POW #5: The Big Knight Switch
“The Big Knight Switch” problem covered concepts such as experimentation, mathematical movement, and strategic formulation. These topics may not be covered in a typical mathematics textbook, but each concept was pertinent to developing conclusions based on multiple outcomes. Much of this problem was derived from the “guess-and-check” method, which is to continue to work towards a final and definite solution. It was an occasionally strenuous process, but the persistence I had to utilize in order to find an answer made it all worth it in the end.
Part 1: Problem Statement
In this problem, I had to analyze three components related to one main question. The initial exercise stated that two black knights and two white knights were placed on opposite ends of a 3-by-3 chessboard. The goal was for the two sets of pieces to “switch” places, or for the white pieces to end up where the black pieces started, and vice versa. The only restrictions were that the pieces could only move one at a time, according to the rules of chess, and all of the knights had to stay within the confines of the 3-by-3 chessboard. In chess, knights can only move two squares in one direction, and then one square in a perpendicular direction. Conversely, knights can also move one square in a certain direction, and then two squares in a perpendicular direction. These rules had to be taken under consideration, too. I first had to test whether this was possible – for the pieces to swap positions – and if so, I had to find the least number of moves to do so. I also had to prove why this solution was in fact the lowest possible number. Once I completed this portion of the overall problem, there were two more prompts I could choose to use, which included:
Part 2: Process
I started to solve the problem by playing around with the pieces, and trying to get some method of formulation to work. The main parameters of the problem seemed to suggest that there were not any numerical values that could be expressed through an equation, so I took the necessary steps of trial-and-error to go about my business. After twelve attempts and coming up with results such as 20, 22 and 26, I realized that the key component to finding the correct answer was the fact that the chess pieces had to mirror one another, as illustrated below:
Part 1: Problem Statement
In this problem, I had to analyze three components related to one main question. The initial exercise stated that two black knights and two white knights were placed on opposite ends of a 3-by-3 chessboard. The goal was for the two sets of pieces to “switch” places, or for the white pieces to end up where the black pieces started, and vice versa. The only restrictions were that the pieces could only move one at a time, according to the rules of chess, and all of the knights had to stay within the confines of the 3-by-3 chessboard. In chess, knights can only move two squares in one direction, and then one square in a perpendicular direction. Conversely, knights can also move one square in a certain direction, and then two squares in a perpendicular direction. These rules had to be taken under consideration, too. I first had to test whether this was possible – for the pieces to swap positions – and if so, I had to find the least number of moves to do so. I also had to prove why this solution was in fact the lowest possible number. Once I completed this portion of the overall problem, there were two more prompts I could choose to use, which included:
- Can the three white knights and three black knights switch places on a 3-by-4 chessboard?
- How many moves will it take a knight to get to the diagonally opposite corner of a given chessboard?
Part 2: Process
I started to solve the problem by playing around with the pieces, and trying to get some method of formulation to work. The main parameters of the problem seemed to suggest that there were not any numerical values that could be expressed through an equation, so I took the necessary steps of trial-and-error to go about my business. After twelve attempts and coming up with results such as 20, 22 and 26, I realized that the key component to finding the correct answer was the fact that the chess pieces had to mirror one another, as illustrated below:
Once I had this thought and noted it on my paper, it occurred to me that I could apply this process to the entire problem. No matter where a given piece was, a piece similar to it would have to move in an opposing direction. Following much frustration and hardship, I finally was able to come up with the best solution. I broke down the problem bit by bit, and with each reflecting move I came closer and closer to the answer. I knew that since there were four pieces on the chessboard, I could separate the overall problem into sections of four moves. If you refer to the last picture in each row, the general pattern seems to convey a diagonal relationship, a perpendicular relationship, a diagonal relationship, and then a perpendicular relationship. I was confident that I was on the right track as soon as I made this observation.
Sure, 16 was the lowest number that I had found thus far in the process, but I needed a way to prove it. This is when I recognized that the diagram I had created – and the steps I had taken to get my answer – were proof enough of the lowest possible solution. I knew that the pieces could not be moved in a more efficient way than I had already done. Again, if you were to look at the last image in each row of my image, the pieces gradually shifted in “knight-like” motions across the perimeter of the chessboard. This observation is displayed to the left:
As evident by the image, each piece always occupies a spot on the outside of the chessboard at the end of each sequence. After studying the relationships between each of the four images, it hit me. I knew how to make sense of it mathematically. In each picture, there are five empty spaces. However, none of the four pieces could possibly move to the middle space, since that would be against the rules of chess. On the contrary, the other four squares could all be possible options for at least one piece to move into if necessary. From the data I collected and the observations I made, I could derive this function from the series of events: pn = o p = number of pieces n = number of available spaces o = lowest possible outcome (least amount of moves) Plugging the numbers from this particular problem into the function I made, I was able to find my final answer. (4)(4) = 16 → 16 = lowest possible outcome (least amount of moves) Now that I had established a function, it was time to put it to the test in a different case or scenario. I had my solution, but I needed a way to show that my findings were not simply a one time occurrence. |
Part 3: Solution
Sure, I had an answer of 16 for the first problem. But this would not cut it. I was in search of a way to highlight the fact that my mathematical thinking could be applied to multiple problems.
