A machine gives out five pennies for each nickel inserted into it. The machine also gives out five nickels for each penny.
Starting out with one penny, is it possible to use the machine in such a manner as to end up with an equal number of nickels and pennies?
As a bonus, is it possible to end up with exactly one dollar?
There does not seem much mystery here.
(1) You start with one coin. On each iteration you put in one coin and get five out, for a net of four more coins. The total number of coins will always be 4n+1, so never the same number of pennies as nickles.
(2) You start with $ 0.01. On each iteration you either retain the same amount (if inserting a nickel), or you gain a net of 0.24 (if you insert a penny and get out 0.25 in nickels). Your total will always be 24n+1, so never $1.00.
One would have to stipulate conditions not described in order to diverge from these outcomes -- e.g. since there must be a finite number of coins to dispense, and some point there may be fewer than five pennies or five nickles to dispense. If that sort of thing is allowed, this no longer belongs in "just math."