Other tags that relate to Thousand Words
Quadratic equations. Mathematical reasoning & proof. Working systematically. Generalising. Graphs. Making and proving conjectures. Inequalities. Complex numbers. Creating and manipulating expressions and formulae. Indices.There are 75 results
Broad Topics > Using, Applying and Reasoning about Mathematics > 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.
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.
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?
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?
Pair Products
Age 14 to 16 Challenge Level:
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
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?
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. . . .
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.
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?
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?”
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?
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?
Lower Bound
Age 14 to 16 Challenge 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 =
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.
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.
Irrational Arithmagons
Age 16 to 18 Challenge Level:
Can you work out the irrational numbers that belong in the circles to make the multiplication arithmagon correct?
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?
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. . . .
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.
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.
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.
Harmonic Triangle
Age 14 to 16 Challenge Level:
Can you see how to build a harmonic triangle? Can you work out the next two rows?
Winning Lines
Age 7 to 16
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
Arithmagons
Age 14 to 16 Challenge Level:
Can you find the values at the vertices when you know the values on the edges?
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?
More Twisting and Turning
Age 11 to 16 Challenge Level:
It would be nice to have a strategy for disentangling any tangled ropes...
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.
Odd Differences
Age 14 to 16 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.
Magic Squares
Age 14 to 18
An account of some magic squares and their properties and and how to construct them for yourself.
Pinned Squares
Age 14 to 16 Challenge Level:
What is the total number of squares that can be made on a 5 by 5 geoboard?
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?
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?
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?
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?
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?
Jam
Age 14 to 16 Challenge Level:
To avoid losing think of another very well known game where the patterns of play are similar.
Take Three from Five
Age 14 to 16 Challenge 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?
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?
Cyclic Triangles
Age 16 to 18 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.
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.
Gnomon Dimensions
Age 14 to 16 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.
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.
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.
NRICH is part of the family of activities in the Millennium Mathematics Project.