Math News: Knights are 24/13 Times Faster Than Kings

[@DrTrefor]

Corrections (thanks to commenters!!) 1) In the video, I was arguing that if you knew a+b is odd AND you could get two squares to the right, then you could get to everything via symmetries. That's true, but the argument as stated only got us to HALF the darksquares (missing for instance (1,1) if you start at (0,0)). Here's the full argument: If you have an (a,b) knight, where, say, b is the odd one then using the "two steps to the left/right/up/down" argument we proved you can get to (0,1) or its symmetric equivalents and from there you can add to get anything like say (1,1)=(0,1)+(1,0) 2) When writing gcd(a,b)=1 I apparently said "denominator" not "divisor" - yikes! 3) At 11:40 I noted that if you know the estimates on the teal and blue triangles, by symmetry you get estimates everywhere. That's true, but my animation covering the full screen swapped the colors by mistake. It shouldn't be alternating, it should be two blue triangles then two teal triangles etc. This was my first video entirely created using Manim - the python library originally created by 3blue1brown. But I knew precisely zero python a month ago. I actually used this video's sponsor brilliant.org/TreforBazett to master the basics of python in about 2 weeks, and after that my foundation felt strong enough to tackle this video. Hope you enjoyed!

[@pH-8]

Was just about to comment with a compliment of your '3b1b level' animation! 🙂 Great video. Thanks.

[@sumedh-girish]

Eyy! Good going, we need more people like you trying out better ways to visualize math. It is the only way we can motivate the community to also embrace making math more approachable than it already is. Thanks to you we are getting one step closer.

[@MrRyanroberson1]

i deeply respect your willingness AND ability to correct errors. Many lack one of the two, some even lack both.

[@Rodhern]

The problem with infinite chess boards is that it takes forever to promote.

[@javierdiaz-s3702]

On the other hand, in an infinitely large chess universe no more than 2 queens are possible…

[@Winium]

Skill issue

@@javierdiaz-s3702 Why not?

[@Macieks300]

9:16 just a nitpick: gcd stands for greatest common divisor, not greatest common denominator

[@DrTrefor]

oh yikes, did I say denominator? Yes, divisor for sure.

[@DarinBrownSJDCMath]

​@@DrTreforI know right? You think hard about what you want to say, proofread your script 4 or 5 times, and even watch it back before publishing to catch anything. And little things still manage to slip through! I know the feeling.

[@mrblakeboy1420]

i’ve only ever heard it as greatest common denominator, it still works

[@robertveith6383]

​@@mrblakeboy1420-- L.C.D. is the "largest common divisor."

[@DrTrefor]

@@DarinBrownSJDCMath I definitely think I'm at my complete stupidest when trying to watch the recording for verbal errors - so bad!

[@charlievane]

so the king is just a couple of tiny donkeys in a robe, a (1,0) and a (1,1)

[@e-pluszak9419]

One caveat, how exactly do we define "average" when the set is infinite? The paper you cited at 9:56 says "average ratio within a box, between king distance and knight distance", which is indeed well defined as the set is finite, I guess implicitely they assume "limit as the box sidelength approaches infinity", but more importantly: for a square box whose sides are parallel to axis and for example a square rotated at 45°, or an Euclidian circle etc. the answer will be massively different

[@DrTrefor]

Ya absolutely. The reason the square boxes are used is because that is what the king gives, so this really is about knight moves relative to the choice of king moves. With a different metric for the "balls" been compared, you get different results for sure.

[@NotSomeJustinWithoutAMoustache]

This visualization at 1:06 with different "starting points" for the knight really helps illustrate why knights on the rim is so dim. In that initial middle placement there were only five 4-move away parts of the board, but in the actual starting placement or at position like 1:10 we see that nearly half the board takes four moves to reach.

[@DarinBrownSJDCMath]

2:33 "A knight on the rim is dim."

[@lunafoxfire]

a knight in the middle is... the shizzle

[@davidhildebrandt7812]

Or the more intense German version: "Springer am Rand bringt Kummer und Schand'", "a knight on the edge brings sadness and disgrace"

[@sqohapoe]

You should say "24/13 times as fast as kings" not "24/13 times faster than kings" when you say 50% faster, it means the bigger number is the smaller number plus 50% of that small number. when you say 24/13 times faster, it means the bigger number is the smaller number plus 24/13 of that small number.

