The Wizard's Cookie Game!

Using G = A(1+r)^N

If you get totally frustrated, you COULD try the equations the Wizard has below under his own section. But it will be more challenging and fun for you to try your own.

The Cookie Game is played like the game of Pig. That is, one die is rolled and a number such as 2 comes up. The players can keep in the game if they want for the next and succeeding rolls. They will gain cookies based upon an exponential growth equation. If, however, the number 2 comes up again, they lose everything they've gained for that series. So they need to know when to stop and hold on to some winnings.

Then for the second round of die throwing, they will lose cookies based on a decay equation that takes into consideration how much they have already won on the previous rolls. If they stay in, there will be less and less cookies subtracted from their original total. Again if they lose, they will lose the maximum from their winnings.

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TO DO:

y = Ar^N

G = A(1 + r)^N for Growth

D = A(1 - r)^M for Decay

Where A is the starting amount, r is the rate of growth or decay and N or M are the number of rolls for each trial.

For example you MIGHT try:

G = (2)2^N.

You will probably find that this number goes up pretty fast, which is good for those who want to GET a lot of cookies, but not so good for the cookie provider, in this case the Wizard! See that after 4 rolls, you would have, what, 32 cookies, for goodness sake! (2 times 2 to the fourth power, which is 2 times 16). So you will need to experiment with more realistic growth and decay curves. Once you have a GOWTH equation, experiment with DECAY equations. It might be a good idea to somehow link what you get for N, the highest roll for growth before quiting, with the second, DECAY equation. If you come up with an interesting equation, the Wizard will post it here with your name, credit etc! If you are a teacher, you are invited to do this with your class and send in a report to the Wizard at cwisnia@mcn.org---you will also have your equations posted!

Hints to consider: Use N = number of die rolls for growth and M for number of die rolls for decay. Your equation may want to be such that the rate of decay will depend upon the number of die rolls N--that is, if N is larger, that will determine where the decay starts, and how fast it decays, even though M is the number of decay rolls of the die. Your equation(s) should not allow for the cookie winner to win a whole lot of cookies---the decay should take away some of them on the average. I mean lets face it, there are a limit to how many cookies any one person may win! Once you have determined your equation(s), do the EXPLORATIONS below to TEST your equations.

What happens if you stay in for 5 rounds of growth, then stay in for 3 rounds of decay? (N = 5; M = 3) What about only 1 round of decay?

What happens if you stay in for 4 rounds growth, then stay in for 4 of decay? (N = 4; M = 4)

When is a good time to quit and keep what you have with your equation? How do you know

Is this game fair to both the player and to the "giver of cookies"?

Suppose you wanted to "load" this game so that it would usually pay out 1-2 cookies. Is this a good equation or should you change it some?

Could you come up with one to pay off 1-2 cookies regularly?

WRITE Your final Equation(s) and e-mail your results to cwisnia@mcn.org

2. The Cookie Game: The Wizard's Version!

The Cookie Game is played like the game of Pig. That is, one die is rolled and a number such as 2 comes up. The players can keep in the game if they want for the next and succeeding rolls. They will gain cookies based upon an exponential growth equation. If, however, the number 2 comes up again, they lose everything they've gained for that series. So they need to know when to stop and hold on to some winnings.

Then for the second round, they will lose cookies based on a decay equation that takes intoi consideration how much they have already won on the previous rolls. If they stay in, there will be less and less cookies subtracted from their original total. Again if they lose, they will lose the maximum from their winnings.

Here ARE the equations

G = (1.5)^N - 1 ( a growth curve)

D = 1.5N(1-.15)^M ( a decay curve)

TOTAL COOKIES earned:

C = G - D

To Explore:

What happens if you stay in for 5 rounds of growth, then stay in for 3 round of decay? (N = 5; M = 3) What about only 1 round of decay? What happens if you only do 1 or 2 growth rolls?

What happens if you stay in for 4 rounds, then stay in for 4 of the second? (N = 4; M = 4)

When is a good time to quit and keep what you have?

What happens if you roll 6 times and have a certain amount of cookies, then you want to know how many you would have after 3, 4, or 5 times of "decay"? How many would you have in each case?