Dr Drew Tiles a Patio (POW #2)
The Problem:
Dr. Drew is tiling a rectangular patio at his house. He’s planning on using square tiles. He has chosen very pretty but inexpensive tiles. His neighbor explains to him that some tiles will have to be more expensive because of durability and integrity. This is because people walking from the garage door to the back door always walk in a diagonal. The tiles along the diagonal will receive heavy wear and tear, therefore they much be the most expensive tile. Dr Drew wants to find the least amount of expensive tiles possible. (Just to clarify this is not a real situation, just one of those weird out there math statements. Don’t worry there aren’t 3,000 watermelons or anything).
Dr. Drew is tiling a rectangular patio at his house. He’s planning on using square tiles. He has chosen very pretty but inexpensive tiles. His neighbor explains to him that some tiles will have to be more expensive because of durability and integrity. This is because people walking from the garage door to the back door always walk in a diagonal. The tiles along the diagonal will receive heavy wear and tear, therefore they much be the most expensive tile. Dr Drew wants to find the least amount of expensive tiles possible. (Just to clarify this is not a real situation, just one of those weird out there math statements. Don’t worry there aren’t 3,000 watermelons or anything).
The Solutions:
We were given a 4 x 6 grid (as shown above) to begin working. It was given as a kick start. I first started by noticing that there are 8 shaded squares. 4 x 6 is 24 and 24/8 is 3. 1/3 was shaded in, that cannot be argued against. I followed this logic and came up with X x Y x 1/3 = Answer. I tested this with the 4 x 6 and got 8. Perfect and simple, right? Wrong. I tested this equation with 63 and (X) 90 (Y) and got 1,872. This number was ten times the answers my classmates were getting. Now I knew that my logic was extremely faulty and tried to come up with more ideas. I couldn’t think of anything so I began drawing out the grid. I got to around 63 x 25 when I knew I needed to stop. I was just wasting time. I started thinking about equations, my classmate came up with the answer 144 but I had no idea how she got there. I didn’t want to copy her formula (she just told me out of the blue), but I now had some direction. 144 is pretty dang close to 63 + 90 (153) and 153 - 144 = 9. You might be asking yourself “Where the hell does 9 fit in?” I can’t say I figured it out right away. It required a lot of thinking. I knew there was something odd about the number choice of 63 and 90. I knew there had to be some sort of connection. What did they have in common? Well if you haven’t already guessed it, the common link is in fact the number 9. 9 was the most common factor between the two. I figured it out. X + Y - Most Common Factor = Answer. 63 + 90 - 9 = 144. I tested this out with 4 and 6. 4 + 6 - 2 (2 being the most common factor) = 8. Remember good ol’ 8? Well there he was, confirming my formula. I asked around the classroom and most other students got a similar answer and equation. It was/is pretty much fool proof. 144 tiles out of 5,670 has to be more expensive.
Further Exploration:
Now that I had the answer I began testing it on everything. Literally EVERYTHING. 2 + 2 - 2 = 2. 3 + 3 - 3 = 3. I noticed a really awesome trend. If two numbers are the same the answer will just be that number. 20 + 20 - 20 = 20. All numbers can be devided by 1 making it a common factor. As long as two numbers share a common number, the equation works extremely well. 4,000 + 22,000 - 2 = 25,998.
Lets say each inexpensive tile is 50 dollars and each expensive tile is 100 dollars. It would cost Dr Drew 276,300 for inexpensive tiles and 14,400 for expensive tiles. Altogether it would cost him 290,700 dollars. Thats a lot of money.
Reflection:
I was definitely not the biggest plan of this problem but it was very different. It was a nice change from the norm. I had to push my thinking and had to use the habits of a mathematician. I had to stay organized, start small, and collaborate. Collaboration was very helpful in solving this problem. I bounced ideas off of multiple people. This problem was very challenging and I hope to have more like it in the future.
We were given a 4 x 6 grid (as shown above) to begin working. It was given as a kick start. I first started by noticing that there are 8 shaded squares. 4 x 6 is 24 and 24/8 is 3. 1/3 was shaded in, that cannot be argued against. I followed this logic and came up with X x Y x 1/3 = Answer. I tested this with the 4 x 6 and got 8. Perfect and simple, right? Wrong. I tested this equation with 63 and (X) 90 (Y) and got 1,872. This number was ten times the answers my classmates were getting. Now I knew that my logic was extremely faulty and tried to come up with more ideas. I couldn’t think of anything so I began drawing out the grid. I got to around 63 x 25 when I knew I needed to stop. I was just wasting time. I started thinking about equations, my classmate came up with the answer 144 but I had no idea how she got there. I didn’t want to copy her formula (she just told me out of the blue), but I now had some direction. 144 is pretty dang close to 63 + 90 (153) and 153 - 144 = 9. You might be asking yourself “Where the hell does 9 fit in?” I can’t say I figured it out right away. It required a lot of thinking. I knew there was something odd about the number choice of 63 and 90. I knew there had to be some sort of connection. What did they have in common? Well if you haven’t already guessed it, the common link is in fact the number 9. 9 was the most common factor between the two. I figured it out. X + Y - Most Common Factor = Answer. 63 + 90 - 9 = 144. I tested this out with 4 and 6. 4 + 6 - 2 (2 being the most common factor) = 8. Remember good ol’ 8? Well there he was, confirming my formula. I asked around the classroom and most other students got a similar answer and equation. It was/is pretty much fool proof. 144 tiles out of 5,670 has to be more expensive.
Further Exploration:
Now that I had the answer I began testing it on everything. Literally EVERYTHING. 2 + 2 - 2 = 2. 3 + 3 - 3 = 3. I noticed a really awesome trend. If two numbers are the same the answer will just be that number. 20 + 20 - 20 = 20. All numbers can be devided by 1 making it a common factor. As long as two numbers share a common number, the equation works extremely well. 4,000 + 22,000 - 2 = 25,998.
Lets say each inexpensive tile is 50 dollars and each expensive tile is 100 dollars. It would cost Dr Drew 276,300 for inexpensive tiles and 14,400 for expensive tiles. Altogether it would cost him 290,700 dollars. Thats a lot of money.
Reflection:
I was definitely not the biggest plan of this problem but it was very different. It was a nice change from the norm. I had to push my thinking and had to use the habits of a mathematician. I had to stay organized, start small, and collaborate. Collaboration was very helpful in solving this problem. I bounced ideas off of multiple people. This problem was very challenging and I hope to have more like it in the future.