Other tags that relate to Tree Graphs
Mathematical reasoning & proof. Topology. Creating and manipulating expressions and formulae. Inequalities. Visualising. Combinatorics. Working systematically. Networks/Graph Theory. Generalising. Number theory.There are 173 results
Broad Topics > Thinking Mathematically > Mathematical reasoning & proof
Tree Graphs
Age 16 to 18
Challenge Level
A connected graph is a graph in which we can get from any vertex to any other by travelling along the edges. A tree is a connected graph with no closed circuits (or loops. Prove that every tree has. . . .
Some Circuits in Graph or Network Theory
Age 14 to 18
Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.
Sprouts Explained
Age 7 to 18
This article invites you to get familiar with a strategic game called "sprouts". The game is simple enough for younger children to understand, and has also provided experienced mathematicians with. . . .
Advent Calendar 2011 - Secondary
Age 11 to 18
Challenge Level
Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.
Breaking the Equation ' Empirical Argument = Proof '
Age 7 to 18
This article stems from research on the teaching of proof and offers guidance on how to move learners from focussing on experimental arguments to mathematical arguments and deductive reasoning.
Knight Defeated
Age 14 to 16
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. . . .
Geometry and Gravity 2
Age 11 to 18
This is the second of two articles and discusses problems relating to the curvature of space, shortest distances on surfaces, triangulations of surfaces and representation by graphs.
A Long Time at the Till
Age 14 to 18
Challenge Level
Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem?
Cube Net
Age 16 to 18
Challenge Level
How many tours visit each vertex of a cube once and only once? How many return to the starting point?
Middle Man
Age 16 to 18
Challenge Level
Mark a point P inside a closed curve. Is it always possible to find two points that lie on the curve, such that P is the mid point of the line joining these two points?
Transitivity
Age 16 to 18
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.
Yih or Luk Tsut K'i or Three Men's Morris
Age 11 to 18
Challenge Level
Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots and a little about prime knots, crossing numbers and. . . .
Where Do We Get Our Feet Wet?
Age 16 to 18
Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.
Iffy Logic
Age 14 to 18
Challenge Level
Can you rearrange the cards to make a series of correct mathematical statements?
Ordered Sums
Age 14 to 16
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. . . .
Sperner's Lemma
Age 16 to 18
An article about the strategy for playing The Triangle Game which appears on the NRICH site. It contains a simple lemma about labelling a grid of equilateral triangles within a triangular frame.
And So on - and on -and On
Age 16 to 18
Challenge Level
Can you find the value of this function involving algebraic fractions for x=2000?
Magic W Wrap Up
Age 16 to 18
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.
Proof: A Brief Historical Survey
Age 14 to 18
If you think that mathematical proof is really clearcut and universal then you should read this article.
Look Before You Leap
Age 16 to 18
Challenge Level
Relate these algebraic expressions to geometrical diagrams.
Euler's Formula and Topology
Age 16 to 18
Here is a proof of Euler's formula in the plane and on a sphere together with projects to explore cases of the formula for a polygon with holes, for the torus and other solids with holes and the. . . .
Classifying Solids Using Angle Deficiency
Age 11 to 16
Challenge Level
Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry
Impossible Sandwiches
Age 11 to 18
In this 7-sandwich: 7 1 3 1 6 4 3 5 7 2 4 6 2 5 there are 7 numbers between the 7s, 6 between the 6s etc. The article shows which values of n can make n-sandwiches and which cannot.
Euclid's Algorithm II
Age 16 to 18
We continue the discussion given in Euclid's Algorithm I, and here we shall discover when an equation of the form ax+by=c has no solutions, and when it has infinitely many solutions.
Three Ways
Age 16 to 18
Challenge Level
If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.
Doodles
Age 14 to 16
Challenge Level
Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?
Exhaustion
Age 16 to 18
Challenge Level
Find the positive integer solutions of the equation (1+1/a)(1+1/b)(1+1/c) = 2
To Prove or Not to Prove
Age 14 to 18
A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.
Symmetric Tangles
Age 14 to 16
The tangles created by the twists and turns of the Conway rope trick are surprisingly symmetrical. Here's why!
Notty Logic
Age 16 to 18
Challenge Level
Have a go at being mathematically negative, by negating these statements.
On the Importance of Pedantry
Age 16 to 18
A introduction to how patterns can be deceiving, and what is and is not a proof.
Direct Logic
Age 16 to 18
Challenge Level
Can you work through these direct proofs, using our interactive proof sorters?
Russian Cubes
Age 14 to 16
Challenge Level
I want some cubes painted with three blue faces and three red faces. How many different cubes can be painted like that?
Pair Squares
Age 16 to 18
Challenge Level
The sum of any two of the numbers 2, 34 and 47 is a perfect square. Choose three square numbers and find sets of three integers with this property. Generalise to four integers.
Picturing Pythagorean Triples
Age 14 to 18
This article discusses how every Pythagorean triple (a, b, c) can be illustrated by a square and an L shape within another square. You are invited to find some triples for yourself.
Interpolating Polynomials
Age 16 to 18
Challenge Level
Given a set of points (x,y) with distinct x values, find a polynomial that goes through all of them, then prove some results about the existence and uniqueness of these polynomials.
Postage
Age 14 to 16
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. . . .
Without Calculus
Age 16 to 18
Challenge Level
Given that u>0 and v>0 find the smallest possible value of 1/u + 1/v given that u + v = 5 by different methods.
More Dicey Decisions
Age 16 to 18
Challenge Level
The twelve edge totals of a standard six-sided die are distributed symmetrically. Will the same symmetry emerge with a dodecahedral die?
Converse
Age 14 to 16
Challenge Level
Clearly if a, b and c are the lengths of the sides of an equilateral triangle then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true?
Proofs with Pictures
Age 14 to 18
Some diagrammatic 'proofs' of algebraic identities and inequalities.
A Computer Program to Find Magic Squares
Age 16 to 18
This follows up the 'magic Squares for Special Occasions' article which tells you you to create a 4by4 magicsquare with a special date on the top line using no negative numbers and no repeats.
The Clue Is in the Question
Age 16 to 18
Challenge Level
Starting with one of the mini-challenges, how many of the other mini-challenges will you invent for yourself?
Leonardo's Problem
Age 14 to 18
Challenge Level
A, B & C own a half, a third and a sixth of a coin collection. Each grab some coins, return some, then share equally what they had put back, finishing with their own share. How rich are they?
NRICH is part of the family of activities in the Millennium Mathematics Project.