How to build your own magic squares.
Five equations... five unknowns... can you solve the system?
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?”
Account of an investigation which starts from the area of an
annulus and leads to the formula for the difference of two squares.
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
A 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.
Robert noticed some interesting patterns when he highlighted square
numbers in a spreadsheet. Can you prove that the patterns will
An algebra task which depends on members of the group noticing the
needs of others and responding.
Prove that the product of the sum of n positive numbers with the sum of their reciprocals is not less than n^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?
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?
The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?
Find b where 3723(base 10) = 123(base b).
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.
To find the integral of a polynomial, evaluate it at some special
points and add multiples of these values.
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 ?
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
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. . . .
Can you hit the target functions using a set of input functions and a little calculus and algebra?
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.
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
For which values of n is the Fibonacci number fn even? Which
Fibonnaci numbers are divisible by 3?
A circle of radius r touches two sides of a right angled triangle,
sides x and y, and has its centre on the hypotenuse. Can you prove
the formula linking x, y and r?
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 the polynomial p(x) with integer coefficients such that one solution of the equation p(x)=0 is $1+\sqrt 2+\sqrt 3$.
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.
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.
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?
The incircles of 3, 4, 5 and of 5, 12, 13 right angled triangles
have radii 1 and 2 units respectively. What about triangles with an
inradius of 3, 4 or 5 or ...?
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.
This shape comprises four semi-circles. What is the relationship
between the area of the shaded region and the area of the circle on
AB as diameter?
Can you find the value of this function involving algebraic
fractions for x=2000?
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?
Derive an equation which describes satellite dynamics.
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.
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 find a rule which relates triangular numbers to square numbers?
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?
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.
First of all, pick the number of times a week that you would like
to eat chocolate. Multiply this number by 2...
By considering powers of (1+x), show that the sum of the squares of
the binomial coefficients from 0 to n is 2nCn
Relate these algebraic expressions to geometrical diagrams.
What angle is needed for a ball to do a circuit of the billiard
table and then pass through its original position?
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 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.