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?

Can the number of $A$s on the whiteboard increase?

What restrictions exist on how we may reduce the number of $A$s?

… and what does that tell you about $p?$

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.