Can you fit polynomials through these points?
Take a few whole numbers away from a triangle number. If you know the mean of the remaining numbers can you find the triangle number and which numbers were removed?
Find all the triples of numbers a, b, c such that each one of them plus the product of the other two is always 2.
Prove that the product of the sum of n positive numbers with the sum of their reciprocals is not less than n^2.
An algebra task which depends on members of the group noticing the needs of others and responding.
Five equations... five unknowns... can you solve the system?
Relate these algebraic expressions to geometrical diagrams.
Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?
A, B & C own a half, a third and a sixth of a coin collection. Each grab some coins, return some, then share equally what they had put back, finishing with their own share. How rich are they?
To break down an algebraic fraction into partial fractions in which all the denominators are linear and all the numerators are constants you sometimes need complex numbers.
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.
Show that for natural numbers x and y if x/y > 1 then x/y>(x+1)/(y+1}>1. Hence prove that the product for i=1 to n of [(2i)/(2i-1)] tends to infinity as n tends to infinity.
The sum of any two of the numbers 2, 34 and 47 is a perfect square. Choose three square numbers and find sets of three integers with this property. Generalise to four integers.
Take a complicated fraction with the product of five quartics top and bottom and reduce this to a whole number. This is a numerical example involving some clever algebra.
Given any two polynomials in a single variable it is always possible to eliminate the variable and obtain a formula showing the relationship between the two polynomials. Try this one.
Find all positive integers a and b for which the two equations: x^2-ax+b = 0 and x^2-bx+a = 0 both have positive integer solutions.
A sequence of polynomials starts 0, 1 and each poly is given by combining the two polys in the sequence just before it. Investigate and prove results about the roots of the polys.
Find all the solutions to the this equation.
Find relationships between the polynomials a, b and c which are polynomials in n giving the sums of the first n natural numbers, squares and cubes respectively.
By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn
If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.
To find the integral of a polynomial, evaluate it at some special points and add multiples of these values.
Can you prove that twice the sum of two squares always gives the sum of two squares?
For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3?
Can you find the value of this function involving algebraic fractions for x=2000?
A point moves around inside a rectangle. What are the least and the greatest values of the sum of the squares of the distances from the vertices?
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. . . .
By proving these particular identities, prove the existence of general cases.
Label this plum tree graph to make it totally magic!
The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . .
Show there are exactly 12 magic labellings of the Magic W using the numbers 1 to 9. Prove that for every labelling with a magic total T there is a corresponding labelling with a magic total 30-T.
In turn 4 people throw away three nuts from a pile and hide a quarter of the remainder finally leaving a multiple of 4 nuts. How many nuts were at the start?
What is the value of the integers a and b where sqrt(8-4sqrt3) = sqrt a - sqrt b?
There is a particular value of x, and a value of y to go with it, which make all five expressions equal in value, can you find that x, y pair ?
Attach weights of 1, 2, 4, and 8 units to the four attachment points on the bar. Move the bar from side to side until you find a balance point. Is it possible to predict that position?
This article explains how to make your own magic square to mark a special occasion with the special date of your choice on the top line.
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Can you find a rule which connects consecutive triangular numbers?
A 2-Digit number is squared. When this 2-digit number is reversed and squared, the difference between the squares is also a square. What is the 2-digit number?
How good are you at finding the formula for a number pattern ?
Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...
If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.
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?
Show that all pentagonal numbers are one third of a triangular number.
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
Can you find a rule which relates triangular numbers to square numbers?
Can you use the diagram to prove the AM-GM inequality?
What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =
This pattern of six circles contains three unit circles. Work out the radii of the other three circles and the relationship between them.
Find all the ways of placing the numbers 1 to 9 on a W shape, with 3 numbers on each leg, so that each set of 3 numbers has the same total.