POW # 5, Possible patches
During POW #5, Possible Patches, we were presented with a man named Ralph Lauren who wanted to cut out pieces of satin for a quilt. He has a certain size sheet and needs to know the maximum amount of certain sized patches he can get out of that single sheet. He needs different size patches and he has different sized satin so each variation of the problem may be a little different, so to solve, we are going to take this one step at a time.
Process
In order to solve the problem, I came up with a few different strategies. First off, I would take the areas of both the total satin size and the certain piece he needed. If we divided those together, we came up with an amount that we knew for sure the maximum would never succeed. For example, if we had a piece of satin that is 10x12 and we were cutting a 4x9 piece from it, we could do 10x12(120) divided by 4x9(36). Remove the decimal and that is the maximum maximum (3). Another strategy I had was to look at the different lengths of the smaller piece and compare them to the large piece as a whole. For example, if there was a quilt that was 18x6 and you needed to remove pieces that were 8x5, I know that the patches can be only cut in one orientation because the 8 unit side of the patch could not be cut from the 6 unit side of the quilt. One more strategy that I used is adding together the side lengths and seeing what the maximum amount of pieces I could get into one side. For example if one side of the satin was 19 and I was making patches that were 4x5, I know I could make 3 patches with the 5 unit side parallel to the 19 unit side, than have another patch cut out in the other orientation to have the 4 unit side parallel to the 19 unit side. In order to solve these problems, you want to be as efficient as possible. When solving the first few problems a strategy that worked for me is to fill out all the corners first and then fill in the middle. While this may not have worked on all the problems to find the maximum, it still helped me out.
Solution/Results
Here are the solutions I got for the various problems that we had to complete and a brief explanation of why I came to that conclusion
- 3x5 patches in a 17x22 satin piece: 24 pieces (I laid out all the pieces in various positions and came out to 24 pieces. This is correct because when I calculated out the absolute maximum, I figured out it was impossible to get more that 24.)
- 9x10 patches in a 17x22 satin piece: 2 pieces (After laying down 2 pieces, there is no way to fit anymore in. This piece resembles a square closely, therefore lacking the wiggle fitting room that a true rectangle provides)
- 5x12 patches in a 17x22 satin piece: 6 pieces (Similar to the first problem, mathematically 6 is the maximum you could fit in the satin piece)
- 10x12 patches in a 17x22 satin piece: 2 pieces (Similar to problem two, this patch is close to a square limiting the versatility of the positioning.
- 3x5 patches in a 4x18 satin piece: 3 pieces (You can not orientate the patch with 5 units parallel to the side of the satin that is 4 units because laying pieces this way would take up more than the side allowed. This is the maximum)
3x5 patches in a 8x9 satin piece: 4 pieces (Just like a few problems above, mathematically there are only room for 4 pieces and I was able to come up with a solution to position these patches)
- 3x5 patches in a 17x22 satin piece: 24 pieces (I laid out all the pieces in various positions and came out to 24 pieces. This is correct because when I calculated out the absolute maximum, I figured out it was impossible to get more that 24.)
- 9x10 patches in a 17x22 satin piece: 2 pieces (After laying down 2 pieces, there is no way to fit anymore in. This piece resembles a square closely, therefore lacking the wiggle fitting room that a true rectangle provides)
- 5x12 patches in a 17x22 satin piece: 6 pieces (Similar to the first problem, mathematically 6 is the maximum you could fit in the satin piece)
- 10x12 patches in a 17x22 satin piece: 2 pieces (Similar to problem two, this patch is close to a square limiting the versatility of the positioning.
- 3x5 patches in a 4x18 satin piece: 3 pieces (You can not orientate the patch with 5 units parallel to the side of the satin that is 4 units because laying pieces this way would take up more than the side allowed. This is the maximum)
3x5 patches in a 8x9 satin piece: 4 pieces (Just like a few problems above, mathematically there are only room for 4 pieces and I was able to come up with a solution to position these patches)
My Own Problems
5x2 patches in a 12x16 satin piece: 18 pieces (To do this one I figured out how to load the maximum on one of the sides and loaded it up from there)
4x3 patches in a 9x13 satin piece: 9 pieces (In order to fill this one up, I filled all the room besides a tiny line on the side)
4x8 patches in a 6x29 satin piece: 3 pieces (You can not fit the 8 unit side parallel to the 6 unit side of the satin piece. Therefore there is only one way to orientate your pieces and this is the maximum you could fit while orientating it this way.)
4x3 patches in a 9x13 satin piece: 9 pieces (In order to fill this one up, I filled all the room besides a tiny line on the side)
4x8 patches in a 6x29 satin piece: 3 pieces (You can not fit the 8 unit side parallel to the 6 unit side of the satin piece. Therefore there is only one way to orientate your pieces and this is the maximum you could fit while orientating it this way.)
Reflection
This problem required a lot of trying over and over again. One of the challenges I had during this problem was to keep on trying even when I just wanted to assume I had the right answer. I realized I couldn't just assume my answer when with one of the problems I thought I had the answer right and when I went back, I actually messed it up. During this problem, I used a couple habits of a mathematician. One of which was solve a smaller problem. I took each problem one by one instead of being overwhelmed by all of the patches that I had to draw out. I also used the habit of conjecturing and testing. I was able to find the answers by just trying out different things and this strategy worked quite well for me.