# Recruiting Workshop: Picture Hanging Puzzles

Today was Math Day at UNI. This is a big event where the department invites lots of prospective students with strong records and declared interest in some flavor of mathematics major to come visit and be amazed by how awesome we are. The day includes a fancy lunch, and (for the students) a little test used as part of scholarship screening, some activities with faculty members, (and for the parents) discussion of financial aid, presentations on possible career choices, and a campus tour.

I ran a short workshop (activity?) for prospective students. I have done this for five years running, now, and I am starting to build up a repertoire of fun demonstrations and investigations for dealing with a students with a high school background. I like to use topology topics because there are lots of mind-bending things one can do with no background: ideas that are interesting, immediately understandable, and hands-on.

The first two years, I ran a donut coloring contest. I stole this idea from Tom Sibley. (I think I heard him speak about it at a Project NExT session.) This is fun: you challenge the students to discover the chromatic number of a torus, basically, using lots of plain cake donuts and colored frosting. I really enjoy this activity, but it was expensive to run, and the students just didn’t take to eating the donuts like I expected. (The rule was you could only have one donut at a time. If you messed up and wanted a new donut, you had to eat the old one. The students gave me this look like I was trying to poison them.)

Last year and the year before I adapted some Martin Gardner stuff from his old Scientific American column. In particular, there is a neat investigation about cylinders and Möbius bands. You can make a bunch of different embeddings of these by taping long strips of paper with different numbers of half-twists, and then you have fun bisecting them all. Just cut down the middle! The results are a little surprising, and if you are careful you can find some patterns. It feels like you are back in kindergarten, but you are doing topology.

This year I felt I needed something new, so I decided to go with some material from the paper Picture Hanging Puzzles by Demaine et al. I first went through some of the ideas here during a Google+ hangout run by Christian Perfect (under the auspices of The Aperiodical.)  If you have an afternoon and want to read an enjoyable paper that starts out like some simple bit of recreational mathematics but eventually hits some deep algorithmic complexity, try this. In fact, don’t sweat the algorithmic complexity bit. Just check out the paper. It is accessible and  interesting.

The material made for a very good short activity. I used about half an hour, but it could easily be an hour-long event. I want to record here what I did today so that next year I can use it again. If it encourages you to try it out, too, all the better. But if you do that, I want to hear how it went for you so we can both improve.

Here is a very quick “lesson plan”:

for 30ish students. time: 30 minutes. materials: ten 21 foot lengths of 1.5in wide satin ribbon in your school colors, and either a chalkboard & chalk, whiteboard & markers, or a large pad of self-adhesive paper to stick to the wall & some markers (in my venue, I had to use the third option).

• Introduce myself very rapidly.
• make a volunteer of someone in the front row and have them help you unroll one length of ribbon.
• grab two volunteers. Ask one to be the wall-and-nail by holding out a fist. Ask the other to solve this problem: using this piece of ribbon as a wire for hanging a giant picture attached to the free ends, show how to hang the picture so it won’t fall down, but if you remove the nail it will fall down. have volunteers demonstrate validity of solution.
• Grab different set of volunteers. solve problem with two nails. have them demonstrate that their solution works.
• Ask same volunteers to find a way to wrap the wire around the nails so that having both nails holds the picture up, but removing either nail makes it fall down.
• when they look stumped, announce that it might take some experimentation.
• tell the students to form teams of three: two students to be the wall and nails, one to hang the picture. start throwing spools of ribbon at them.
• run around bursting with excitement, enthusiasm and encouragement for 5-7 minutes checking in with all of the groups.
• When the start to get a feel for the problem, put up on display several diagrams of interesting non-solutions. Ask for each which nail removals cause the picture to fall down and which don’t. when they get stuck, have them experiment with their teams.
• come back to whole group discussion and sort out why the non-solutions on the wall don’t work by picking out “bad nails”

Holy cow. I am writing this down from memory, and I realize that we did a lot and we went fast! At this point, i was about 15-20 minutes into the activity.

• go back to teams and try to solve the two nails puzzle. give them five minutes.
• come back to large group discussion. show them a solution by drawing the diagram. point out relation to Borromean rings. Quickly point out connection to free group on two generators by labeling the solution. Say the word commutator. (Essentially, reproduce figure 5 from the paper.) bring up two teams and make two copies of the solution. demonstrate effectiveness by “pulling one nail from each”
• Send them back to their teams to try this: find a way to wind the wire so that three nails holds the picture, but removing any one makes it fall.
• They will not finish this in five minutes. to conclude. tell them the answer in terms of the group theory language. bring up three teams and make three copies of the solution. demonstrate by “pulling nails”
• tell them they have just started studying topology, and briefly say what that subject is.
• Send them home with a thank you and ask them to think about this: how would you wrap the wire around three nails so that the picture still hangs if you remove any one of them, but will fall if you remove any subset of two?

Yeah. If you are still reading, you get the idea that this was hands-on, frenetic, and fun. I enjoyed myself, and I think the students did, too.

The funny thing about these workshops is that they are edutainment not education. So as long as they get the sense that mathematics is interesting, and spending time in my presence is fun, the thing is a success.

## 2 thoughts on “Recruiting Workshop: Picture Hanging Puzzles”

1. suevanhattum says:

Do you think this would work well with 4th to 8th grade girls? (I might be doing a math circle soon.)

2. TJ says:

Yes, I do. But I would do things differently, as you might guess. If it is a math circle, I might pose more questions, and answer almost none of them myself. And I might leave out the whole group theory angle if I am talking to 4th graders. A motivated 8th grader might have already seen some rudiments of formal algebra, and then it might be worth mentioning. Certainly I think my bright, motivated 4th grade kid is not going to get much out of the algebraic representation here. But he can certainly see the alternating terms with mixing “going right” and “going left.”

Also, I think my ribbons were a little too long. If I were buying materials from scratch, I would likely make them only 15 feet long. This makes the 3 nail problem a little tight. But 21 feet is just too long for the two nail problem we spent most of our time discussing.