THe Gumball Dilemma
The "Gumball Dilemma" problem, is a problem that tests students knowledge it conjecturing, and in probability of different scenarios based off of variables changing. The problem starts out by two twins walking with their mom when they come across a gumball machine. The kids want gumballs of the same color. The machine costs 1 cent per gumball, but there are two different colors of gumballs. How much would the mom have to spend in order to make sure the twins get the same color of gumball. Then, we are to explore the same problem with different numbers of kids and different amounts of gumballs. For example, triplets with a 5 color machine, or 6 kids with a 4 color machine.
In order to solve this problem, I made tables. These tables had 4 different variables represented by the four columns. (even though one out of the four was not included in the equation, and was only to represent a picture of them) We have C that represents the amount of different colors in the problem, K which is the amount of kids, D which represents the column where I make a drawing of the current situation, and P represents the price of guaranteeing that all the kids in the problem got the same number of gumballs. In the picture above, you can see that I took different numbers of C, and tested what would happen to P if we raised K by one for each ascending situation. Than I would draw out the picture to help me solve it. After completing this side, I proceeded to doing the same thing on the other side of the paper where the constant variable I was testing was K instead of C, and we were seeing the effects of changing C.
In order to solve this problem, I made tables. These tables had 4 different variables represented by the four columns. (even though one out of the four was not included in the equation, and was only to represent a picture of them) We have C that represents the amount of different colors in the problem, K which is the amount of kids, D which represents the column where I make a drawing of the current situation, and P represents the price of guaranteeing that all the kids in the problem got the same number of gumballs. In the picture above, you can see that I took different numbers of C, and tested what would happen to P if we raised K by one for each ascending situation. Than I would draw out the picture to help me solve it. After completing this side, I proceeded to doing the same thing on the other side of the paper where the constant variable I was testing was K instead of C, and we were seeing the effects of changing C.
After doing both of these, I was able to find an equation that represented the P, which is the maximum amount you would have to pay for any situation, in terms of C and K. This equation was KC-(C-1)=P. I am confident that my equation is one of the ones that work based off my frequent tests of all the different occurrences with K being 1-5 and C being 1-5. All of these 30 + tests that I ran all worked under that equation. If you plug equation KC-(C-1), to the problem of 3 kids and 3 different colors, you will get 3(3)-(3-1). This equals 9-2 which is the same as 7. If you look at my table above, you will see that the same row with 3 kids and 3 colors, will cost 7 cents maximum.
I believe I deserve a 10/10 on this POW because I worked hard to do many different tests to prove my equation. My drawings are neat along with the rest of my work and display the exact information needed to solve the problem, which is exactly what I did. Habits of a Mathematician that I used include Looking for Patterns and Staying Organized. I looked for patterns by scanning my results for the changes of P when I was increasing and decreasing the other variables. This was useful in helping me find the total equation in the end. I stayed organized by making tables to do my work instead of just sprawling different equations all over my paper.