There's Something Here I Just Know It - Take Away Game

I don't know the name of this game, but I have always enjoyed it.  The game play is where two players have a bunch of objects in front of them (I used pennies) and they take turns taking away pennies.  They can take away up to a certain amount on their turn and the person who takes the last penny loses.

Here is the version I played:

We started with 43 pennies and the players could take away up to 5 pennies on their turn.  I won every single time.  The reactions I got from the students is something I want to bottle.  They were fighting over who got to play next.  They got their calculators and were doing crazy things there.  They were watching me closely.  They wanted to watch what I did over and over again.  They were playing against each other to try out their strategies (without being asked!).  And the same question again and again, "How are you winning?"  (answer at the bottom of the post)

There were other questions formulating in their heads that they weren't asking aloud.  Questions such as how many pennies should be remaining on the 2nd to last turn? and Does the number of pennies a player take depend on how many were taken on the last turn? 

They were shouting out ideas to each other during game play.  "She always takes 1 more than you do!", "She's doing math in her head!", "Just take the same amount she takes.",  "If she takes 5 you take 1!"

Once a student determined what I was doing and was able to beat me (okay so I didn't win every single time), I changed the rules.  Now it was 41 pennies and each player could take away up to 4 pennies.  If that student won again I knew they understood the how-to-win rules.  

Here's the thing.  This felt like a 1st act.  But I'm not exactly sure what the 2nd and 3rd acts are.  I'm not even sure of the math topic this is covering.  Patterns??  Linear Equations?  Developing formulas?? All of the above??  Please help me here.  What do I do next with my students?

Here is how I develop a version of the game.  I decide the maximum number of pennies a person could take on their turn.  Let's call this p.  Then I need to determine the total number of pennies needed to play this game.  There are many answers to this, but here is the formula g = t(p+1) + 1 where g is the total pennies in the game, and t is the number of turns each player would have before the end of the game.  

*Spoiler Alert*  Below is how to win.  

Remember that number p?  Add 1 to that.  That is the sum of the pennies that need to be taken during 2 turns.  For example, if you are playing a game where you can take up to 4 pennies, then 5 pennies need to be taken between you and the other player.  Allow the other player to go first.  If they take 1 penny, you take 4.  If they take 2 pennies, you take 3.  If they take 3 pennies, you take 2. And if they take 4 pennies you take 1.  

Sometimes the other player will insist that you go first.  If so just make sure you get back to being the one that takes the 5th penny.