The Daughter of The Tennis Ball Situation
This project was awarded a silver medal at the 2004 Hudson County Science Fair and the United States Metric Assosiation Award.
Abstract
The Daughter of The Tennis Ball Situation
I attended the Rutgers Young Scholars Program in Discrete Mathematics 2003 over the summer. There, I met professor Evan Wantland, the coordinator for Number Theory. I was motivated for this project by professor Wantland through his teachings of combinatorics and permutations. During one of his colloquiums I stumbled upon his experiment, which was an adaptation of the original The Tennis Ball Situation. The original experiment dealt with 2 balls being thrown in and 1 ball being thrown out. His adaptation was 3 balls being thrown in and 2 balls being thrown out. This experiment’s adaptation to the original concept was 4 balls being thrown in and 3 balls being thrown out. As the number of balls being thrown in increases the number of possibilities increases. The experiment was conducted by using combinatorics and permutations. The combination part of the experiment generated the sequence: 1,4,28,220,1820,15504,134596,1184040,10518300,94143280,
847660528,7669339132,69668534468,635013559600,5804731963800,
53194089192720,488526937079580,41432089765583440,
382460951663844400. The permutation section of the experiment generated the sequence: 1, 24, 168, 1320, 10920. The results of my project show that out of the two number sequences that were formed, permutations created a new number sequence and combinatorics generated one that already exists.
Abstract: Layman
The Daughter of The Tennis Ball Situation is an adaptation from the original problem of The Tennis Ball Situation. Using ascending numbered balls (i.e. 1, 2, 3, 4), I will hypothetically throw 4 numbered balls into a box, in which a monster lives. The monster, which hates tennis balls, only has three arms to throw the balls back. So, the monster throws back 3 tennis balls at random back to me. By doing this for about 9 to 10 levels (via combinatorics and permutations), two number series will be created. These two number series may be ones that are already in existence, for which this will act as an application, or a new number series, which this will act as its primary application.
Introduction
The objective is to explore the "4-tennis ball problem" (at each round 4 balls are thrown inside a box from witch an imaginary monster throws back three balls at a time). To solve this problem I used the concepts of combinatorics and permutations. Also, trees were used to represent all of the possibilities for the tennis balls thrown back. The hypothesis stated that the numbers in the number series would be those of a new number sequence. This means that an application is produced to support the creation and existence of the new number series.
What purpose does this experiment hold? Well, everyday, people use number sequences for scientific studies as well as for artistic purposes. The Fibonacci Sequence, for example, is used to find out different patterns to color a house, as well as to represent cell growth exponentially.
So, by understanding the value of number sequences, the appreciation and value of this experiment increases dramatically. In terms of this experiment, the concern was not the actual applications for the number series generated. Instead, it was just to simply create one. Once the sequence in generated, as a future study, research will be done to find real life applications.
Experimental
Materials
- Paper (about 50-100 sheets)
- Writing Utensils (about 5-10)
Procedure
1) Start with 1 as your initial stage (0 balls in, 0 balls out.)
2) Throw 4 numbered balls into the box (i.e. 1,2,3,and 4) and count the number of possibilities for the 3 balls thrown back (using combinatorics and permutations.)
3) Throw next 4 numbered balls and count the possibilities for the next 3 balls thrown.
4) Repeat step 3 until the desired stage is reached.
Results
COMBINATIONS
1,4,28,220,1820,15504,134596,1184040,10518300,94143280,
847660528,7669339132,69668534468,635013559600,5804731963800,
53194089192720,488526937079580,41432089765583440,
382460951663844400
PERMUTATIONS
1, 24, 168, 1320, 10920, 93024, 807576, 710424063109800, 564859680, 5085963168, 46016034790
Discussion
COMBINATIONS
Stage One- 1
In stage 1, there is only one possibility. Since there are 0 balls inside the box, the monster can throw 0 balls out. So, his only option is to not throw anything out.
Stage Two- 4
1, 2, 3 1, 2, 4
1, 3, 4 2, 3, 4
Stage Three- 28
4,5,6 4,5,7 4,5,8 5,6,7
5,6,8 6,7,8 2,5,6 2,5,6,
2,5,7 2,5,8 2,6,8 2,7,8
4,6,7 4,6,8 4,7,8 1,5,6
1,5,7 1,5,8 1,6,7 1,6,8
1,7,8 3,5,6 3,5,7 3,5,8
3,6,7 3,6,8 3,7,8 5,7,8
Stage One- 1
In stage 1, there is only one possibility. Since there are 0 balls inside the box, the monster can throw 0 balls out. So, his only option is to not throw anything out.
