POW 2
Kick It!
POW Write Up Kick It!
The football team named the Free Thinkers chose a different system for scoring points during a game. They decided that each field goal is worth 5 points, and each touchdown is worth 3 points. In their games, the only way to score points is to either score a field goal or a tough down. There are no other ways of scoring points that exist such as extra points or safeties. Therefore, points can only be accumulated by adding different combinations of 5 and 3. Someone on the free thinker’s team noticed that because of this rule not every whole number is a possible total score for their team. Immediately she noticed that 1 was impossible along with 2 and 4. Based on her thinking, she speculates that there exists some highest impossible whole number for their team’s total score. The task is to find this highest impossible score.
The first task that I was assigned to do was to find the highest impossible score for the Free Thinkers team and to justify why every number that follows that highest impossible number will work. To begin, I made a table with three columns labeled field goals, touchdowns and total points. Then, I made an initial list in the total points from 0 – 10 to see which numbers could be made by combining 3 and 5. I began to put check marks by numbers that were possible and the number or each touchdown and field goal in their column, and an x by the impossible numbers. I knew that 0 was obviously possible and that 1 and 2 were impossible because they are both less than 3. Next, I noted that 3, 5, 6 and 10 were possible where 4 and 7 were the only impossible after 2. Immediately after seeing these possibilities, I made a note that all multiples of 3 and 5 were going to be possible scores and put check marks by all those numbers up to 20.
I then continued working on my table of possibilities and saw that every possible score from 8 on was possible. I then decided to make the statement that 7 was my highest impossible score. My table, however, only went to 20 so I began to look for a way to prove that every number after 7 was possible. To begin to prove that 7 was the highest impossible score I began with the string of possible scores 8, 9 and 10. I stated that because these 3 numbers in sequence are all possible, adding 3 to each of them individually to infinity will cover every possible whole number from 8 on. Because I am adding 3 each time I know that they will be possible scores because the team gets 3 points for each field goal. This means that each time I add 3, it is like the team scored a field goal. This reasoning led me to come to the conclusion that 7 was definitely the highest impossible score for the Free Thinkers.
To take this problem one step further, I decided to try to see what the highest impossible score would be in an actual football game with a touchdown worth 6 points and a field goal worth 3 points. I began the same way with a table including columns for each possible score and started with the total scores of 0 – 20. Due to my findings in the previous situation, I knew that all multiples of 3 and 6 were possibilities so I marked those as possible first. I made my table of possible and impossible total scores all the way to 24 and noticed that every third number was possible. This is where I saw a flaw in my problem. Since 6 was a multiple of 3, I would not have a highest impossible number because every third number worked and there will never be a string of 3 possible numbers for me to use to prove a highest impossible score as in the first scenario. I tested my theory all the way to 50 and the pattern continued on with every third number being a possible with the two in-between as impossible. I then concluded that there would be no highest impossible number for the point system that I picked because one of the possible points was a multiple of the other.
In my table I noticed several patterns. One was in the column for field goals. After the highest impossible score of 7, the number of field goals began to repeat the pattern 1, 3, 0, 2, 4. I tested out this pattern and it worked well for my table that went to 22 but I am not quite certain that this pattern is anything but a coincidence because there are more than one ways to make the total scores other than the way I chose to combine 3 and 5. This means that for another person, they could have found a totally different pattern based on the combinations that they chose. I also noticed a pattern in the column for touchdowns that was a little more complex yet still worked for the entirety of my table. Starting with 8, there was a pattern of 1, 0, 2, 1, 0. Then starting at 13 the pattern was 2, 1, 3, 2, 1. This symmetrical pattern with each number increasing by one worked when I carried on through 22 but I had the same thoughts as with the first pattern. These patterns could help me in my table, but for another person, they may have one that is completely different.
I then began to put together a theory that in order to be able to find a highest impossible score, the scores for a touchdown and a field goal had to be relatively prime so there were no possible same factors between them, and one would not be a multiple of the other. I tested this theory with 5 and 7 to find that 23 was the highest impossible score.
When first given this problem I thought it would be very black and white. I thought there would just be one answer and one way to get there. After completing my work this problem left me wanting to continue on and find other pattern and look at other possible combinations of numbers and for this reason I consider this a great problem. It started off slow with making the table and finding ways to add the two numbers, but after I found that everything after 7 worked, my brain took off and I was so caught up in the problem that I discovered things I never thought would come from this problem. For example, reading the problem I never would have guessed that it was at all related to numbers being relatively prime. I came to this conclusion on my own which was amazing.
