Group Photo

Stage: 2, 3, 4 and 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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.

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How Many Dice?

Stage: 3 Challenge Level: Challenge Level:1

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. . . .

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Flagging

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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?

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Greetings

Stage: 3 Challenge Level: Challenge Level:1

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. . . .

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Snowman

Stage: 4 Challenge Level: Challenge Level:1

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?

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Cube Paths

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Penta Colour

Stage: 4 Challenge Level: Challenge Level:1

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

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Scratch Cards

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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?

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Permute It

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Take the numbers 1, 2, 3, 4 and 5 and imagine them written down in every possible order to give 5 digit numbers. Find the sum of the resulting numbers.

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Tri-colour

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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?

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Painting Cubes

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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?

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W Mates

Stage: 5 Challenge Level: Challenge Level:1

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.

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Knight Defeated

Stage: 4 Challenge Level: Challenge Level:1

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. . . .

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Magic W

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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.