In this problem, we were tasked with cutting circular “pies” into the maximum number of pieces that you could with a certain number of cuts. For instance, a pie with three cuts could have six to seven pieces, depending on how you cut it. We had to find the maximum number of pieces a circle could be cut into.
To do this problem, I started small - literally. I made some small circles and began experimenting with one, two, and three cuts. After a few failed attempts at four cuts, I realised that I needed to make my circles bigger, which was a key component to my figuring out the rest of the circles. At this point, I also found a pattern in the max number of pieces. The number of pieces went up with the number of cuts added to the previous number of pieces. This is my table that I had up to this point, including this pattern:
To do this problem, I started small - literally. I made some small circles and began experimenting with one, two, and three cuts. After a few failed attempts at four cuts, I realised that I needed to make my circles bigger, which was a key component to my figuring out the rest of the circles. At this point, I also found a pattern in the max number of pieces. The number of pieces went up with the number of cuts added to the previous number of pieces. This is my table that I had up to this point, including this pattern:
This was how I double checked my answers for future circles, like the one with 5 cuts. I originally got 15 pieces, but the pattern said I would get 16 (11+5). I tried a little bit more, and then, low and behold, got 16 pieces. The same process happened with the rest of the circles as well.
During this process, I learned something about how the intersections of lines worked. I learned that every line had to cross every other line in a unique place. So no crossing through already existing intersections. That was how I ended up maximizing my piece total. Because of this, I also began counting my intersections.
In the end, I only drew circles for one through six cuts, and used the pattern for the rest. Here is the table I ended up with, including intersections, and the pattern I found for intersections.
During this process, I learned something about how the intersections of lines worked. I learned that every line had to cross every other line in a unique place. So no crossing through already existing intersections. That was how I ended up maximizing my piece total. Because of this, I also began counting my intersections.
In the end, I only drew circles for one through six cuts, and used the pattern for the rest. Here is the table I ended up with, including intersections, and the pattern I found for intersections.
Once I had this table, I began looking for an equation. My main focus was the piece total, but I also looked at the intersection totals. I found the equations through trial and error, and by experimenting with the numbers. The variables I used were c, which was number of cuts, p, which was number of pieces, and i, which was number of intersections. Here is the piece total equation:
The equation uses c as the upper limit so the equation can be used to find the number of pieces in any case. I used zero as the lower limit (instead of one), so that I could account for the zero-th case.
I also used summation notation for the intersection equation, which you can see below.
I also used summation notation for the intersection equation, which you can see below.
This equation is almost the same as the piece total one (because the pattern is almost the same), except it subtracts c, rather than adding one. It does this because of how the pattern starts one case later in the intersection data than it does in the piece data.
In this POW, I think I used the habits of a mathematician “start small,” and “be confident, patient, and persistent.” I used start small in how I started the problem, as I said above. I was confident, patient, and persistent, in the way I used the pattern after I found it. I was confident that it worked, and patient/persistent in re-doing certain circles until I knew I had gotten the maximum number of pieces.
In this POW, I think I used the habits of a mathematician “start small,” and “be confident, patient, and persistent.” I used start small in how I started the problem, as I said above. I was confident, patient, and persistent, in the way I used the pattern after I found it. I was confident that it worked, and patient/persistent in re-doing certain circles until I knew I had gotten the maximum number of pieces.