Resources tagged with: Mathematical reasoning & proof

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Broad Topics > Thinking Mathematically > Mathematical reasoning & proof

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.

Particularly General

Age 16 to 18 Challenge Level:

By proving these particular identities, prove the existence of general cases.

Proof Sorter - Geometric Sequence

Age 16 to 18 Challenge Level:

Can you correctly order the steps in the proof of the formula for the sum of the first n terms in a geometric sequence?

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.

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.

Generally Geometric

Age 16 to 18 Challenge Level:

Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.

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?

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.

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?

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?

Common Divisor

Age 14 to 16 Challenge Level:

Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.

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.

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.

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.

Geometric Parabola

Age 14 to 16 Challenge Level:

Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.

Look Before You Leap

Age 16 to 18 Challenge Level:

Relate these algebraic expressions to geometrical diagrams.

Water Pistols

Age 16 to 18 Challenge Level:

With n people anywhere in a field each shoots a water pistol at the nearest person. In general who gets wet? What difference does it make if n is odd or even?

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.

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.

Iff

Age 14 to 18 Challenge Level:

Take a triangular number, multiply it by 8 and add 1. What is special about your answer? Can you prove it?

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

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. . . .

Plus or Minus

Age 16 to 18 Challenge Level:

Make and prove a conjecture about the value of the product of the Fibonacci numbers $F_{n+1}F_{n-1}$.

DOTS Division

Age 14 to 16 Challenge Level:

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

Our Ages

Age 14 to 16 Challenge Level:

I am exactly n times my daughter's age. In m years I shall be ... How old am I?

Multiplication Square

Age 14 to 16 Challenge Level:

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

Square Mean

Age 14 to 16 Challenge Level:

Is the mean of the squares of two numbers greater than, or less than, the square of their means?

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!

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.

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. . . .

Target Six

Age 16 to 18 Challenge Level:

Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.

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.

Always Perfect

Age 14 to 16 Challenge Level:

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

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?

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.

Fractional Calculus III

Age 16 to 18

Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.

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.

Number Rules - OK

Age 14 to 16 Challenge Level:

Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...

Composite Notions

Age 14 to 16 Challenge Level:

A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.

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 Squares II

Age 14 to 18

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

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. . . .

Thousand Words

Age 16 to 18 Challenge Level:

Here the diagram says it all. Can you find the diagram?

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.

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.

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?

Why 24?

Age 14 to 16 Challenge Level:

Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.