The football team named the Free Thinkers chose a different system for scoring points during a game. They decided that each field goal is worth 5 points, and each touchdown is worth 3 points. In their games, the only way to score points is to either score a field goal or a tough down. There are no other ways of scoring points that exist such as extra points or safeties. Therefore, points can only be accumulated by adding different combinations of 5 and 3. Someone on the free thinker’s team noticed that because of this rule not every whole number is a possible total score for their team. Immediately she noticed that 1 was impossible along with 2 and 4. Based on her thinking, she speculates that there exists some highest impossible whole number for their team’s total score. The task is to find this highest impossible score.
The first task that I was assigned to do was to find the highest impossible score for the Free Thinkers team and to justify why every number that follows that highest impossible number will work. To begin, I made a table with three columns labeled field goals, touchdowns and total points. Then, I made an initial list in the total points from 0 – 10 to see which numbers could be made by combining 3 and 5. I began to put check marks by numbers that were possible and the number or each touchdown and field goal in their column, and an x by the impossible numbers. I knew that 0 was obviously possible and that 1 and 2 were impossible because they are both less than 3. Next, I noted that 3, 5, 6 and 10 were possible where 4 and 7 were the only impossible after 2. Immediately after seeing these possibilities, I made a note that all multiples of 3 and 5 were going to be possible scores and put check marks by all those numbers up to 20.
I then continued working on my table of possibilities and saw that every possible score from 8 on was possible. I then decided to make the statement that 7 was my highest impossible score. My table, however, only went to 20 so I began to look for a way to prove that every number after 7 was possible. To begin to prove that 7 was the highest impossible score I began with the string of possible scores 8, 9 and 10. I stated that because these 3 numbers in sequence are all possible, adding 3 to each of them individually to infinity will cover every possible whole number from 8 on. Because I am adding 3 each time I know that they will be possible scores because the team gets 3 points for each field goal. This means that each time I add 3, it is like the team scored a field goal. This reasoning led me to come to the conclusion that 7 was definitely the highest impossible score for the Free Thinkers.
To take this problem one step further, I decided to try to see what the highest impossible score would be in an actual football game with a touchdown worth 6 points and a field goal worth 3 points. I began the same way with a table including columns for each possible score and started with the total scores of 0 – 20. Due to my findings in the previous situation, I knew that all multiples of 3 and 6 were possibilities so I marked those as possible first. I made my table of possible and impossible total scores all the way to 24 and noticed that every third number was possible. This is where I saw a flaw in my problem. Since 6 was a multiple of 3, I would not have a highest impossible number because every third number worked and there will never be a string of 3 possible numbers for me to use to prove a highest impossible score as in the first scenario. I tested my theory all the way to 50 and the pattern continued on with every third number being a possible with the two in-between as impossible. I then concluded that there would be no highest impossible number for the point system that I picked because one of the possible points was a multiple of the other.
In my table I noticed several patterns. One was in the column for field goals. After the highest impossible score of 7, the number of field goals began to repeat the pattern 1, 3, 0, 2, 4. I tested out this pattern and it worked well for my table that went to 22 but I am not quite certain that this pattern is anything but a coincidence because there are more than one ways to make the total scores other than the way I chose to combine 3 and 5. This means that for another person, they could have found a totally different pattern based on the combinations that they chose. I also noticed a pattern in the column for touchdowns that was a little more complex yet still worked for the entirety of my table. Starting with 8, there was a pattern of 1, 0, 2, 1, 0. Then starting at 13 the pattern was 2, 1, 3, 2, 1. This symmetrical pattern with each number increasing by one worked when I carried on through 22 but I had the same thoughts as with the first pattern. These patterns could help me in my table, but for another person, they may have one that is completely different.
I then began to put together a theory that in order to be able to find a highest impossible score, the scores for a touchdown and a field goal had to be relatively prime so there were no possible same factors between them, and one would not be a multiple of the other. I tested this theory with 5 and 7 to find that 23 was the highest impossible score.
When first given this problem I thought it would be very black and white. I thought there would just be one answer and one way to get there. After completing my work this problem left me wanting to continue on and find other pattern and look at other possible combinations of numbers and for this reason I consider this a great problem. It started off slow with making the table and finding ways to add the two numbers, but after I found that everything after 7 worked, my brain took off and I was so caught up in the problem that I discovered things I never thought would come from this problem. For example, reading the problem I never would have guessed that it was at all related to numbers being relatively prime. I came to this conclusion on my own which was amazing.
Work for POW 2
My extension for the problem