Stage Two- 24
1,2,3 1,2,4 1,3,4 2,3,4
1,3,2 1,4,2 1,4,3 2,4,3
2,1,3 2,1,4 3,1,4 3,2,4
2,3,1 2,4,1 3,4,1 3,4,2
3,2,1 4,1,2 4,1,3 4,2,3
3,1,2 4,2,1 4,3,1 4,3,2
Stage Three- 168
4,5,6 4,5,7 4,5,8 5,6,7 1,5,7 1,7,8
4,6,5 4,7,5 4,8,5 5,7,6 1,7,5 1,8,7
5,4,6 5,4,7 5,4,8 6,5,7 5,1,7 7,1,8
5,6,4 4,7,5 5,8,4 6,7,5 5,7,1 7,8,1
6,4,5 7,4,5 8,4,5 7,5,6 7,1,5 8,1,7
6,5,4 7,5,4 8,5,4 7,6,5 7,5,1 8,7,1
5,6,8 6,7,8 2,5,6 2,5,6, 1,5,8 3,5,6
5,8,6 6,8,7 2,6,5 2,6,5 1,8,5 3,6,5
6,5,8 7,6,8 5,2,6 5,2,6 5,1,8 5,3,6
6,8,5 7,8,6 5,6,2 5,6,2 5,8,1 5,6,3
8,5,6 8,6,7 6,2,5 6,2,5 8,1,5 6,3,5
8,6,5 8,7,6 6,5,2 6,5,2 8,5,1 6,5,3
2,5,7 2,5,8 2,6,8 2,7,8 1,6,7 3,5,7
2,7,5 2,8,5 2,8,6 2,8,7 1,7,6 3,7,5
5,2,7 5,2,8 6,2,8 7,2,8 6,1,7 5,3,7
5,7,2 5,8,2 6,8,2 7,8,2 6,7,1 5,7,3
7,2,5 8,2,5 8,2,6 8,2,7 7,1,6 7,3,5
7,5,2 8,5,2 8,6,2 8,7,2 7,6,1 7,5,3
4,6,7 4,6,8 4,7,8 1,5,6 1,6,8 3,5,8
4,7,6 4,8,6 4,8,7 1,6,5 1,8,6 3,8,5
6,4,7 6,4,8 7,4,8 5,1,6 6,1,8 5,3,8
6,7,4 6,8,4 7,8,4 5,6,1 6,8,1 5,8,3
7,6,4 8,4,6 8,4,7 6,1,5 8,1,6 8,3,5
7,4,6 8,6,4 8,7,4 6,5,1 8,6,1 8,5,3
3,6,7 3,6,8 3,7,8 5,7,8 3,7,6 3,8,6
3,8,7 5,8,7 6,3,7 6,3,8 7,3,8 7,5,8
6,7,3 6,8,3 7,8,3 7,8,5 7,3,6 8,3,6
8,3,7 8,5,7 7,6,3 8,6,3 8,7,3 8,7,5
During the progression of the project I realized that there was a pattern in both of the experiments.
COMBINATORICS
PERMUTATIONS
Conclusion
At the conclusion of the experiment, two number series were created. The number series that was created by the combinatorics part is one that already exists. It is called “Binomial Coefficients Binomials.” M. Abramowitz and I. A. Stegun originally created this number series. The Daughter of The Tennis Ball Situation will now serve as an application for that sequence.
As for the permutations part of the experiment, the sequence that was created is brand new. This sequence has not yet been discovered. So, my experiment will serve as it’s primary application.
Acknowledgements
I would like to acknowledge the people who supported me throughout the experimentation. First, I would like to thank my parents. They have supported me throughout my life, and here was no difference. Next, I would like to thank Dr. Evan Wantland, my professor of number theory at The Rutgers Young Scholars Program in Discrete Mathematics 2004. He introduced me to this new style of mathematics and counting. Finally, I would like to thank Mr. Jeremy Stanton. His perseverance throughout my experiment really affected my project for the better. Knowing that he thought of me as being responsible enough to handle the experimentation on my own, I wanted to succeed for him and not let him down. Thank you to everyone who helped me along my way to scientific discovery.