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Practice Paper 3 Question 8

A whiteboard has \(p\) \(``A"\) symbols and \(m\) \(``B"\) symbols written on it. You choose any two symbols to erase and replace them with another one according to the following rules:

  • \(A A \Rightarrow B\)
  • \(A B\text{ or } BA \Rightarrow A\)
  • \(B B \Rightarrow B\)

If you continue to apply replacements for as long as possible, which values of \(p\) and \(m\) result in a single \(A\) remaining at the end?

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  • Hint 1
    Can the number of \(A\)s on the whiteboard increase?
  • Hint 2
    What restrictions exist on how we may reduce the number of \(A\)s?
  • Hint 3
    ... and what does that tell you about \(p?\)
  • Hint 4
    The question essentially asks to remove all \(B\)s. Couild there be a situation when you cannot remove a B? Justify.


The number of \(A\)s can never be increased, so \(p\geq1.\) Also, as the number of \(A\)s may decrease only by \(2\) at a time via the first rule, \(p\) must be odd (since we need one \(A\) at the end). In contrast, there is no constraint on the initial number of \(B\)s. Since the number of \(A\)s is always greater than \(0,\) we will always be able to apply a replacement if there is at least one \(B.\) Every replacement that involves a \(B\) reduces the number of \(B\)s by one. Consequently, the number of \(B\)s will be reduced to \(0\) no matter how many we started with.

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