Other tags that relate to Tree Graphs
Inequalities. Creating and manipulating expressions and formulae. Working systematically. Networks/Graph Theory. Combinatorics. Making and proving conjectures. Mathematical reasoning & proof. Number theory. Generalising. Visualising.There are 77 results
Broad Topics > Thinking Mathematically > Generalising
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.
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. . . .
Maximum Scattering
Age 16 to 18 Challenge Level:
Your data is a set of positive numbers. What is the maximum value that the standard deviation can take?
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?
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.
Multiplication Arithmagons
Age 14 to 16 Challenge Level:
Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?
Rational Roots
Age 16 to 18 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.
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.
Chocolate 2010
Age 14 to 16 Challenge Level:
First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...
Loopy
Age 14 to 16 Challenge 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?
Absurdity Again
Age 16 to 18 Challenge Level:
What is the value of the integers a and b where sqrt(8-4sqrt3) = sqrt a - sqrt b?
One, Three, Five, Seven
Age 11 to 16 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.
Train Spotters' Paradise
Age 11 to 16
Dave Hewitt suggests that there might be more to mathematics than looking at numerical results, finding patterns and generalising.
Hypotenuse Lattice Points
Age 14 to 16 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?
Attractive Tablecloths
Age 14 to 16 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?
Pick's Theorem
Age 14 to 16 Challenge Level:
Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.
Steel Cables
Age 14 to 16 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?
Converging Means
Age 14 to 16 Challenge 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. . . .
Arithmagons
Age 11 to 16 Challenge Level:
Can you find the values at the vertices when you know the values on the edges?
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.
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.
Painted Cube
Age 14 to 16 Challenge 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.
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?
Nim
Age 14 to 16 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.
Sums of Pairs
Age 11 to 16 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?”
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.
Of All the Areas
Age 14 to 16 Challenge Level:
Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?
What's Possible?
Age 14 to 16 Challenge Level:
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
Mystic Rose
Age 14 to 16 Challenge Level:
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
Square Pizza
Age 14 to 16 Challenge 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?
Magic Squares
Age 14 to 18
An account of some magic squares and their properties and and how to construct them for yourself.
Beelines
Age 14 to 16 Challenge Level:
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
Building Gnomons
Age 14 to 16 Challenge Level:
Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.
Jam
Age 14 to 16 Challenge Level:
To avoid losing think of another very well known game where the patterns of play are similar.
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.
A Tilted Square
Age 14 to 16 Challenge Level:
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
Plus Minus
Age 14 to 16 Challenge Level:
Can you explain the surprising results Jo found when she calculated the difference between square numbers?
Partly Painted Cube
Age 14 to 16 Challenge 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?
Sliding Puzzle
Age 11 to 16 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.
Polycircles
Age 14 to 16 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?
Shape and Territory
Age 16 to 18 Challenge Level:
If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?
Pentanim
Age 7 to 16 Challenge 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.
Fibonacci Factors
Age 16 to 18 Challenge Level:
For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3?
Cuboid Challenge
Age 11 to 16 Challenge Level:
What's the largest volume of box you can make from a square of paper?
NRICH is part of the family of activities in the Millennium Mathematics Project.