Resources tagged with: Mathematical reasoning & proof

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Broad Topics > Mathematical Thinking > Mathematical reasoning & proof

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.

Look Before You Leap

Age 16 to 18 Challenge Level:

Relate these algebraic expressions to geometrical diagrams.

Diverging

Age 16 to 18 Challenge Level:

Show that for natural numbers x and y if x/y > 1 then x/y>(x+1)/(y+1}>1. Hence prove that the product for i=1 to n of [(2i)/(2i-1)] tends to infinity as n tends to infinity.

Unit Interval

Age 14 to 18 Challenge Level:

Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?

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.

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.

Polynomial Relations

Age 16 to 18 Challenge Level:

Given any two polynomials in a single variable it is always possible to eliminate the variable and obtain a formula showing the relationship between the two polynomials. Try this one.

Dodgy Proofs

Age 16 to 18 Challenge Level:

These proofs are wrong. Can you see why?

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?

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.

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.

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. . . .

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.

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?

Binomial

Age 16 to 18 Challenge Level:

By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn

Proofs with Pictures

Age 14 to 18

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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.

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

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.

Quadratic Harmony

Age 16 to 18 Challenge Level:

Find all positive integers a and b for which the two equations: x^2-ax+b = 0 and x^2-bx+a = 0 both have positive integer solutions.

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?

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.

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.

How Many Solutions?

Age 16 to 18 Challenge Level:

Find all the solutions to the this equation.

Contrary Logic

Age 16 to 18 Challenge Level:

Can you invert the logic to prove these statements?

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.

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?

Road Maker

Age 14 to 18 Challenge Level:

Which of these roads will satisfy a Munchkin builder?

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.

Telescoping Functions

Age 16 to 18

Take a complicated fraction with the product of five quartics top and bottom and reduce this to a whole number. This is a numerical example involving some clever algebra.

Janine's Conjecture

Age 14 to 16 Challenge Level:

Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .

Sixational

Age 14 to 18 Challenge Level:

The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . .

Can it Be

Age 16 to 18 Challenge Level:

When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?

Proof of Pick's Theorem

Age 16 to 18 Challenge Level:

Follow the hints and prove Pick's Theorem.

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?

Notty Logic

Age 16 to 18 Challenge Level:

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

Little and Large

Age 16 to 18 Challenge Level:

A point moves around inside a rectangle. What are the least and the greatest values of the sum of the squares of the distances from the vertices?

Iffy Logic

Age 14 to 18 Challenge Level:

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

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.

Mind Your Ps and Qs

Age 16 to 18 Short Challenge Level:

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

Direct Logic

Age 16 to 18 Challenge Level:

Can you work through these direct proofs, using our interactive proof sorters?

Polite Numbers

Age 16 to 18 Challenge Level:

A polite number can be written as the sum of two or more consecutive positive integers. Find the consecutive sums giving the polite numbers 544 and 424. What characterizes impolite numbers?

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.

Integral Inequality

Age 16 to 18 Challenge Level:

An inequality involving integrals of squares of functions.

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?

Basic Rhythms

Age 16 to 18 Challenge Level:

Explore a number pattern which has the same symmetries in different bases.

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. . . .

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. . . .