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?
In fact, this machine always increases your number of coins by 4. So the total number of coins always equals 1 mod 4. Number of pennies can never equal number of nickels.
And what about the $? Exchanging a nickel for 5 pennies does not change the value of the money that you have. But exchanging a penny for 5 nickels gives you an extra 24 cents. So the only amounts that are achievable are equal to 1 mod 24:
1 cent, 25 cents, 49 cents, 73 cents, 97 cents, 123 cents, etc.
$1.00 is not achievable.