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  #### Resources tagged with Creating and manipulating expressions and formulae similar to Circles in Circles:
  
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  ### There are 113 results
  
  Broad Topics > Algebraic expressions, equations and formulae > Creating and manipulating expressions and formulae
  
  ### Circles in Circles
  
  ##### Age 16 to 18 Challenge Level:
  
  This pattern of six circles contains three unit circles. Work out the radii of the other three circles and the relationship between them.
  
  ### Three Four Five
  
  ##### Age 14 to 16 Challenge Level:
  
  Two semi-circles (each of radius 1/2) touch each other, and a semi-circle of radius 1 touches both of them. Find the radius of the circle which touches all three semi-circles.
  
  ### Just Touching
  
  ##### Age 16 to 18 Challenge Level:
  
  Three semi-circles have a common diameter, each touches the other two and two lie inside the biggest one. What is the radius of the circle that touches all three semi-circles?
  
  ### Incircles
  
  ##### Age 16 to 18 Challenge Level:
  
  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 ...?
  
  ### Algebra from Geometry
  
  ##### Age 11 to 16 Challenge Level:
  
  Account of an investigation which starts from the area of an annulus and leads to the formula for the difference of two squares.
  
  ### Screen Shot
  
  ##### Age 14 to 16 Challenge Level:
  
  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. . . .
  
  ### Sitting Pretty
  
  ##### Age 14 to 16 Challenge Level:
  
  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?
  
  ### The Pillar of Chios
  
  ##### Age 14 to 16 Challenge Level:
  
  Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.
  
  ### Nicely Similar
  
  ##### Age 14 to 16 Challenge Level:
  
  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?
  
  ### Semi-square
  
  ##### Age 14 to 16 Challenge Level:
  
  What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?
  
  ### The Medieval Octagon
  
  ##### Age 14 to 16 Challenge Level:
  
  Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.
  
  ### Generating Triples
  
  ##### Age 14 to 16 Challenge Level:
  
  Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?
  
  ### There and Back
  
  ##### Age 14 to 16 Challenge Level:
  
  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?
  
  ### Magic Squares for Special Occasions
  
  ##### Age 11 to 16
  
  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.
  
  ### Inside Outside
  
  ##### Age 14 to 16 Challenge Level:
  
  Balance the bar with the three weight on the inside.
  
  ### Complex Partial Fractions
  
  ##### Age 16 to 18 Challenge Level:
  
  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.
  
  ### Reciprocals
  
  ##### Age 16 to 18 Challenge Level:
  
  Prove that the product of the sum of n positive numbers with the sum of their reciprocals is not less than n^2.
  
  ### ' Tis Whole
  
  ##### Age 14 to 18 Challenge Level:
  
  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?
  
  ### Salinon
  
  ##### Age 14 to 16 Challenge Level:
  
  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?
  
  ### Pareq Calc
  
  ##### Age 14 to 16 Challenge Level:
  
  Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .
  
  ### System Speak
  
  ##### Age 16 to 18 Challenge Level:
  
  Five equations... five unknowns... can you solve the system?
  
  ### Reasonable Algebra
  
  ##### Age 14 to 16 Challenge Level:
  
  Use algebra to reason why 16 and 32 are impossible to create as the sum of consecutive numbers.
  
  ### Algebra Match
  
  ##### Age 11 to 16 Challenge Level:
  
  A task which depends on members of the group noticing the needs of others and responding.
  
  ### Simplifying Doughnut
  
  ##### Age 14 to 18 Challenge Level:
  
  An algebra task which depends on members of the group noticing the needs of others and responding.
  
  ### Triangles Within Pentagons
  
  ##### Age 14 to 16 Challenge Level:
  
  Show that all pentagonal numbers are one third of a triangular number.
  
  ##### Age 14 to 16 Challenge Level:
  
  Robert noticed some interesting patterns when he highlighted square numbers in a spreadsheet. Can you prove that the patterns will continue?
  
  ### Fibonacci Factors
  
  ##### Age 16 to 18 Challenge Level:
  
  For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3?
  
  ### Polynomial Interpolation
  
  ##### Age 16 to 18 Challenge Level:
  
  Can you fit polynomials through these points?
  
  ### Always Two
  
  ##### Age 14 to 18 Challenge Level:
  
  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.
  
  ### Steel Cables
  
  ##### Age 14 to 16 Challenge Level:
  
  Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
  
  ### Poly Fibs
  
  ##### Age 16 to 18 Challenge Level:
  
  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.
  
  ### W Mates
  
  ##### Age 16 to 18 Challenge Level:
  
  Show there are exactly 12 magic labellings of the Magic W using the numbers 1 to 9. Prove that for every labelling with a magic total T there is a corresponding labelling with a magic total 30-T.
  
  ### Training Schedule
  
  ##### Age 14 to 16 Challenge Level:
  
  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?
  
  ### Pythagoras Proofs
  
  ##### Age 14 to 16 Challenge Level:
  
  Can you make sense of these three proofs of Pythagoras' Theorem?
  
  ### 2-digit Square
  
  ##### Age 14 to 16 Challenge Level:
  
  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?
  
  ### Interactive Number Patterns
  
  ##### Age 14 to 16 Challenge Level:
  
  How good are you at finding the formula for a number pattern ?
  
  ### Terminology
  
  ##### Age 14 to 16 Challenge Level:
  
  Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
  
  ### Hand Swap
  
  ##### Age 14 to 16 Challenge Level:
  
  My train left London between 6 a.m. and 7 a.m. and arrived in Paris between 9 a.m. and 10 a.m. At the start and end of the journey the hands on my watch were in exactly the same positions but the. . . .
  
  ### Lower Bound
  
  ##### Age 14 to 16 Challenge Level:
  
  What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =
  
  ### Balance Point
  
  ##### Age 14 to 16 Challenge Level:
  
  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?
  
  ### Pair Products
  
  ##### Age 14 to 16 Challenge Level:
  
  Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
  
  ### Magic Sums and Products
  
  ##### Age 11 to 16
  
  How to build your own magic squares.
  
  ### More Polynomial Equations
  
  ##### Age 16 to 18 Challenge Level:
  
  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.
  
  ### Gutter
  
  ##### Age 14 to 16 Challenge Level:
  
  Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?
  
  ### Ball Bearings
  
  ##### Age 16 to 18 Challenge Level:
  
  If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.
  
  ### Sums of Pairs
  
  ##### Age 11 to 16 Challenge Level:
  
  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?”
  
  ### Matchless
  
  ##### Age 14 to 16 Challenge Level:
  
  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 ?
  
  ### Absurdity Again
  
  ##### Age 16 to 18 Challenge Level:
  
  What is the value of the integers a and b where sqrt(8-4sqrt3) = sqrt a - sqrt b?
  
  ### Old Nuts
  
  ##### Age 16 to 18 Challenge Level:
  
  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?
  
  ### Pair Squares
  
  ##### Age 16 to 18 Challenge Level:
  
  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.
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