Resources tagged with: Mathematical reasoning & proof

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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.

Magic Squares II

Age 14 to 18

An article which gives an account of some properties of magic squares.

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!

Contrary Logic

Age 16 to 18
Challenge Level

Can you invert the logic to prove these statements?

Dodgy Proofs

Age 16 to 18
Challenge Level

These proofs are wrong. Can you see 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?

Road Maker

Age 14 to 18
Challenge Level

Which of these roads will satisfy a Munchkin builder?

Proof of Pick's Theorem

Age 16 to 18
Challenge Level

Follow the hints and prove Pick's Theorem.

More Number Sandwiches

Age 11 to 16
Challenge Level

When is it impossible to make number sandwiches?

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?