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.
Balance the bar with the three weight on the inside.
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?
Find all the solutions to the this equation.
A task which depends on members of the group noticing the needs of others and responding.
Kyle and his teacher disagree about his test score - who is right?
Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?
The sums of the squares of three related numbers is also a perfect square - can you explain why?
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.
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?
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 ?
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.
An algebra task which depends on members of the group noticing the needs of others and responding.
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 explain why a sequence of operations always gives you perfect squares?
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.
Can you fit polynomials through these points?
If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.
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?
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.
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.
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.
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.
Find the five distinct digits N, R, I, C and H in the following nomogram
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...
115^2 = (110 x 120) + 25, that is 13225 895^2 = (890 x 900) + 25, that is 801025 Can you explain what is happening and generalise?
Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
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.
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
Account of an investigation which starts from the area of an annulus and leads to the formula for the difference of two squares.
The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?
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?
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?
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. . . .
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?”
Find b where 3723(base 10) = 123(base b).
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. . . .
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?
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?
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.