### An Intuitive Explanation of Monsky's Theorem

Monsky's Theorem states that you cannot cut a square into an odd number of triangles with equal area. Despite being a seemingly simple problem about geometry, the solution draws from completely unrelated fields of mathematics, whether that be algebraic number theory or graph theory. In this blog post, I attempt to explain these connections in an intuitive manner. No higher-math prerequisites are required, although prior experience with proofs would be somewhat helpful.

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### Comments

The explanations and illustrations are great. However, I find it easy to lose sight of what the overall goal is. For example, we start looking at triangles with a vertex at (0, 0), but it is only multiple paragraphs later that you explain why we can restrict ourselves to these triangles. Comparatively, in the original PDF the overall goal is clear, but the details are much more opaque. But all in all, good blogpost!

5.9Very cool. (All the best with your birthday party btw. I hope people appreciate the effort you've gone to with cake sharing.) This is a nitpick but it would be more common to use the word "four" in a sentence like "... these are the primary 4"; this differentiates it from the numerals used just above, and makes it clear that it's counting things, not using a number. But as I said, this is a nitpick in an excellently written article. (Confused by this though - "Essentially, the theorem states that there’s a unique norm over the real numbers that matches our desiblue 2-adic norm for every rational number." What's desiblue? Typo or word I'm not familiar with?)

7.3Lost track of the details after the 2-adic metric was introduced, but I felt the gist of the flow at a high level. Very cool! I plan to revisit this after reviewing some algebra

8.1I must admit that after the first couple minutes I skimmed much of the remainder, reading the first sentence of each paragraph, as the subject is more advanced that I was ready to invest the time needed to absorb. I think the presentation, tone, pacing are all excellent.

7Wow. First of all, happy birthday! Then, I must admit, it is an unexpectedly interesting and complicated solution development for such a simply put statement. Just a perfect recipe for a math explainer. And it was delicious, you executed it quite well!

8.1Great article due to the breadth and depth covered (no pun intended). The article reads in a friendly way and the introduction was to the point: I'm witnessing my friend deliberate that part of their birthday party where they'll try and cut their minecraft cake into equal parts based on the number of guests. That being said, the overall structure went from a casual conversation with a friend, to realizing this conversation was held in the millenium falcon that just went lightspeed when it got to 2-adic norm. For instance, we went from selecting theorems to a p-adic analysis, and from no metric space into an ultrametric space. How about first formally defin ultrametric space? (I know it was said, but how does this look like in symbolic notation in terms of a metric space (X, d) is a set X with a metric d defined on X). Overall, I think creating a video about this would be brilliant! Best of luck.

6.4This presentation is absolutely outstanding and very well written, and does a beautiful job of explaining this very statisfying proof. I think there are a few instances of sentences that could be phrased better. I think more definitions for terminology should have been given. Also, say explicitly when a condition is replaced by a sufficient condition. Below are some things I would change. Note that I can't fit everything in the 5000 characters limit. Important results should have their statement and proof distinguished from the presentation (like paragraph with different style, a title, maybe in a box). "What do we know about triangulations? Well, we know that we have n triangles in the triangulation": this does not make sense, as the part "What do we know about triangulations?" seems to inquire about triangulations in general, while the second part "we know that we have n triangles in the triangulation" talks about the specific triangulation task at hand. This is a small thing, but there are quite a few passages in the presentation that require a bit more polishing, like this one, to make the reading experience better. "For every rational number a, there exists an integer n such that a = 2^n ⋅ p/q where both p and q are odd": the case a=0 is not distinguished; moreover, it is important to specify that n is unique. Later it is said that "the 2-adic norm is definite [...]. In fact, ∥0∥_2 is defined as 0", which sounds strange to the reader, as it wasn't previously mentioned that ∥0∥_2 is defined as 0. "the theorem states that there’s a unique norm over the real numbers that matches our desiblue 2-adic norm for every rational number": the extension is not unique. Also, I couldn't find the word "desiblue" anywhere, is it a typo? Importantly, it should be mentioned that the properties of the 2-adic norm can be preserved when extending it to the reals, as they are used later. "remember that we want to show that ∥ad−bc∥ is not equal to 2/n for any odd n": it was never stated that we intend on proceeding this way. Especially since it is obvious to the reader that some triangle with a vertex in (0,0) can have area 1/n. (Later this is acknowledged once colors are introduced, however it could have been phrased better from the start). "Let’s assume for now that it needs to be greater than 1/2": actually we are just taking a sufficient condition. "because that allows for a greater variety of values": I'm not sure what this means. "this essentially means that we must have ∥ad−bc∥_2 ≥ 1": I think this step should be explained; I guess the 2-adic norm is always a power of 2, and the next power of 2 after 1/2 is 1. However the reader may rightfully ask if it is allowed to assume that the extension of the norm also takes values only powers of 2. (Notice for example that ∥sqrt(2)∥_2 = 1/sqrt(2) which is between 1/2 and 1). This can be fixed by saying that ∥ad−bc∥_2 ≥ 1 is taken as sufficient condition. "there’s actually a more elusive way to confirm that the ∥ad−bc∥_2 is the one on the right side, and it’s to instead require that ∥bc∥_2 < ∥ad∥_2": this is poorly phrased. A clearer way to illustrate the point is to say that a sufficient condition for the inequality is ∥bc∥_2 < ∥ad∥_2. The following step, in which another sufficient condition is given, is also phrased unclearly. "Let’s call points of the type (a,b) blue points, and points of the type (c,d) green points.": this step is confusing, as it is unclear what "points of the type (a,b)" means. It seems they are those that satisfy the previous inequalities, however those inequalities depend on (c,d). Everything is then stated clearly in the next sentence; I think that that should have been the definition from the beginning, which would then be shown to be a sufficient condition for the other inequalities. Moreover, it should be mentioned that the conditions for blue and green points are mutually exclusive (that is: no point can be both blue and green). The "low-resolution coloring": I think the picture isn't accurate for the presentation, as the vertices of the square do not match the colors they should be, and similarly for the colors on the edges. Maybe this is due to which point in the pixel is taken as representative; but I think that making the picture reflect the important properties of the coloring would be better (even if the square becomes a rectangle). "A triangle that has degree 0 cannot have any red-blue edges [...]": it was not mentioned that we are considering the subgraph of the dual graph with only red-blue edges. The phrase "An example with only the blue-red edges connected is shown below" doesn't actually suggest we are making this choice. The reader thinks we are considering the dual graph (in its entirety). "Note that the only connection the outside vertex can have with the inside of the square is through the bottom": it should have been clarified in advance what colors appear on each edge of the square (as was done for the vertices).

8.4This one is great! I would write "shoelace formula" without caps, since it is not named after a mathematician named Shoelace. (I would just call it a "determinant formula" or something, as the important new thing in the shoelace/trapezoid formula is that it generalizes to polygons.)

8.5What does “ desiblue” mean? I have never heard that word before, and searching the web doesn’t turn up anything relevant. The following sentence is quite unclear: “Well, assume for the sake of contradiction that ∥bc∥₂ is the term that ends up on the right side; then, we would have to have ∥ad∥₂≤∥bc∥₂, which contradicts our assumption!” It does not contradiction any of the things that were stated to be an assumption, and certainly not the thing that was assumed within that very sentence. Instead it contradicts something that was called a requirement. Frankly I didn’t make it all the way through the article. The complexity level kept ramping up, and the text did not provide sufficient clarity for my understanding to keep up.

5.6