The second component of the overall activity involved a 4-by-3 chessboard in the More Knights Switching section. I had to find the solution to the least amount of moves it would take three white knights and three black knights to swap places on this particular chessboard. After formulating a function for the first part, I was actually hesitant to implement it for this question. I wanted to see if I could play around with the problem, and still be efficient about the way I went about solving it. Just like the initial exercise, I experimented numerous times with the order in which the pieces moved, where they moved, and what spaces each piece occupied. I attempted to find a solution for a recorded 37 times until I found a final answer I was confident with. Nonetheless, I made some notes along the way as to where certain pieces should be located at various points during the process:
Sure, I had an answer of 16 for the first problem. But this would not cut it. I was in search of a way to highlight the fact that my mathematical thinking could be applied to multiple problems.
The second component of the overall activity involved a 4-by-3 chessboard in the More Knights Switching section. I had to find the solution to the least amount of moves it would take three white knights and three black knights to swap places on this particular chessboard. After formulating a function for the first part, I was actually hesitant to implement it for this question. I wanted to see if I could play around with the problem, and still be efficient about the way I went about solving it. Just like the initial exercise, I experimented numerous times with the order in which the pieces moved, where they moved, and what spaces each piece occupied. I attempted to find a solution for a recorded 37 times until I found a final answer I was confident with. Nonetheless, I made some notes along the way as to where certain pieces should be located at various points during the process:
Now that I had solved the problem, I (again) needed to prove that my answer was in fact correct. I had a function that I produced after the first problem, so I could plug the values in for this problem. However, this one was slightly more complicated. As one might recall, my function was:
pn = o
p = number of pieces
n = number of available spaces
o = lowest possible outcome (least amount of moves)
The catch with this particular scenario was that each of the spaces was being used at all times, or had the potential to be used. This threw off the previous equation and its relevance to this question. Following some thoughts about how I could approach this problem, I used what I already knew to help me:
pn = o + (a)(b)
a = additional spaces
b = additional pieces
Placing the values from the problem into the equation gave me...
(4)(4) = 16 + (a)(b)
16 + (a)(b)
16 + (3)(2)
16 + (6) = 22
→ 22 = lowest possible outcome (least amount of moves)
While there is not necessarily direct proof that the values I substituted into the function is technically valid, I thought it was a smart way to incorporate new information with old data. By multiplying the number of spaces by the number of pieces, I used the same methodology as I had used to check my answer in the first problem.
The final part to this problem (A Knight Goes Travelling) involved finding how many moves it would take a single knight to reach the opposite corner of a chessboard of an unknown size. There were no complexities to this problem, and I could not really apply the formula I had created before to this prompt. All that mattered was that I kept track of each piece of data, and looked for trends in my discoveries. Some examples of my drawings are below:
pn = o
p = number of pieces
n = number of available spaces
o = lowest possible outcome (least amount of moves)
The catch with this particular scenario was that each of the spaces was being used at all times, or had the potential to be used. This threw off the previous equation and its relevance to this question. Following some thoughts about how I could approach this problem, I used what I already knew to help me:
pn = o + (a)(b)
a = additional spaces
b = additional pieces
Placing the values from the problem into the equation gave me...
(4)(4) = 16 + (a)(b)
16 + (a)(b)
16 + (3)(2)
16 + (6) = 22
→ 22 = lowest possible outcome (least amount of moves)
While there is not necessarily direct proof that the values I substituted into the function is technically valid, I thought it was a smart way to incorporate new information with old data. By multiplying the number of spaces by the number of pieces, I used the same methodology as I had used to check my answer in the first problem.
