# Resources tagged with: Generalising

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### There are 78 results

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.

### 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?

### 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.

### 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.

### All Tangled Up

##### Age 14 to 18Challenge Level

Can you tangle yourself up and reach any fraction?

### 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.

### 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?

### 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?

### 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?

### 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?

### Integral Sandwich

##### Age 16 to 18Challenge Level

Generalise this inequality involving integrals.

### 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?

### The Bridges of Konigsberg

##### Age 11 to 18Challenge Level

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

### 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. . . .

### 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?

### Attractive Tablecloths

##### Age 14 to 16Challenge Level

Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?

### 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.

### Fractional Calculus II

##### Age 16 to 18

Here explore some ideas of how the definitions and methods of calculus change if you integrate or differentiate n times when n is not a whole number.

### 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.

### Fractional Calculus I

##### Age 16 to 18

You can differentiate and integrate n times but what if n is not a whole number? This generalisation of calculus was introduced and discussed on askNRICH by some school students.

### Magic Squares

##### Age 14 to 18

An account of some magic squares and their properties and and how to construct them for yourself.

### 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?

### 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.

### 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 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?

### 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?

### Incircles

##### Age 16 to 18Challenge Level

The incircles of 3, 4, 5 and of 5, 12, 13 right angled triangles have radii 1 and 2 units respectively. What about triangles with an inradius of 3, 4 or 5 or ...?

### 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. . . .

### 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?

### 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?

### 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.

### 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.

### 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.

### 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?”

### 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?

### 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?

### 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?

### Harmonic Triangle

##### Age 14 to 16Challenge Level

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

### 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?

### More Twisting and Turning

##### Age 11 to 16Challenge Level

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

### 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?