Reasoning, convincing and proving

  • Paradoxes
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    Paradoxes

    A paradox is a statement that seems to be both untrue and true at the same time. This article looks at a few examples and challenges you to investigate them for yourself.
  • Sprouts Explained
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    Sprouts explained

    This article invites you to get familiar with a strategic game called "sprouts". The game is simple enough for younger children to understand, and has also provided experienced mathematicians with significant food for thought.
  • Go Forth and Generalise
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    Go forth and generalise

    Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.
  • Euclid's Algorithm II
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    Euclid's algorithm II

    We continue the discussion given in Euclid's Algorithm I, and here we shall discover when an equation of the form ax+by=c has no solutions, and when it has infinitely many solutions.

  • con Tricks
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    Con tricks

    Here are some examples of 'cons', and see if you can figure out where the trick is.
  • Powerful properties
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    Powerful properties

    Yatir from Israel wrote this article on numbers that can be written as $ 2^n-n $ where n is a positive integer.
  • Mouhefanggai
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    Mouhefanggai

    Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai.
  • Volume of a Pyramid and a Cone
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    Volume of a pyramid and a cone

    These formulae are often quoted, but rarely proved. In this article, we derive the formulae for the volumes of a square-based pyramid and a cone, using relatively simple mathematical concepts.