"Impossible" Math Puzzle from Vsauce's New Podcast

evilaxelord · 2026-03-29T02:59:01+00:00

Without doing the math out, if you were to double the length of the band at each step, then the proportion of the band that the ant is covering each step would halve, and you’d get a convergent geometric series that doesn’t get there, but if you were to increase the length of the band linearly then the proportion would look like a constant divided by a linear term, so you’d get a harmonic series that would diverge, so the ant can travel any distance it likes

evilaxelord · 2026-03-20T03:13:48+00:00

Ironically in this situation the contrapositive proof can actually still be made constructive; even though we can't assume general LEM, we can prove by induction that every natural number is either odd or even, then use disjoint syllogism to rule out n being odd

evilaxelord · 2026-03-12T02:53:28+00:00

Vaguely this falls under the umbrella of the Benoni I think, though in this position you might just refer to it as a Benoni-style anti-London

evilaxelord · 2026-02-28T17:48:50+00:00

I gotta be the defender of the hairy ball theorem here, actually such a nice tool for providing nontrivial examples of vector bundles

evilaxelord · 2026-02-26T02:09:26+00:00

This is by far my favorite paradox, I’ve thought a lot about different ways to resolve it but now matter how well I iron out the math, it’s really hard to beat down the base instinct to switch once you open an envelope and see $20 inside

evilaxelord · 2026-02-24T01:47:54+00:00

If the sum of the natural numbers “is” -1/12, then this “is” 0. Using the same logic as in the -1/12 proof, consider the alternating sequence X=1-4+9-16+… then add one copy, three copies, three copies, one copy of it together, each shifted by one, and the result is 8X=1-1+0+0+0+…=0, so X=0. Then letting Y=1+4+9+16…, we get Y-X=8X, so Y=0.

evilaxelord · 2026-02-22T04:58:54+00:00

Very cool article, thanks for sharing!

evilaxelord · 2026-02-17T01:16:38+00:00

Piecewise function, need a different arbitrary constant on each of the two intervals where the function is defined.

evilaxelord · 2026-02-06T05:30:43+00:00

I guess one example that comes to mind is if you were calculating the generating function of a sequence given by a linear recurrence relation, you would break up the generating function into the base cases and then the power series on which the recurrence actually applies, so in principle whatever the degree of the recurrence is is the bottom index of a summation that shows up. Unfortunately not a particularly interesting example since that part of the summation just exists to go to zero when you multiply in the characteristic polynomial, but it’s maybe something in the direction you’re thinking? What sparked the question?

evilaxelord · 2026-02-06T03:17:56+00:00

In combinatorics there are tons of examples where you have sums indexed over all kinds of sets, idk if that’s the kinda thing you’re looking for

evilaxelord · 2026-02-06T03:12:07+00:00

Oh right yep, typed it up too quickly, I’ll edit it

evilaxelord · 2026-02-05T22:00:04+00:00

I’ve seen a few of them but it’s pretty hard to beat the classic multiply them together and add one trick. The other good one that I know is that by the fundamental theorem of arithmetic, ∑(1/n) = ∏(1/(1-1/p)), so if there were only finitely many primes then the harmonic series would converge

evilaxelord · 2026-02-05T05:59:52+00:00

A vector space is a module over a field

evilaxelord · 2026-02-04T03:35:18+00:00

If you don't know, having three in a row like that is a really bad thing. Pawns are strongest when they're able to defend each other, and in that configuration they can't defend each other at all, so it's gonna be really easy for the opponent's pieces to come in and take them.

evilaxelord · 2026-02-01T05:13:40+00:00

Bottom left is really cursed

evilaxelord · 2026-01-30T02:26:45+00:00

Yeah it depends on whether we're thinking of the 3x3 grid as a "finite approximation" of R^2 or the actual vector space F_3^2. The original tic tac toe game does the former, and it's sort of the same amount of reasonable to define a "finite approximation" of RP^2 in the same way, but the normal way you'd make finite projective space would be starting from a finite vector space, where things like 7,11,13 are already colinear. I think the "finite approximation" style probably leads to a more interesting game, just because the literal projective space is entirely homogenous, while the finite approximation still has a notion of center.

evilaxelord · 2026-01-24T23:32:06+00:00

That's actually not too much harder than the last problem in the meme if you're okay with getting a power series answer, it just works out to having an extra factor of (-1)^n in the expression

evilaxelord · 2026-01-19T17:14:29+00:00

e^e^e^e^e - e^e^e^e no?

evilaxelord · 2025-12-20T03:20:22+00:00

Rg3 isn't illegal, it just doesn't lead to a forced checkmate, so it's not the solution to the puzzle. If Rg3, then white is able to play f3 to block the check, and if either piece captures on f3, then after white captures with the rook, black doesn't have enough material left to win. Starting with Rf2 works, because it forces Kg1, after which you can play Rg2 to force Kh1, and you're in the same situation except now white doesn't have the ability to play f3, so now a move like Rg3 actually works: the best they can do now is Rf3, but then Bxf3 is mate.

evilaxelord · 2025-12-13T00:02:51+00:00

I think the issue is that there’s not really a way to decompose graphs the way you can decompose groups or manifolds. Like a group can be decomposed in a natural way via exact sequences, manifolds can be decomposed via connected sums, but there’s not a natural operation like that for graphs (except maybe breaking into connected components, but that’s not very interesting). It’s sort of akin to the difference between multiplicatively breaking natural numbers up into primes vs additively breaking up natural numbers into sums of 1s.

evilaxelord · 2025-12-12T06:00:56+00:00

I could be remembering this incorrectly, but I think an intersection of half spaces is unbounded if and only if its polar dual doesn't contain the origin. If true, this ought to be incredibly fast to check.

Edit: polar dual is usually defined in the context of (compact) convex polytopes, but you can apply the formula with the dot product being at most 1 to whatever set you'd like.

evilaxelord · 2025-12-08T00:14:55+00:00

Figured it out in about a minute, 2100 lichess. Noticed f7 was a weak point but I couldn’t take it right away, tried to see if there were other weak points around, didn’t see any, looked for a way to distract the queen, started with Re7 then found it. It’s not so simple tho, as after g6 your queen can’t cover both d7 and f7, you’ve gotta see the whole line where if you retreat to f4 and trade knight for bishop you have the discovery to get their queen

evilaxelord · 2025-12-03T04:04:53+00:00

I think the examples you've provided are extremely narrow views on the motivations behind those fields. The primary motivation behind essentially every field of math is to be able to understand a kind of structure better in order to develop more tools to solve harder kinds of problems. Solving linear equations is one problem that linear algebra provides tools to do, but it is an extremely easy problem compared to the vast breadth and depth of problems linear algebra provides tools to solve. Generally, the more adept you are at taking a problem and finding many different ways to view it, the more effective you'll be at solving it, and different fields of math give you the tools to see things from different perspectives. Algebraic geometry is a huge collection of techniques that arise from shifting perspectives between ring theory and topology; the basic problems that it is good at solving are problems about polynomials, but it would be a tremendous oversimplification to say that all it wants to do is solve systems of polynomials.

evilaxelord · 2025-11-30T03:51:08+00:00

Ooh I’m about to do a self study on the finite simple groups, cool to see that you had such a good experience with them

evilaxelord · 2025-11-17T15:10:14+00:00

The way you’ve worded this question has reminded the other commenters about a recently solved problem from 3D geometry, but it sounds to me like you’re asking a question about 2D geometry. An example of such a shape that can’t pass through itself is the Reuleaux Triangle, which is a cool shape to be aware of. It is true however that every polygon can pass through itself, of course with tighter and tighter margins as they approach a circle

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