[@robertveith6383]

"Times faster" is a meaningless phrase anyway. Do not use it.

[@robertveith6383]

*@ Dr. Trefor Bazett* -- "Times faster" is a meaningless phrase, as is "times largrer/slower/smaller." Do not use any of them. As someone referred to it, if applicable, use "X times as fast," for example.

[@rickkaye5864]

Small point - Around 8:20 in the video, you say that if you can travel 2 squares to the right, by symmetry you can get 2 squares up, down and to the left and from there to any of the dark squares. But, that's not true in general. The 2, 4 super knight can reach 2 squares to its right but can't get to the dark square that is 1 square to its right and 1 square up.

[@rickkaye5864]

@@DrTrefor My reasoning was as follows. If we number the rows 1...N and the 2,4 knight starts on an even numbered row, every move will take it to another even numbered row. Therefore, it cannot get to a black square (or any square) on an odd numbered row. Am I missing something?

[@galoomba5559]

Yeah, the other condition (a+b is odd) implies that, the video just glossed over it.

[@DrTrefor]

Ah yes, I see what you are saying now. Yes, my argument was incomplete, it only got to HALF the dark squares and you need the a+b is odd property to get to the rest like say the one 1 sqaure up and to the right. If you have an (a,b) knight, then subtracting jumps of size 2 get's you to (0,1) and by symmetry (1,0) and adding those gets you to (1,1).

[@mathematicskid]

Sorry, but the proof around 9:00 is flawed. Being able to move by 2 doesn't mean you can reach all squares of the same color, only half. This is fixable by noting that if the knight moves by (2a, 2b+1) and its variants, and it can move by 2, it can move by (2a, 2b+1) - a * (2, 0) - b * (0, 2) = (0, 1) and thereby reach cell (c, d) by c * (1, 0) + d * (0,1), or by noting that any desired square of the opposite color has some square it can move to that's in the same half of the same color and some square that's in the opposite half (since the cells (-a, b) and (-b, a) away differ by (a-b, a-b), and a-b = a+b - 2b = odd - even = odd), and that as a result, any cell in the opposite color is reachable, and the opposite half of the same color is reachable through any cell of the opposite color.

[@maxthexpfarmer3957]

in addition it doesnt take into account the edge of a board, eg a 18,19 leaper is useless on an 8x8 board

[@DrTrefor]

Oh thank you, yes quite right and I added a note in the pinned comment to this effect. Once you get the "move over by 2" argument, getting to (1,0) or whatever is (almost) immediate, but I think you're right that it is worth explicitly stating this.

[@maxthexpfarmer3957]

at 8:32 i would like to point out that being able to go 2 squares in any direction does not necessarily give the ability to go to any dark square, since going 1 square diagonally could still be out of the picture

[@DrTrefor]

True, my argument only got to half the dark squares using the "over by 2" property alone. But if you combine that with the "a+b is odd" property THEN you can get to any of the sqaures.

[@hotdogskid]

If youre testing thumbnails out you gotta try using manim in the thumbnail its like clickbait but for recreational math people lol

[@polyhistorphilomath]

Workin' on our knight moves. Tryin' to make some front page drive-in news.

[@GMPranav]

Super knights are cool for math research, but they would be terrifying in a real chess game lol

[@DanielDugovic]

As a chess player, there are considerations relevant to chess, such as: - Given a pair of squares, is the king faster or the knight faster? (How does this change if we first randomly move the piece many times to determine its starting square?) - How many routes exist between two squares? - If optimal routes are blocked off, how fast is the second-shortest route?

[@lfx2407]

the region drawn at 11:40 is probably wrong, if you want to express that blue region is limited by the speed of diagonal travel and green is limited by the speed of orthogonal travel. above the blue region you drawn first is another blue region, symmetric to the first one under the reflection of y=x.

[@DrTrefor]

oh nice catch, added a note about this to the pinned comment - thank you!

[@anikethdesai]

Please make videos on the Laplacian Operator and the Hessian Matrix and why it's used to find Maxima and Minima of a surface. Thank You, absolutely love your videos!!

