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?
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?
Let S1 = 1 , S2 = 2 + 3, S3 = 4 + 5 + 6 ,........ Calculate S17.
Balance the bar with the three weight on the inside.
Prove that 3 times the sum of 3 squares is the sum of 4 squares. Rather easier, can you prove that twice the sum of two squares always gives the sum of two squares?
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 all the triples of numbers a, b, c such that each one of them plus the product of the other two is always 2.
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?
How to build your own magic squares.
Use algebra to reason why 16 and 32 are impossible to create as the sum of consecutive numbers.
Account of an investigation which starts from the area of an annulus and leads to the formula for the difference of two squares.
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.
Can you fit polynomials through these points?
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.
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.
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3?
By proving these particular identities, prove the existence of general cases.
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.
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?
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?
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?
Can you use the diagram to prove the AM-GM inequality?
A task which depends on members of the group noticing the needs of others and responding.
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?
The sums of the squares of three related numbers is also a perfect square - can you explain why?
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.
Find all the solutions to the this equation.
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. . . .
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 good are you at finding the formula for a number pattern ?
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.
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.
Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?
The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?
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.
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?
Relate these algebraic expressions to geometrical diagrams.
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. . . .
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 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.
Find b where 3723(base 10) = 123(base b).