Cool Party Trick! - Combinatorics Game

Here is a fun game you can play anywhere, anytime, with anyone. I call it, "Count to 16", although the name is slightly misleading. The reason I call it this is to give the impression that the game is easy to learn, which it is.

The game goes like this:

1. There are two players.
2. The two players each have pennies.
3. One player goes first, the other player goes second, they alternate turns.
4. On the player's turn, the player must put in either one, two, or three pennies in a pile.
5. The first player to make the pile 16 pennies, wins! (the pile starts at zero pennies)

Now I will show you how I usually present this game to someone who is not familiar with combinatorics.

Me: "Hello, do you want to play a game?"
Somebody: "Sure, what is it?"
Me: "It's called, Count to 16. It's really easy."
Somebody: "How does it work?"
Me: "Okay, there is a pile of pennies. Each of us have pennies. We each put in 1, 2, or 3 pennies into the pile when it is our turn. We alternate turns. The first person to make the pile 16 pennies wins. Do you want to play?"
Somebody: "Uh, okay."
Me: "Great, my hands will represent the pile, and my fingers will represent the pennies."

Note: This is when I hold out my hands with my fist closed, representing zero pennies in the pile.

Me: "Do you want to go first or second?"
Note: This is a crucial element to the game, which I will explain later.

Somebody: "Uh, I'll go first."
Me: "Okay, do you want to put in 1, 2, or 3 pennies into the pile?"
Somebody: "I will put in 3."

Me: "Okay, I put in 1."
Note: The hands below have a total of 4 fingers showing, representing a total of 4 pennies in the pile.

Me: "How many do you want to put in?"
Somebody: "I will put in 3."
Note: Total count is now 7.

Me: "Okay, I put in 1."

Somebody: "I put in 2."

Me: "I put in 2."
Note: Total count is 12. My hands roll over to 2 because I only have 10 fingers.

Somebody: "uh, uh, it doesn't look good for me. Because no matter what I put in, 1, 2, or 3 pennies, you can put in the correct number to get to 16 pennies."
Me: "yeah." (hehe)

The trick is whoever goes second, will always be able to win. At least, if they know how.

Me: "Do you want to play again?"
Somebody: "Okay, but this time I'll go second."

Me: "Okay, I'll put in 1."

Somebody: "I'll put in 3."
Note: Total count 4.

Me: "I'll put in 3."

Somebody: "I'll put in 3."
Note: Total count is now 10.

Me: "I'll put in 2."
Note: Total count 12.

Somebody: "Ah, you got me again. No matter what I put in, you can get to 16."
Me: "yeah." (hehe)

The secret is that whoever is able to get to 12 will be able to win. But how can you guarantee you will get to 12? If you are able to get to 8, you can guarantee yourself to get to 12. But how can you guarantee you will be able to get to 8? If you can get to 4 you will be able to get to 8.

This is a classic combinatorics game that teaches basic concepts in programming. The idea is to get to 4, 8, 12, and then 16 to win the game. Below is a chart showing how many pennies to put in depending on how many pennies are currently in the pile.

Note: If the current pile has 4 or 8 pennies, you have to try and throw the other person off so you can get to 8 or 12 pennies.