[@DrTrefor]

Oh that would be a good one. I've been meaning to expand my multivariable calculus series:)

[@anikethdesai]

@DrTrefor Also, please if you don't mind, can you explain the Hessian Matrix in a Linear Transformation point of view? I've been trying to think of it that way and it's been troubling me for a month now

[@AJMansfield1]

2:40 This centrality 'number of options' behavior isn't really relevant for the question, though, what matters is if there's situations that would admit shorter paths if the board were unbounded. And as far as I can tell, that only happens between a corner and the square diagonally inset from it -- i.e. it takes 4 moves to go from A1 to B2 instead of 2 -- everywhere else the shortest path still has exactly the same number of moves.

[@brboLikus]

Great video! I wonder what the equivalent result for boards of higher dimensions looks like... (Also, just a small issue - at 6:30, the value at the origin (0,0) should really be 0, not 2.)

[@samu6982]

At 5:50 I think 3A is wrong (-1, 0) should be (0, -1) = (2, 1) + (-1, -1) + (-1, -1)

[@DrTrefor]

Oh nice catch. Ya I think the point and arrows are correctly at (0,-1) but the label was incorrectly swapped.

[@francescoghizzo]

Is there a mathematical reason why mating with a knight and a bishop is so damn difficult?

[@DrTrefor]

I think think the main idea is that the set of squares a knight can jump to and a bishop can go to don't line up nicely. With two bishops, for instance, you can create a diagonal patch the enemy king can't walk through.

[@swayamshah6891]

yeah. i think its called a skill issue (only kidding i can barely do two bishops)

[@PrimordialOracleOfManyWorlds]

suppose you use 3D space, what will happen?

[@DrTrefor]

Ya the core idea of a sumset applies equally well in 3D or nD space. It is in a sense harder to have initial sets A that go out and cover the entire space

[@PrimordialOracleOfManyWorlds]

@@DrTrefor cool af.

[@choco_jack7016]

i hate how you draw the knight paths vertical first for all 8 spaces.

[@ciCCapROSTi]

6:50 I don't like that you put the 2 in the origin. You didn't overwrite any of the already set down labels, so origin should have stayed the 0 label. Also if we consider superknights, we should also consider superkings, which can step in a ring exactly N away.

[@m3morizes]

The generalized super knights remind me of the two squares problem. The regular knight corresponds to the Gaussian 1+2I (and its associates) which have length √5. Not really relevant, but was on my mind.

[@MrRyanroberson1]

Your rigor here was poor. You said "... with coefficients 1" for the sum sets, but that would mean for a knight you could have at most 8A, since 9A has no valid elements (of 9 distinct knight moves). Likewise you said "... 2 up, down, left, or right .. access all dark squares", which does not follow generally, such as from the 1,1 superknight, which is like a slow bishop.

[@DarinBrownSJDCMath]

The terms in the sum are not required to be distinct. That would be called an h-fold restricted sumset.

[@DarinBrownSJDCMath]

Ah, I did hear him say "different [i.e. distinct] elements" in the video. So that was misleading, you're right.

[@DrTrefor]

I added a note in the pinned comment about how to use the "2 to the right" property combined with the a+b is odd property to get to the rest of the squares. And as @DarinBrownSJDCMath notes, the definition of sumset used here is for non-distinct elements of A so even though an element starts with coefficient 1 it can be added to another of the same element.

[@cobo1678]

Spent the afternoon working on this problem after seeing the title of this video, I got that the knight is about 1.21x the speed of the king on an 8x8 board, not your 1.3. Time to check my work and see where I went wrong

[@drdca8263]

Hm, if instead of averaging over the regions… I was thinking like, what are the constants c and C such that for sufficiently large m, m K is a subset of (m c) N_{a,b} , and m N_{a,b} is a subset of (m C) K

[@YesMeRaul]

Oh yeah, my favorite math teacher does a video on chess

[@tyleradams6048]

🎶Workin' on our knight mooooves🎶

[@MuhamamdUmar-lp4ee]

Sir can you please make some more videos on maths related to chess ♟️♟️

[@DrTrefor]

I had too much fun with this one, I might have to!

[@MuhamamdUmar-lp4ee]

@DrTrefor ❤️❤️❤️❤️

[@galoomba5559]

the way you're drawing the knight moves gives me pain

[@dansimpson6844]

Bob Seeger wrote a song about this.

[@System.Error.]

Someone send this to Hikaru 😅

[@System.Error.]

