The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?
Find all the solutions to the this equation.
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?
Find the five distinct digits N, R, I, C and H in the following nomogram
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?
Prove that the product of the sum of n positive numbers with the sum of their reciprocals is not less than n^2.
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?
Kyle and his teacher disagree about his test score - who is right?
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?
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?
However did we manage before calculators? Is there an efficient way to do a square root if you have to do the work yourself?
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?
Balance the bar with the three weight on the inside.
By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn
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.
What is the value of the integers a and b where sqrt(8-4sqrt3) = sqrt a - sqrt 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. . . .
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.
If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.
An algebra task which depends on members of the group noticing the needs of others and responding.
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.
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
A task which depends on members of the group noticing the needs of others and responding.
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...
Can you fit polynomials through these points?
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.
By proving these particular identities, prove the existence of general cases.
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
How to build your own magic squares.
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.
How good are you at finding the formula for a number pattern ?
Account of an investigation which starts from the area of an annulus and leads to the formula for the difference of two squares.
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. . . .
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 ?
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 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?
Can you find a rule which relates triangular numbers to square numbers?
What is the total number of squares that can be made on a 5 by 5 geoboard?
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.
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. . . .
Show that all pentagonal numbers are one third of a triangular number.
Can you show that you can share a square pizza equally between two people by cutting it four times using vertical, horizontal and diagonal cuts through any point inside the square?
Can you find a rule which connects consecutive triangular numbers?
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. . . .
Use algebra to reason why 16 and 32 are impossible to create as the sum of consecutive numbers.
Can you find the value of this function involving algebraic fractions for x=2000?
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.