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.
How good are you at finding the formula for a number pattern ?
The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?
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. . . .
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
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?
For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3?
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.
What is the value of the integers a and b where sqrt(8-4sqrt3) = sqrt a - sqrt b?
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?
Prove that the product of the sum of n positive numbers with the sum of their reciprocals is not less than n^2.
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.
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?
Brian swims at twice the speed that a river is flowing, downstream from one moored boat to another and back again, taking 12 minutes altogether. How long would it have taken him in still water?
Find the five distinct digits N, R, I, C and H in the following nomogram
Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
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 =
Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
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.
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.
A task which depends on members of the group noticing the needs of others and responding.
The heptathlon is an athletics competition consisting of 7 events. Can you make sense of the scoring system in order to advise a heptathlete on the best way to reach her target?
Robert noticed some interesting patterns when he highlighted square numbers in a spreadsheet. Can you prove that the patterns will continue?
An algebra task which depends on members of the group noticing the needs of others and responding.
If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.
Can you make sense of these three proofs of Pythagoras' Theorem?
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.
By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn
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.
Can you fit polynomials through these points?
Use algebra to reason why 16 and 32 are impossible to create as the sum of consecutive numbers.
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.
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 proving these particular identities, prove the existence of general cases.
A moveable screen slides along a mirrored corridor towards a centrally placed light source. A ray of light from that source is directed towards a wall of the corridor, which it strikes at 45 degrees. . . .
Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?
Account of an investigation which starts from the area of an annulus and leads to the formula for the difference of two squares.
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.
However did we manage before calculators? Is there an efficient way to do a square root if you have to do the work yourself?
Fifteen students had to travel 60 miles. They could use a car, which could only carry 5 students. As the car left with the first 5 (at 40 miles per hour), the remaining 10 commenced hiking along the. . . .
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.
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. . . .
Five equations... five unknowns... can you solve the system?
Find b where 3723(base 10) = 123(base b).
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?”