The sums of the squares of three related numbers is also a perfect square - can you explain why?
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?
For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3?
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.
Find all the solutions to the this equation.
However did we manage before calculators? Is there an efficient way to do a square root if you have to do the work yourself?
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. . . .
The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?
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.
Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
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.
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.
Can you fit polynomials through these points?
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.
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?
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...
Find the five distinct digits N, R, I, C and H in the following nomogram
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
Use algebra to reason why 16 and 32 are impossible to create as the sum of consecutive numbers.
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?
An algebra task which depends on members of the group noticing the needs of others and responding.
Balance the bar with the three weight on the inside.
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.
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?
Can you make sense of these three proofs of Pythagoras' Theorem?
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.
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.
How to build your own magic squares.
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?
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?
Account of an investigation which starts from the area of an annulus and leads to the formula for the difference of two squares.
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?
To find the integral of a polynomial, evaluate it at some special points and add multiples of these values.
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.
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?
Find b where 3723(base 10) = 123(base b).
Five equations... five unknowns... can you solve the system?
If the hypotenuse (base) length is 100cm and if an extra line splits the base into 36cm and 64cm parts, what were the side lengths for the original right-angled triangle?
If a sum invested gains 10% each year how long before it has doubled its value?
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?
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?
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.