# Resources tagged with: Generalising

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Broad Topics > Thinking Mathematically > Generalising ### Rational Roots

##### Age 16 to 18Challenge Level

Given that a, b and c are natural numbers show that if sqrt a+sqrt b is rational then it is a natural number. Extend this to 3 variables. ### Magic Squares II

##### Age 14 to 18

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

##### Age 16 to 18Challenge Level

What is the value of the integers a and b where sqrt(8-4sqrt3) = sqrt a - sqrt b? ### Shape and Territory

##### Age 16 to 18Challenge Level

If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle? ### Interpolating Polynomials

##### Age 16 to 18Challenge 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. ### Generally Geometric

##### Age 16 to 18Challenge Level

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

##### Age 16 to 18Challenge Level

Can you work out the irrational numbers that belong in the circles to make the multiplication arithmagon correct? ### Can it Be?

##### Age 16 to 18Challenge Level

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

##### Age 16 to 18

This article by Alex Goodwin, age 18 of Madras College, St Andrews describes how to find the sum of 1 + 22 + 333 + 4444 + ... to n terms. ### Fibonacci Factors

##### Age 16 to 18Challenge Level

For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3? ### All Tangled Up

##### Age 14 to 18Challenge Level

Can you tangle yourself up and reach any fraction? ### Integral Sandwich

##### Age 16 to 18Challenge Level

Generalise this inequality involving integrals. ### 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. ### What's Possible?

##### Age 14 to 16Challenge Level

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make? ### Loopy

##### Age 14 to 16Challenge Level

Investigate sequences given by $a_n = \frac{1+a_{n-1}}{a_{n-2}}$ for different choices of the first two terms. Make a conjecture about the behaviour of these sequences. Can you prove your conjecture? ### Multiplication Square

##### Age 14 to 16Challenge Level

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice? ### Why Stop at Three by One

##### Age 16 to 18

Beautiful mathematics. Two 18 year old students gave eight different proofs of one result then generalised it from the 3 by 1 case to the n by 1 case and proved the general result. ### Problem Solving, Using and Applying and Functional Mathematics

##### Age 5 to 18Challenge 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. ### Pair Products

##### Age 14 to 16Challenge Level

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice? ### Pareq Calc

##### Age 14 to 16Challenge Level

Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . . ### Converging Means

##### Age 14 to 16Challenge Level

Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the. . . . ### Magic Squares

##### Age 14 to 18

An account of some magic squares and their properties and and how to construct them for yourself. ### Janine's Conjecture

##### Age 14 to 16Challenge 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. . . . ### Multiplication Arithmagons

##### Age 14 to 16Challenge Level

Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons? ### Maximum Scattering

##### Age 16 to 18Challenge Level

Your data is a set of positive numbers. What is the maximum value that the standard deviation can take? ### For Richer for Poorer

##### Age 14 to 16Challenge Level

Charlie has moved between countries and the average income of both has increased. How can this be so? ### Plus Minus

##### Age 14 to 16Challenge Level

Can you explain the surprising results Jo found when she calculated the difference between square numbers? ### Games Related to Nim

##### Age 5 to 16

This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning. ### Generating Triples

##### Age 14 to 16Challenge Level

Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more? ### Painted Cube

##### Age 14 to 16Challenge Level

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces? ### Winning Lines

##### Age 7 to 16

An article for teachers and pupils that encourages you to look at the mathematical properties of similar games. ### Partly Painted Cube

##### Age 14 to 16Challenge Level

Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use? ### Jam

##### Age 14 to 16Challenge Level

A game for 2 players ### Nim

##### Age 14 to 16Challenge Level

Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The loser is the player who takes the last counter. ### Mystic Rose

##### Age 14 to 16Challenge Level

Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes. ### Nim-like Games

##### Age 7 to 16Challenge Level

A collection of games on the NIM theme ### Arithmagons

##### Age 11 to 16Challenge Level

Can you find the values at the vertices when you know the values on the edges? ### Sums of Pairs

##### Age 11 to 16Challenge Level

Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?” ### Sliding Puzzle

##### Age 11 to 16Challenge Level

The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves. ### Take Three from Five

##### Age 11 to 16Challenge Level

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him? ### Cuboid Challenge

##### Age 11 to 16Challenge Level

What's the largest volume of box you can make from a square of paper? ### Cyclic Triangles

##### Age 16 to 18Challenge Level

Make and prove a conjecture about the cyclic quadrilateral inscribed in a circle of radius r that has the maximum perimeter and the maximum area. ### Searching for Mean(ing)

##### Age 11 to 16Challenge Level

If you have a large supply of 3kg and 8kg weights, how many of each would you need for the average (mean) of the weights to be 6kg? ### More Twisting and Turning

##### Age 11 to 16Challenge Level

It would be nice to have a strategy for disentangling any tangled ropes... ### Harmonic Triangle

##### Age 14 to 16Challenge Level

Can you see how to build a harmonic triangle? Can you work out the next two rows? ### Semi-square

##### Age 14 to 16Challenge Level

What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle? ### Odd Differences

##### Age 14 to 16Challenge Level

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = nÂ² Use the diagram to show that any odd number is the difference of two squares. ### Lower Bound

##### Age 14 to 16Challenge Level

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 = ### Pentanim

##### Age 7 to 16Challenge Level

A game for 2 players with similarities to NIM. Place one counter on each spot on the games board. Players take it is turns to remove 1 or 2 adjacent counters. The winner picks up the last counter. ### Square Pizza

##### Age 14 to 16Challenge Level

Can you show that you can share a square pizza equally between two people by cutting it four times using vertical, horizontal and diagonal cuts through any point inside the square?