Well, I have to say that it’s pretty hard to start writing when it’s been three months since the last post. Eek! I can’t believe it has been so long. And did I really not post the solution to the last puzzle?!? I am so ashamed. But that means there’s only one thing to do–post the damn answer. Okay, so if you recall, we were trying to cleverly measure out 45 minutes using fuses that burn for exactly one hour and exactly half an hour. The fuses are non-uniform, and you can’t predict how long one will burn by its length. If you cut a one-hour fuse in half, for example, and light the pieces, one could burn for 5 minutes and the other for 55. So what can you do?
Of course the thing to notice here is that a fuse is designed to burn for a set amount of time when you light it at one end. If you light the fuse at two ends, the fuse will be consumed twice as fast, and therefore in half the originally planned time. So–light the one-hour fuse at both ends and it will burn for half an hour. There are now several ways to time out 45 minutes using your one-hour and half-hour fuses. One thing to do is light one of each type of fuse at one end, and let them burn until the half-hour fuse is consumed. At this point, light the other end of the one-hour fuse. It had half an hour left, and now it is burning at two ends, so it will take 15 minutes to burn. Half an hour plus 15 minutes is 45 minutes.
Alright. That was the last math mostly puzzle. I am already missing WESU and the Somewhat Science crew, as well as my excellent co-host Jed Rendleman. Now that I am all moved in to my new house in Colorado Springs, though, I am making plans to check in with KRCC, the excellent station at Colorado College. Hopefully they will be interested in letting me do some math on the radio. I will report back with any news. In the mean time, I want to share another puzzle of sorts that came to me from a friend. It involved trading quilt squares in a group of 5 people. This is a little different from the other puzzles in that it comes from an actual problem that my friend’s mother was having. It wasn’t designed to have a cute little solution or anything like that. I did solve it, and it was fun, but it wasn’t entirely simple and cute. Anyway, I’ll share it and let you think it through. Here’s the email I got from Judy:
My son, Martin gave me your email address. Hopefully he already told you I’m struggling with organizing the group project for some quilters, and can’t seem to get it to work out “exactly” the way I want it to. Here’s the scoop if you feel like a puzzle. I’d certainly appreciate the help and insight.
I am organizing a round robin quilt exchange. That means a group of quilters each make a small quilt block and trade them with each other over a period of time until everyone in the group has had a chance to add a border to everyone else’s quilt.
On the first trade day, each person passes their quilt block to another person in the group and receives a block from someone else in that group, so they can add a border to that block. After a month or so, the blocks are passed to another person in that group, so they can add a border to the block and so on until everyone in the group has had a chance to sew a border on every block. At that time the block is returned to the person that made the center quilt block.
I’m having trouble setting up the passing order. The last time we did this activity we passed our project to the same person each time until we got our own quilt back. If I passed my quilt to “Martha” then every time we traded, I gave my quilt to “Martha,” which means she always had to follow my work in this game. It is kind of limiting, since part of the fun of doing these trades, is getting to meet the other people in the group.
I’d like to arrange the trades so no one ever gets their quilt from the same person, but I can’t quite figure it out. I tried it with 5 people in a group and, while everyone gets the quilt to sew on, the quilts are passed twice to the same person. I’m hoping there is an equation to figure this out, instead of my trial by error method.
I think we will pretty much always have five people in a group, but it is possible that we would have groups with 4 or 6 people. Will a formula work for that, too?
Judy and I both thought to organize our work on the problem in a table, which made the problem into sort of a Latin square with conditions. A Latin square is essentially a Sudoku puzzle. The idea was to make a 5 by 5 grid. Each row in the grid would correspond to a quilt square. The people would be numbered 1 through 5. A row would describe the path of the quilt square through the 5 people in the group. If we could successfully fill in the grid so that each row and column contains the numbers 1 through 5 this would give a way to pass the quilts so that each person worked on each quilt and everybody always had a quilt to work on. There are lots of ways to do this–for example
The problem is that this grid describes the situation of Judy’s first quilt passing experiment–where each person just passes to the person next to them until the quilts have seen every person in the group. But that’s not what we want. How do we translate the condition that each person should pass to a different person each time into a condition on the Latin square? Well, one person passing to another corresponds to a sequence of two numbers in a row (like if 23 appears in a row that means person 2 passed a quilt to person 3). So if we never want to repeat passes, we have to create a Latin square in which no sequence of two numbers is repeated in different rows.
Okay, so I will tell you that I either found such a Latin square or proved it was impossible. And I addressed the problems of 4 and 6 people. But I’m out of time for now and don’t want to spoil the fun in case someone wants to try it out for themselves. For now I’ll leave you hanging.
This seems like a reasonable place to put in a plug for Martin Gilmore, my friend and Judy’s son, who is a really fabulous singer/songwriter/guitar player based in Denver. Besides his solo work, he plays with the bluegrass band Long Road Home and the rock band Bimarinal. And besides the fun I had with this puzzle, Martin has brought me hours of fun as a guest on Live@Lunch (back in my KRFC days).