Can you fit polynomials through these points?
Five equations... five unknowns... can you solve the system?
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 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.
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.
Prove that the product of the sum of n positive numbers with the sum of their reciprocals is not less than n^2.
Find all the solutions to the this equation.
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.
An algebra task which depends on members of the group noticing the needs of others and responding.
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.
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.
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?
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?
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 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.
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.
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.
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. . . .
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?
For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3?
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.
The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?
Can you produce convincing arguments that a selection of statements about numbers are true?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
If you know the perimeter of a right angled triangle, what can you say about the area?
The sums of the squares of three related numbers is also a perfect square - can you explain why?
Can you find the value of this function involving algebraic fractions for x=2000?
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.
How to build your own magic squares.
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.
There are unexpected discoveries to be made about square numbers...
Robert noticed some interesting patterns when he highlighted square numbers in a spreadsheet. Can you prove that the patterns will continue?
What is special about the difference between squares of numbers adjacent to multiples of three?
By proving these particular identities, prove the existence of general cases.
Can you explain what is going on in these puzzling number tricks?
Account of an investigation which starts from the area of an annulus and leads to the formula for the difference of two squares.
A task which depends on members of the group noticing the needs of others and responding.
Find the five distinct digits N, R, I, C and H in the following nomogram
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
Can you explain why a sequence of operations always gives you perfect squares?
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?
Kyle and his teacher disagree about his test score - who is right?
Find b where 3723(base 10) = 123(base b).
A mother wants to share a sum of money by giving each of her children in turn a lump sum plus a fraction of the remainder. How can she do this in order to share the money out equally?
Label this plum tree graph to make it totally magic!
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. . . .