# Resources tagged with: Mathematical reasoning & proof

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Broad Topics > Thinking Mathematically > Mathematical reasoning & proof ### 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. ### 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. ### 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? ### Plus or Minus

##### Age 16 to 18 Challenge Level:

Make and prove a conjecture about the value of the product of the Fibonacci numbers $F_{n+1}F_{n-1}$. ### 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? ### There's a Limit

##### Age 14 to 18 Challenge Level:

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely? ### 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. ### Golden Eggs

##### Age 16 to 18 Challenge Level:

Find a connection between the shape of a special ellipse and an infinite string of nested square roots. ### 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. ### 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? ### Particularly General

##### Age 16 to 18 Challenge Level:

By proving these particular identities, prove the existence of general cases. ### 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 ### 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? ### 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! ### 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. ### Triangle Incircle Iteration

##### Age 14 to 16 Challenge Level:

Keep constructing triangles in the incircle of the previous triangle. What happens? ### Iffy Logic

##### Age 14 to 18 Challenge Level:

Can you rearrange the cards to make a series of correct mathematical statements? ### 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}. ### Mind Your Ps and Qs

##### Age 16 to 18 Short Challenge Level:

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

##### Age 14 to 16 Challenge Level:

Can you see how this picture illustrates the formula for the sum of the first six cube numbers? ### 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. ### Contrary Logic

##### Age 16 to 18 Challenge Level:

Can you invert the logic to prove these statements? ### 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. . . . ### 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. ### Thousand Words

##### Age 16 to 18 Challenge Level:

Here the diagram says it all. Can you find the diagram? ### 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? ### 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. ### 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. ### Proofs with Pictures

##### Age 14 to 18

Some diagrammatic 'proofs' of algebraic identities and inequalities. ### 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. ### 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. ### 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. ### Can it Be?

##### Age 16 to 18 Challenge Level: ### 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. ### Proof of Pick's Theorem

##### Age 16 to 18 Challenge Level:

Follow the hints and prove Pick's Theorem. ### 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? ##### 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. ### 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. ### 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? ### 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. ### Impossible Triangles?

##### Age 16 to 18 Challenge Level:

Which of these triangular jigsaws are impossible to finish? ### 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? ### Direct Logic

##### Age 16 to 18 Challenge Level:

Can you work through these direct proofs, using our interactive proof sorters? ### 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. ### 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. ### 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. ### Dodgy Proofs

##### Age 16 to 18 Challenge Level:

These proofs are wrong. Can you see why?