Pick up - Algebra Game

The first couple lessons I shared were from my Algebra 2 class, so I figured it was time to share something from Algebra 1.  This lesson revolves around a game that I think a student taught me years ago, but was similar to a game that we played in my college game theory class.  The kids have a lot of fun, because they get to compete and figure out strategy.  I call the game Pick up.

 I split students up into groups of twos, and give each group a few pieces of scrap paper.  I tell each group to rip up the paper to get 21 scraps of similar size.  On one of those scraps they write the words "Math Fun." Then I give them directions for the game, which goes like this:

• The 21 pieces are placed down on a desk or the floor with the "Math Fun" piece showing and visible the whole game, like above.
• Players take turns picking up at least 1, and up to 3 pieces at a time.
• Whoever must pick up the Math Fun piece loses.

Then I let the students play.  They get to practice for awhile to get a feel for some basic strategy, but soon we start a tournament. Students that don't win the first round play off for a wildcard spot or two later in the bracket.  Excitement builds, someone wins candy and then we begin discussion.

I prompt them with questions like "Well, what worked?"  The winner will definitely have figured out what to do at the end, but they won't need to step it all the way back to the start to win games, so they don't.  Then we start discussion around the question "Well, in what situation are you sure to win?"  We decide to not count the Math Fun piece because it's pretty much irrelevant and we go through each situation that occurs at the end of a player's turn assuming their opponent is playing perfectly.  It's fun for them think their way through it, and my students have been able to figure out the situations below with minimal prompting.

1- You lose, opponent takes 1.
2- You lose, opponent takes 2.
3- You lose, opponent takes 3
4- You WIN, opponent has to leave you with 1, 2 or 3.
5- You lose, opponent takes 1, leaving you with 4.
6- lose
7- lose
8- win
9- lose
10- lose
11- lose
12- win

At #8, they might see the pattern, at 12 they are sure of it.
"So, could we.... write a linear equation that would tell us the winning numbers?"
"If we counted the Math Fun piece, how would our winning situations change?  How would the equation change?"
blahblahblah Slope, blahblahblah y-intercept.  Hooray for math!

Extensions:
     Which player has the advantage if both play perfectly?
     What if you could take up to 4 pieces?  Only 2? 10?

Super Extension:
     What if you split it into 2 piles, with two special pieces, and could only take from one at a time?