Reasoning, convincing and proving

  • Quad in Quad
    problem

    Quad in quad

    Age
    14 to 18
    Challenge level
    filled star filled star empty star

    Join the midpoints of a quadrilateral to get a new quadrilateral. What is special about it?

  • Fixing the Odds
    problem

    Fixing the odds

    Age
    14 to 16
    Challenge level
    filled star filled star filled star
    You have two bags, four red balls and four white balls. You must put all the balls in the bags although you are allowed to have one bag empty. How should you distribute the balls between the two bags so as to make the probability of choosing a red ball as small as possible and what will the probability be in that case?
  • Tri-Colour
    problem

    Tri-colour

    Age
    11 to 14
    Challenge level
    filled star filled star empty star
    Six points are arranged in space so that no three are collinear. How many line segments can be formed by joining the points in pairs?
  • More marbles
    problem

    More marbles

    Age
    11 to 14
    Challenge level
    filled star filled star filled star
    I start with a red, a blue, a green and a yellow marble. I can trade any of my marbles for three others, one of each colour. Can I end up with exactly two marbles of each colour?
  • Find the fake
    problem

    Find the fake

    Age
    14 to 16
    Challenge level
    filled star empty star empty star
    There are 12 identical looking coins, one of which is a fake. The counterfeit coin is of a different weight to the rest. What is the minimum number of weighings needed to locate the fake coin?
  • Legs Eleven
    problem

    Legs eleven

    Age
    11 to 14
    Challenge level
    filled star filled star empty star
    Take any four digit number. Move the first digit to the end and move the rest along. Now add your two numbers. Did you get a multiple of 11?
  • Marbles
    problem

    Marbles

    Age
    11 to 14
    Challenge level
    filled star filled star empty star
    I start with a red, a green and a blue marble. I can trade any of my marbles for two others, one of each colour. Can I end up with five more blue marbles than red after a number of such trades?
  • Always the Same
    problem

    Always the same

    Age
    11 to 14
    Challenge level
    filled star filled star filled star
    Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number. Cross out the numbers on the same row and column. Repeat this process. Add up you four numbers. Why do they always add up to 34?
  • Rule of Three
    problem

    Rule of three

    Age
    11 to 14
    Challenge level
    filled star empty star empty star
    If it takes four men one day to build a wall, how long does it take 60,000 men to build a similar wall?
  • No Right Angle Here
    problem

    No right angle here

    Age
    14 to 16
    Challenge level
    filled star filled star filled star
    Prove that the internal angle bisectors of a triangle will never be perpendicular to each other.