Summer of Math Exposition

It seems a bit too complex for high school students; I would prefer a more traditional approach. In addition, I didn’t see any guidance on memorizing procedures. Still, it could be an interesting topic for university students who are just starting out in mathematics.

The visualizations of what amounts to graph isomorphisms was nice!

Another way to state the problem is “how many equations are needed to specify an element of S_26 ?”

An initial confusion I had was the order of quantifiers. For each permutation, there exists a collection of sets of equations, where each set of equations specifies the permutation. You care about the smallest such set. But might this be different for each permutation? The answer is no by symmetry (the symmetric group acts on the equations as well as the solutions). You might make this clearer in the video: for every possible permutation, there exists an analogous set of 11 equations to the ones presented that specify that exact permutation.

It’s a friendly animation style, and it introduced and layered concepts well.

In the proof of no valid solution for 10 equations, I didn’t understand the constraint that 25 letters be used

A nice fun puzzle with some clearly explained math to back it. Well done.

This was a fun topic for exploring automorphisms/symmetries as a concept, and the visualizations made things clear.

Tricky. The video is very well made and fulfils all criteria — the puzzle is novel, the number of necessary equations is interesting etc — but I feel that within the spirit of the competition, it isn’t complete without providing a solution strategy. While you derived a solution for the six letter variant, your pointers and some other observations I made did not yield significant progress on the main problem in the time I was willing to allocate to it.

A fun and novel video!

Asking rhetorical questions encouraged the viewer to think about the problem first, which helped with engagement and clarity. I actually did try to solve it and found it pretty fun.

At first I was going to say it didn’t really “break algebra” since you had constraints outside of the letter equations. But you mentioned that anyway.

The downside for me was that I wasn’t entirely sure what the purpose of the puzzle was other than coolness. I think as a result of that I didn’t learn much maths, but the graphs were a nice touch.

But still, great job!!

I really enjoyed this video! The puzzle was fun, and as you revealed more and more of how to create/solve them, I became more and more encouraged to solve the problem. You did a great job of showing how creativity and re-visualizing the problem has a big impact, and I think that even if a typical highschooler would not be able to come up with it all on their own, some of the tricks they might be able to figure out. The puzzle itself is a bit contrived (but then again, it is a puzzle), and it wasn’t clear to me at first if the challenge was to create your own equations or to solve the one you had made. ALSO, you should consider having the solution at the end! :)

“Bending the rules of mathematics” is really stretching what you’re actually doing in this video, but appart from this technicality, your video was great :-)

Factual mistake. AA=A has two solutions. 1 and 0.

You also put constraints on what the letters can be. This is additional info which undermines the premise of your claim

amazing animations, incredible explanations. maybe expand on a generalisation of the min equations you need to find n unknowns (given these restrictions)?

I like this because I think it is an example of hypergraphs. I don’t know much about them, but the notation you used to describe relationships between three nodes made it easier to understand how I might deal with a hypergraph of any number of node connectivity: just add a set of symbols to each edge. The notation you used is a little different than what I’m describing because it doesn’t put identifiers on each equation. And it shouldn’t because each equation is indistinguishable, I think, besides being either multiplication or addition and having an orientation regarding the terms. Well, that’s just me trying to understand how useful the notation is. I think that was the most interesting part for me.

I was able to understand the logic behind the 10 solution being impossible. Though I had to look again after seeing the two node equation in the final graph. It seems to be another unique graph type. And so would a triangle island. However, I suppose they aren’t allowed for a reason. Probably because… Well I’m not sure. It would free up something to include one triangle isolated on its own. But it is still not enough to solve the 10 problem.

I wonder if these puzzles are just ways to train our brains to understand notation and logic. That could be a subgoal to understanding some other application. I’m not sure how hypergraphs are used for the greater economy, but it isn’t really necessary to give an example. I do wonder how this puzzle is connected to other bits of math I know.

I thought the labeled graph asymmetry obstruction to having a unique solution was really clever!

What a fun, novel problem solved in an interesting way. I wasn’t with you at first, but once you started diving into examples, I was quite entertained. I don’t think this is going to be something that will change my life, but if a student is in a graph-theory class, this feels like a very approachable problem to help them learn a little more about the subject. Very well executed and presented.

Very nice animations