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#### Resources tagged with Generalising similar to Binary Sequences:

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### Janine's Conjecture

##### Stage: 4 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. . . .

### Shape and Territory

##### Stage: 5 Challenge Level:

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

### Multiplication Square

##### Stage: 4 Challenge Level:

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

### Magic Squares II

##### Stage: 4 and 5

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

### Rational Roots

##### Stage: 5 Challenge 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.

### AMGM

##### Stage: 4 Challenge Level:

Can you use the diagram to prove the AM-GM inequality?

### Generally Geometric

##### Stage: 5 Challenge Level:

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

### Magic Squares

##### Stage: 4 and 5

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

### Fractional Calculus III

##### Stage: 5

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

### Nim-interactive

##### Stage: 3 and 4 Challenge 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 winner is the player to take the last counter.

### Fibonacci Factors

##### Stage: 5 Challenge Level:

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

### Nim

##### Stage: 4 Challenge 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.

### Nim-like Games

##### Stage: 2, 3 and 4 Challenge Level:

A collection of games on the NIM theme

### Sums of Pairs

##### Stage: 3 and 4 Challenge 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?”

### Integral Sandwich

##### Stage: 5 Challenge Level:

Generalise this inequality involving integrals.

### Plus Minus

##### Stage: 4 Challenge Level:

Can you explain the surprising results Jo found when she calculated the difference between square numbers?

### Games Related to Nim

##### Stage: 1, 2, 3 and 4

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.

### Interpolating Polynomials

##### Stage: 5 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.

### Pair Products

##### Stage: 4 Challenge Level:

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

### Steel Cables

##### Stage: 4 Challenge Level:

Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?

### Attractive Tablecloths

##### Stage: 4 Challenge 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?

### Odd Differences

##### Stage: 4 Challenge Level:

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.

### One, Three, Five, Seven

##### Stage: 3 and 4 Challenge Level:

A game for 2 players. Set out 16 counters in rows of 1,3,5 and 7. Players take turns to remove any number of counters from a row. The player left with the last counter looses.

### In a Spin

##### Stage: 4 Challenge Level:

What is the volume of the solid formed by rotating this right angled triangle about the hypotenuse?

### Winning Lines

##### Stage: 2, 3 and 4

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

### Irrational Arithmagons

##### Stage: 5 Challenge Level:

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

### Sliding Puzzle

##### Stage: 1, 2, 3 and 4 Challenge 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.

### Hypotenuse Lattice Points

##### Stage: 4 Challenge Level:

The triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN?

### Cyclic Triangles

##### Stage: 5 Challenge 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.

### Mystic Rose

##### Stage: 4 Challenge Level:

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

### Jam

##### Stage: 4 Challenge Level:

A game for 2 players

### Arithmagons

##### Stage: 4 Challenge Level:

Can you find the values at the vertices when you know the values on the edges?

### Multiplication Arithmagons

##### Stage: 4 Challenge Level:

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

### Generating Triples

##### Stage: 4 Challenge 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?

### Fractional Calculus I

##### Stage: 5

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.

### Building Gnomons

##### Stage: 4 Challenge Level:

Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.

### Incircles

##### Stage: 5 Challenge 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 ...?

### Polycircles

##### Stage: 4 Challenge Level:

Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

### Pentanim

##### Stage: 2, 3 and 4 Challenge Level:

A game for 2 players with similaritlies 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.

### Sum the Series

##### Stage: 5

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.

### Absurdity Again

##### Stage: 5 Challenge Level:

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

### Jam

##### Stage: 4 Challenge Level:

To avoid losing think of another very well known game where the patterns of play are similar.

### Gnomon Dimensions

##### Stage: 4 Challenge Level:

These gnomons appear to have more than a passing connection with the Fibonacci sequence. This problem ask you to investigate some of these connections.

### Chord

##### Stage: 5 Challenge Level:

Equal touching circles have centres on a line. From a point of this line on a circle, a tangent is drawn to the farthest circle. Find the lengths of chords where the line cuts the other circles.

### Maximum Scattering

##### Stage: 5 Challenge Level:

Your data is a set of positive numbers. What is the maximum value that the standard deviation can take?

### Beelines

##### Stage: 4 Challenge Level:

Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?

### What's Possible?

##### Stage: 4 Challenge Level:

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

### Why Stop at Three by One

##### Stage: 5

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

##### Stage: 1, 2, 3, 4 and 5 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.