Project 366 #16: 160112 An Interesting Problem
Today was a quiet photo day, as I spent a few hours walking round checking serial numbers.
This is a piece of paper I had kicking round on my desk.
I was working on a jigsaw, and separating the edge pieces from the centre pieces. That got me thinking as to what proportion of the pieces should be edge pieces.
When you think about it, it does get interesting. The proportion is dependent on both the total number of pieces (the fraction gets smaller as the number of pieces goes up) and of the aspect ratio of the puzzle (a 100 piece puzzle that is 10x10 has 36% edge pieces, a 100 piece puzzle that is 50x2 has 100% edge pieces).
I reckon that the fraction of edge pieces is represented in the expression above, where a and b are side length ratios and x is the total number of pieces.
This suggests that as the number of pieces tends to infinity the fraction of edge pieces tends to zero, and that's kind of counter intuitive. It does make sense though. As the number of pieces becomes infinitely large, the relative thickness of the border becomes infinitely thin, and so it does incline to zero.
Number crunchers and algebra hounds can please feel free to correct me!
Project 366 #16: 160112 An Interesting Problem
Today was a quiet photo day, as I spent a few hours walking round checking serial numbers.
This is a piece of paper I had kicking round on my desk.
I was working on a jigsaw, and separating the edge pieces from the centre pieces. That got me thinking as to what proportion of the pieces should be edge pieces.
When you think about it, it does get interesting. The proportion is dependent on both the total number of pieces (the fraction gets smaller as the number of pieces goes up) and of the aspect ratio of the puzzle (a 100 piece puzzle that is 10x10 has 36% edge pieces, a 100 piece puzzle that is 50x2 has 100% edge pieces).
I reckon that the fraction of edge pieces is represented in the expression above, where a and b are side length ratios and x is the total number of pieces.
This suggests that as the number of pieces tends to infinity the fraction of edge pieces tends to zero, and that's kind of counter intuitive. It does make sense though. As the number of pieces becomes infinitely large, the relative thickness of the border becomes infinitely thin, and so it does incline to zero.
Number crunchers and algebra hounds can please feel free to correct me!