Just like his double disambiguous what video reaction

[@getsu0]

The Knight is just appealing to neurodivergence I gotta say it

[@aieousavren]

Just like real life

[@Blingsss]

From zero Python to creating this? Talk about an upgrade! The math behind this video is just as fascinating as the animations. Anyone else inspired to pick up coding or a new challenge? SolutionInn has been my go to for academic motivation, but this video might just spark a side project!

[@kikkisu]

There was no need to use AI for this thumbnail.

[@DrTrefor]

Perhaps! I'm actually A/B testing an AI thumbnail and one I made manually, going to be interesting to see which gets more engagement:)

[@EvTheBadConlanger]

@@DrTrefor Using AI thumbnails is understandable, but it can come off as lacking integrity. Not the most serious crime, but an ick for many. That's just my opinion anyway.

[@SirTurdle]

@@DrTrefor I recommend making the man-made one be visually appealing without being “perfect”. IMO, i liked this thumbnail anyways

[@neshirst-ashuach1881]

Theres also no reason *not* to use an AI image. It looks nice and was presumably cheap and easy to produce.

[@muhammadgheith2492]

@@neshirst-ashuach1881disagree on the premise that it’s “cheap” to produce. Sure it costs nothing (save for the art theft discussion which I shouldn’t be ignored), but the power cost of generative AI is much worse than your standard art piece. These things run on server that take an enormous amount of power. If you’re ever interested, check out Cody Johnstons video on AI slop

[@altair-tf8fp]

holy hell

[@iamdigory]

I don't think you should necessarily feel bad about using ai. The arguments are weak, but it does make your channel look cheep

[@hyoominoid01]

why is the top comment arguing about an ai thumbnail bro watch the chess video 😭 🙏

[@purplenanite]

Please don't use the AI thumbnail

[@DrTrefor]

Actually running an experiment on this video doing A/B testing with an AI thumbnail and my own manually made one - will see which ends up being more popular!

[@Luca-bv9fh]

@@DrTrefor I would not take the data on this at face value, e.g. I only clicked this video to see if anyone had commented on it being AI, and I'm glad to see people calling it out. You mention making the video in Manim, this is familiar to people who watch maths videos and I'm sure they would also respond well to a thumbnail made with it

[@LJMlover]

​@@DrTrefor I personally didn't notice the ai thumbnail

[@davidtitanium22]

I clicked when the thumbnail is not AI, but i'm glad since I have a tendency to just block channels that uses AI thumbnails/profile cause i just assume the content is low effort slop

[@sebas31415]

Please, for the love of all that is holy, dont use AI. It 1: looks bad 2: is based on artist's works without permission and 3: the prevelant usege of AI has proven to be unsubstainable to the environment. I have been a subscriber for years, and I've never felt more disappointed at you. I know you are doing "A/B" testing. But using AI makes me question your legitimacy as both a content creator and a mathematician. Goodbye; im unsubbing

[@GrifGrey]

"legitimacy as a mathematician" is crazy

[@drdca8263]

Were you actually subscribed, or did you just see a thumbnail with ML model generated graphics and claim to have been subscribed and to be unsubscribing?

[@sebas31415]

@@drdca8263 i have been subscribed since 2022. I unsubscribed. I loved his videos on the math of traffic and the one about how bubbles can be used as a model for surface area optimization. He's the reason I know about hyperbolic trig functions.

[@drdca8263]

@ Alright, you’ve convinced me. I apologize for my accusation-or-something-like-one .

[@sebas31415]

@drdca8263 your ok! I just dispise ai. Maybe later I'll resubscribe! Have a good day/evening/night

[@angelajohnson4666]

You are correct.

[@simski_99]

who asked?

[@MrVontar]

I don't even think 1 chess move ahead 🥲

[@MrVontar]

This inspired me to try playing, apparently I missed a win against a 2865 bot lol. Not bad considering I never even play

[@MrVontar]

This is kind of interesting cause you can think of chess as an infinite series almost

[@MrVontar]

I beat a 3200 bot!!

[@neshirst-ashuach1881]

Thanks for the video, and Ignore the people whinging about AI - the image looked fine and the vast majority of people wont care where it came from.

[@rishabhshah8754]

well, fine is subjective

[@neshirst-ashuach1881]

@rishabhshah8754 Obviously - all art is subjective. But its a youtube thumbnail, not the Cistine chapel. And for that, the image is perfectly fine; its certainly not worse than the normal thumbnails on this channel (unless you're a weirdo who just hates any AI product no matter what).