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#### Resources tagged with Combinatorics similar to W Mates:

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Broad Topics > Decision Mathematics and Combinatorics > Combinatorics

### W Mates

##### Stage: 5 Challenge Level:

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.

### Plum Tree

##### Stage: 4 and 5 Challenge Level:

Label this plum tree graph to make it totally magic!

### Magic Caterpillars

##### Stage: 4 and 5 Challenge Level:

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

### Magic W

##### Stage: 4 Challenge Level:

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.

### Jluuis or Even Asutguus?

##### Stage: 5 Challenge Level:

Sixth challenge cipher

### Snowman

##### Stage: 4 Challenge Level:

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?

### 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.

### 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).

### Deep Roots

##### Stage: 4 Challenge Level:

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

### Lost in Space

##### Stage: 4 Challenge Level:

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?

### Ip?

##### Stage: 5 Challenge Level:

Seventh challenge cipher

### A Fine Thing?

##### Stage: 5 Challenge Level:

Second challenge cipher

### Stage 5 Cipher Challenge

##### Stage: 5 Challenge Level:

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

### Semicircle

##### Stage: 5 Challenge Level:

Fourth challenge cipher

### Olympic Magic

##### Stage: 4 Challenge Level:

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?

### Snooker Frames

##### Stage: 5 Challenge Level:

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?

### 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?

### Up a Semitone?

##### Stage: 5 Challenge Level:

Fifth challenge cipher

### Vital?

##### Stage: 5 Challenge Level:

Third challenge cipher

### Knight Defeated

##### Stage: 4 Challenge Level:

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. . . .

### Magic W Wrap Up

##### Stage: 5 Challenge Level:

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.

##### Stage: 5 Challenge Level:

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?

### Counting Binary Ops

##### Stage: 4 Challenge Level:

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.

### 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.

##### Stage: 4 Challenge Level:

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?

### Russian Cubes

##### Stage: 4 Challenge Level:

How many different cubes can be painted with three blue faces and three red faces? A boy (using blue) and a girl (using red) paint the faces of a cube in turn so that the six faces are painted. . . .

### One Basket or Group Photo

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

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.

### Penta Colour

##### Stage: 4 Challenge Level:

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?

### Modular Knights

##### Stage: 5 Challenge Level:

Try to move the knight to visit each square once and return to the starting point on this unusual chessboard.

### Doodles

##### Stage: 4 Challenge Level:

A 'doodle' is a closed intersecting curve drawn without taking pencil from paper. Only two lines cross at each intersection or vertex (never 3), that is the vertex points must be 'double points' not. . . .

### Euler's Officers

##### Stage: 4 Challenge Level:

How many different solutions can you find to this problem? Arrange 25 officers, each having one of five different ranks a, b, c, d and e, and belonging to one of five different regiments p, q, r, s. . . .

### Symmetric Tangles

##### Stage: 4

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

### Ordered Sums

##### Stage: 4 Challenge Level:

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. . . .

### Factorial Fun

##### Stage: 5 Challenge Level:

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

### Postage

##### Stage: 4 Challenge Level:

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. . . .

### Cube Net

##### Stage: 5 Challenge Level:

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

### N000ughty Thoughts

##### Stage: 4 Challenge Level:

Factorial one hundred (written 100!) has 24 noughts when written in full and that 1000! has 249 noughts? Convince yourself that the above is true. Perhaps your methodology will help you find the. . . .

### 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.

### Molecular Sequencer

##### Stage: 4 and 5 Challenge Level:

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

### Scratch Cards

##### Stage: 4 Challenge Level:

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?

### Snooker

##### Stage: 5 Challenge Level:

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