Conjecturing and generalising

  • Chord
    problem

    Chord

    Age
    16 to 18
    Challenge level

    Equal touching circles have centres on a line. From a point of this line on a circle, a tangent is drawn to the farthest circle. Find the lengths of chords where the line cuts the other circles.

  • Fibonacci Factors
    problem

    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?

  • Least of All
    problem

    Least of All

    Age
    16 to 18
    Challenge level
    A point moves on a line segment. A function depends on the position of the point. Where do you expect the point to be for a minimum of this function to occur.
  • Patterns of inflection
    problem

    Patterns of Inflection

    Age
    16 to 18
    Challenge level

    Find the relationship between the locations of points of inflection, maxima and minima of functions.

  • Speedy summations
    problem

    Speedy Summations

    Age
    16 to 18
    Challenge level

    Watch the video to see how to add together an arithmetic sequence of numbers efficiently.

  • Ball bearings in a metal wheel.
    problem

    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.

  • Polite Numbers
    problem

    Polite Numbers

    Age
    16 to 18
    Challenge level

    A polite number can be written as the sum of two or more consecutive positive integers, for example 8+9+10=27 is a polite number. Can you find some more polite, and impolite, numbers?

  • Generally Geometric
    problem

    Generally Geometric

    Age
    16 to 18
    Challenge level

    Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.

  • Can it be?
    problem

    Can It Be?

    Age
    16 to 18
    Challenge level

    When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?

  • Maximum Scattering
    problem

    Maximum Scattering

    Age
    16 to 18
    Challenge level

    Your data is a set of positive numbers. What is the maximum value that the standard deviation can take?

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