How good are you at finding the formula for a number pattern ?
What is the value of the integers a and b where sqrt(8-4sqrt3) = sqrt a - sqrt b?
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.
The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?
The sums of the squares of three related numbers is also a perfect square - can you explain why?
Find all the solutions to the this equation.
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.
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...
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?
By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn
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?
How to build your own magic squares.
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
Balance the bar with the three weight on the inside.
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?
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.
Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?
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.
This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?
For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3?
The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?
Prove that the product of the sum of n positive numbers with the sum of their reciprocals is not less than n^2.
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
Find the five distinct digits N, R, I, C and H in the following nomogram
Can you find the value of this function involving algebraic fractions for x=2000?
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.
To find the integral of a polynomial, evaluate it at some special points and add multiples of these values.
Find the missing angle between the two secants to the circle when the two angles at the centre subtended by the arcs created by the intersections of the secants and the circle are 50 and 120 degrees.
Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .
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 ?
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.
By proving these particular identities, prove the existence of general cases.
Can you explain why a sequence of operations always gives you perfect squares?
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. . . .
Can you fit polynomials through these points?
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.
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.
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.
Five equations... five unknowns... can you solve the system?
Kyle and his teacher disagree about his test score - who is right?
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. . . .
Can you use the diagram to prove the AM-GM inequality?
If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.
However did we manage before calculators? Is there an efficient way to do a square root if you have to do the work yourself?
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?
Robert noticed some interesting patterns when he highlighted square numbers in a spreadsheet. Can you prove that the patterns will continue?
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?
An algebra task which depends on members of the group noticing the needs of others and responding.
Relate these algebraic expressions to geometrical diagrams.
This pattern of six circles contains three unit circles. Work out the radii of the other three circles and the relationship between them.