How Likely Is That?
Page 4

Shelly Berman shelly@mathforum.org


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The Game of Greed

The goal in the Game of Greed is always to get the highest score. The real problem is to find a strategy that will get the highest score over the long run. That is, any particular strategy may not work well on any particular turn, but will that strategy yield the highest score when used consistently for every turn?

Level 1

In groups of three, players take turns flipping a coin.

On your turn, you flip the coin.
If you get a tail on any flip, your score for the round is 0, and your turn is over. You must pass the coin to the next player.

If you get a head on this first flip, you get 1 point and can choose to flip again or stop. If you stop now you get the 1 point for that round.

If you make a second flip and get a tail, your score for the round is 0, and your turn is over.

If you get a second head, you get an additional point and can choose to flip again.

Continuing to flip will add 1 more point for each head you get in that turn, BUT if you get a tail before you stop you get NO points for that round.

Play a few rounds and see what strategy you think will get you the highest score.

Level 2

The play starts the same way. You get a 0 score and pass the coin if you get a tail. If you get a head the first time, you get 1 point and can choose to flip again.

In level 2 play, getting a tail still gets you a 0 score for the round and your turn is over. However, every time you get a head it is worth twice as much as the last one. You get 2 additional points for the second head, the next one is worth 4 points, then 8, and so on. If you get a tail before you stop voluntarily, you get NO points for that round.

Play a few rounds and see if you'd use the same or a different strategy as level 1 play to get the highest score.

Level 3

Instead of flipping one coin at a time, you can choose to flip as many coins as you like when it is your turn. If you get ANY tails, your score for that turn is 0. If you get all heads, they double in value for each head.
What would your score be if you got 1 head? 2 heads? 3 heads? 4 heads? Can you guess what your score would be if you flipped 10 coins and they all came up heads!

Now what strategy seems most likely to get the highest score?

Level 4

Instead of using coins, we will use number cubes for this level of Greed. When it is your turn you roll a number cube. If you roll a 1, your turn is over, you get a score of 0 for the turn, and you pass the cube.
If you roll any other number, you get as many points as the number rolled and can choose to stop or roll again.
As long as you get any number other than 1, that value is added to the points you can get for that round if you stop. You can choose to continue to roll, BUT if you roll a 1, no matter how many points you had for your turn until then, your score for the turn is 0.

Play a few rounds and discuss what seems to be a good winning strategy.

Level 5

Like level 4, this is played with number cubes, and the scoring stays the same you get the value for any numbers except 1, and a 1 gives you a score of zero for the turn. In level 5, you can choose to roll as many cubes at a time for your turn.

Does this change your strategy for a best score?

Level 6

You begin each turn in level 6 by throwing a pair of number cubes. If you throw a double-1, a double-3 or a double-6, your score is 0 and your turn is ended. For any other combination of numbers, you get the product of the two numbers and can choose to roll again. As before, you can continue to toss until you throw a losing double or until you choose to stop.

When would you decide to stop on any particular turn?

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