Polyplane and Magic Polygons

I'm in New Jersey this week, so thought I'd break format and give an update on a couple of topics that connect to the trip!

Polyplane

Back in March, I wrote about a straws-and-string method for constructing nested polyhedral frames. I dragged my feet a little on finishing my model to the point that most of the coordinates I could think to enter in the submission form for Glen Whitney and Alex Kontorovich's Polyplane exhibit were taken. However, the massive model I built some years back that had since decomposed fit the bill, so I spent a chunk of the other weekend augmenting a model featured in this blog into that massive model. Below are photos of the model before and after adding the purple layer:

Since the spot in the exhibit for a deltoidal icositetrahedron was already crowded, I've interpreted this model as the frame given by the purple and green edges instead of just the purple edges, splitting each kite face into two triangular faces. This inflates the number of faces up to \(48\) and edges up to \(72\), while the number of vertices remains \(26\). I handed it off in New Jersey this week just in time for it to tag along to Florida with the rest of the exhibit.

Magic Polygons

I went to Math League today and ran a session for some bright 4th and 5th graders on Magic (Perimeter) Polygons, a close relative of JRMF's Magic Flowers activity. Some worksheets I made for the activity are available here.

As a starting point, I introduced the idea of a magic triangle by filling out a triangle incorrectly with the numbers \(1-6\) and asking them to fix it by rearranging the numbers so that all three sides summed to the same value. Here's my wrong triangle and a correction given by a student with magic number \(9\):

I then tasked students with finding more magic triangles with different magic numbers. Some quickly found triangles with magic numbers \(9,10,11,\) and \(12\) and a couple gave nice arguments based on the corner values getting used twice in edge sums to justify that \(9\) was the smallest magic number possible and \(12\) the biggest. They then worked on triangles with edge length \(4\). There were no arguments for why \(17,19,20,21,\) and \(23\) were possible while \(18\) and \(22\) were not.

I sort of dragged them into looking at the relationship between a magic triangle labeled with \(1-6\) and the triangle you get by subtracting each vertex from \(7\), since the result was another magic triange. A student identified \(10\) as being the number to subtract from for triangles labeled \(1-9\) and was able to get the triangle with magic number \(19\) that he had not found yet from his triangle with magic number \(21\). I left the "why" of how this worked as a challenge.

Several of the students came up with ways to take a lot of the guesswork out by targeting magic numbers. They noticed that if they wanted magic number \(m\), then the three edges must sum to \(3m\), and so \(1+2+3+...+n+a+b+c = 3m\), where \(1-n\) are the labels and \(a,b,c\) are the corner labels counted a second time. Since they could compute \(1+2+3+...+9 = 45\), for example, this gave them the ability to target a magic number like \(19\) by noting that \(3 \cdot 19 = 57\), so the sum of the corners must be \(a+b+c = 57-45 = 12\). They then chose three values that summed to \(12\) like \(3,4,5\) and then attempted to fill in the rest of each edge to get \(19\).

Many of the students had already started looking at the more complicated puzzles by this point, but I explained there were two sorts of challenges they could look at:

1. We've seen that a few different magic triangles exist when the edge length is \(3\) or \(4\). What about \(5\), \(6\), \(7\), and so on? Is there always at least one magic triangle of a given size? I hinted that there is a trick for using their solutions for small triangles to solve the larger triangles, but said no more.

2. We've so far focused on the perimeters of triangles, but we could look at the perimeters of squares, pentagons, hexagons, and so on. How do things change for these? Many student noticed they could always create a magic triangle with corners \(1,2,3\). Is it possible to make a square with corners \(1,2,3,4\), a pentagon with corners \(1,2,3,4,5\), and so on? I hinted that this might not work every time, but said no more.

Students worked a little longer and then we wrapped up. I stayed for the next talk and felt bad that a fair number of the students were continuing to work on the puzzles instead of paying attention to the presentation -- but only a little bad, since the whole point was for kids to get caught up in a problem.

If I could wipe their memories and run it again, I would have more than four copies of the edge length \(3\) triangle and more than five of the edge length \(4\) triangle, likely putting each set on its own page. I flipped some triangles upside down to save paper, but I would not do that with the extra room. In the end, that choice largely cost ease of communication, since students would have to look at two triangles a little more closely to decide if they were labeled the same or to transfer numbers from one to another.

While providing exactly the number of triangles as there are magic numbers is good as a closed, concrete task, I think this group would have benefitted from a little more ambiguity: How many magic numbers are there? Do I have them all? If so, how do I know that's all of them? I tried to retroactively insert the ambiguity by suggesting that perhaps I didn't give them enough triangles to find them all, but it would have felt more natural if that ambiguity were tacitly reinforced by the handout itself. I had hemmed and hawed about ending the handout with hexagons to save paper, but the fact that students were working after the activity had formally ended indicated that the bigger puzzle book was probably better.

This post was sponsored by the Julia Robinson Mathematics Festival