Resources tagged with: Mathematical reasoning & proof

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There are 174 results

Broad Topics > Thinking Mathematically > Mathematical reasoning & proof

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?

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.

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.

Cross-country Race

Age 14 to 16
Challenge Level

Eight children enter the autumn cross-country race at school. How many possible ways could they come in at first, second and third places?

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?

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?

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.

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. . . .

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.

Network Trees

Age 14 to 18
Challenge Level

Explore some of the different types of network, and prove a result about network trees.

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. . . .

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

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.

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. . . .

Mechanical Integration

Age 16 to 18
Challenge Level

To find the integral of a polynomial, evaluate it at some special points and add multiples of these values.

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. . . .

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.

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.

N000ughty Thoughts

Age 14 to 16
Challenge Level

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

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

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?

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. . . .

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.

Problem Solving, Using and Applying and Functional Mathematics

Age 5 to 18
Challenge Level

Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.

Proofs with Pictures

Age 14 to 18

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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. . . .

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.

An Alphanumeric

Age 16 to 18

Freddie Manners, of Packwood Haugh School in Shropshire solved an alphanumeric without using the extra information supplied and this article explains his reasoning.

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!

Iffy Logic

Age 14 to 18
Challenge Level

Can you rearrange the cards to make a series of correct mathematical statements?

Mind Your Ps and Qs

Age 16 to 18 Short
Challenge Level

Sort these mathematical propositions into a series of 8 correct statements.

Contrary Logic

Age 16 to 18
Challenge Level

Can you invert the logic to prove these statements?

Notty Logic

Age 16 to 18
Challenge Level

Have a go at being mathematically negative, by negating these statements.

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.

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?

Dodgy Proofs

Age 16 to 18
Challenge Level

These proofs are wrong. Can you see why?

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.

Binomial Coefficients

Age 14 to 18

An introduction to the binomial coefficient, and exploration of some of the formulae it satisfies.

More Number Sandwiches

Age 11 to 16
Challenge Level

When is it impossible to make number sandwiches?

The Bridges of Konigsberg

Age 11 to 18
Challenge Level

Investigate how networks can be used to solve a problem for the 18th Century inhabitants of Konigsberg.

Triangular Intersection

Age 14 to 16 Short
Challenge Level

What is the largest number of intersection points that a triangle and a quadrilateral can have?

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?

Look Before You Leap

Age 16 to 18
Challenge Level

Relate these algebraic expressions to geometrical diagrams.

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.

Find the Fake

Age 14 to 16
Challenge Level

There are 12 identical looking coins, one of which is a fake. The counterfeit coin is of a different weight to the rest. What is the minimum number of weighings needed to locate the fake coin?

An Introduction to Number Theory

Age 16 to 18

An introduction to some beautiful results in Number Theory.

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?

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.

The Great Weights Puzzle

Age 14 to 16
Challenge Level

You have twelve weights, one of which is different from the rest. Using just 3 weighings, can you identify which weight is the odd one out, and whether it is heavier or lighter than the rest?

A Knight's Journey

Age 14 to 18

This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.