The final part to this problem (A Knight Goes Travelling) involved finding how many moves it would take a single knight to reach the opposite corner of a chessboard of an unknown size. There were no complexities to this problem, and I could not really apply the formula I had created before to this prompt. All that mattered was that I kept track of each piece of data, and looked for trends in my discoveries. Some examples of my drawings are below:
On top of the images, it was important for me to display my data in a list, so that it would be easier for me to identify patterns. I calculated the distances up to a 1-by-12 chessboard:
The main observation I made based on the table is that different values for the knight’s movements corresponded with certain chessboard size ranges. For instance, chessboard sizes 5-by-5 through 7-by-7 all required at least 4 movements, chessboard sizes 8-by-8 through 10-by-10 required a minimum of 6 movements, and chessboard sizes 11-by-11 and 12-by-12 suggested that it would take the piece 8 movements to get across.
Looking at the data, it might be fair to say that a knight moving across a 13-by-13 chessboard would need to take at least 8 movements, because of the pairings of 3 in the right column as far as the ranges go.
Part 4: Extensions
Reflection
“The Big Knight Switch” problem was crucial in my ability to understand concepts surrounding strategic formulation, proving mathematical solutions, and taking multiple routes in order to solve a problem. I had to overcome adversity to come to my final answer(s), and it was definitely a difficult process sometimes. Unlike other Problems of the Week in the past, I could not use a formula, equation or function to help me solve the problem, but only to prove my answer once I had finished. This assignment required experimentation, and physically maneuvering each component to its fullest extent. If one piece was not used properly, that single move could affect the rest of the process, which did not make things any easier on me. To be successful in a problem like this, it takes grit and determination, and I think I was able to display those skills effectively.
I honestly believe that I deserve a 10 out of 10 for my work on this problem because of the challenges that I faced, and having to move around those respective obstacles. I could have stuck with the final solution as my answer and called it quits, but I dissected what the problem was asking of me, and proved why my answer was the correct one. I created a solid function that correlated with the problem, and applied it to multiple cases. For the last part of the problem, I clearly labeled the steps I took in the forms of both images and numbers, and I found a way to make sure that all three of these components cohered.
The most prominent Habit of a Mathematician shown in my work was Conjecture & Test. While somewhat obvious, it was clear and bright in the context of each question. Without this Habit of a Mathematician, I would not have made nearly as much progress as I did in the experimental side of things, which really benefited me in the long run. If I would have tried to develop an equation to solve each problem or try to think about a detailed plan to get each piece to the end without trying it out first, I would not have been as successful as I was. It is important to be comfortable with taking some risks in regards to math, and I was not afraid to step outside of the box and express some unconventional ideas every once in awhile. Overall, I am really happy with the effort I put into the problem(s), and I have no regrets about the procedure as a whole.
Looking at the data, it might be fair to say that a knight moving across a 13-by-13 chessboard would need to take at least 8 movements, because of the pairings of 3 in the right column as far as the ranges go.
Part 4: Extensions
- Simple: What is the least amount of moves that two black knights and two white knights could take to switch places on a 2-by-3 chessboard?
- Moderate: What is the least amount of moves that two black knights and two white knights could take to switch places on a 13-by-13 chessboard?
- Difficult: What is the least amount of moves that two black knights and two white knights could take to switch places on an 5-by-8 chessboard?
Reflection
“The Big Knight Switch” problem was crucial in my ability to understand concepts surrounding strategic formulation, proving mathematical solutions, and taking multiple routes in order to solve a problem. I had to overcome adversity to come to my final answer(s), and it was definitely a difficult process sometimes. Unlike other Problems of the Week in the past, I could not use a formula, equation or function to help me solve the problem, but only to prove my answer once I had finished. This assignment required experimentation, and physically maneuvering each component to its fullest extent. If one piece was not used properly, that single move could affect the rest of the process, which did not make things any easier on me. To be successful in a problem like this, it takes grit and determination, and I think I was able to display those skills effectively.
I honestly believe that I deserve a 10 out of 10 for my work on this problem because of the challenges that I faced, and having to move around those respective obstacles. I could have stuck with the final solution as my answer and called it quits, but I dissected what the problem was asking of me, and proved why my answer was the correct one. I created a solid function that correlated with the problem, and applied it to multiple cases. For the last part of the problem, I clearly labeled the steps I took in the forms of both images and numbers, and I found a way to make sure that all three of these components cohered.
The most prominent Habit of a Mathematician shown in my work was Conjecture & Test. While somewhat obvious, it was clear and bright in the context of each question. Without this Habit of a Mathematician, I would not have made nearly as much progress as I did in the experimental side of things, which really benefited me in the long run. If I would have tried to develop an equation to solve each problem or try to think about a detailed plan to get each piece to the end without trying it out first, I would not have been as successful as I was. It is important to be comfortable with taking some risks in regards to math, and I was not afraid to step outside of the box and express some unconventional ideas every once in awhile. Overall, I am really happy with the effort I put into the problem(s), and I have no regrets about the procedure as a whole.