Initially, they are not all heads or tails. Thus, there are at least one heads and one tails, and the third coin is unknown. WLOG, we will assume the third coin is a heads. Thus, there are two heads and 1 tails.
Lets say we get them to flip
a single coin (say the nickel), hoping that this is the one coin different from the others. We are right 33% of the time and end the game with 3 heads. The rest of the time, we flip one of the heads, so we end up with two tails (one of which is the nickel) and one heads.
If we didn't end the game (66% of the time), then we can get them to flip
one of the remaining two coins (say the dime) hoping it is heads. If we are right (50% of the time) we have ended the game. Otherwise, the dime was tails and we now have two heads (dime and quarter) and one tails (nickel).
The last move is to
flip the nickel again.
So, the simplified strategy is simply
1. Flip nickel
2. Flip dime
3. Flip nickel
What ever starting configuration, this will guarantee to get you all three the same at some point.
Here is a quick demonstration of all possibilities. Say that the nickel is value "X" where X can be heads or tails. The following are the possible configurations for Nickel, Dime, Quarter:
Note that "XXX" is not a valid configuration since we would already be done. Also, note that since we have abstracted out "heads" and "tails", we don't need to consider "YXX" since that is symmetrical to "XYY".
Here is what happens when we apply the above strategy to these configurations:
* XYY -> YYY
* XXY -> YXY -> YYY
* XYX -> YYX -> YXX -> XXX
You can see that the first finished in one move, the second in two, and the third in three.