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.