Animating Nicomachuss 2000 year old mathematical gem (Mathologer Christmas video) Mathologer 2 2022-12-24 | Short videos don't seem to work terribly well on the main channel :( And so here it is the short Mathologer Christmas video for 2022 on Mathologer 2. I hope that all you real Mathologer die hards will find it anyway. I finally got around to animating Nicomachus's theorem, one of my all-time favorite mathematical gems. The wiki page is probably the best and most accessible place to look for more information about Nicomachus's theorem tinyurl.com/3h8dk53rTemper R. Haring suggests to continue the patterns to 4th powers like this:If the rule is that each row consists of 1 then 2 then 3 numbers and each have to be consecutive uneven numbers then, for n^4 you get: 1 = 1^47 + 9 = 2^425 + 27 + 29 = 3^461 + 63 + 65 + 67 = 4^4etc.with the first number of each row being p(p^2-1)+1.Franz Biscuit adds:Interesting. So for the general case, the pattern appears to be p^n-p+1 yields the first number of the pth row for the nth power.Music: Earth, the pale blue dot by Ardie SonEnjoy!burkard
Who survives Solitaire? Mathologer 2 2022-01-23 | This is the answer to one of the challenges in the latest Mathologer video youtu.be/BMPa0FA65Fk Note we are restricting us to asking which pegs can be the sole survivors in the middle of the board. If surviving anywhere else is also an option then some other pegs can also survive.The music in this clip is I promise by Ian Post.Enjoy!
The proof that proves them all (x-post from Instagram) Mathologer 2 2021-12-22 | Here is a second proof of the fact that the angle sum in a 5-pointed star is 180 degrees. You can use the procedure that this video is about to figure out the angle sum of any polygon whatsoever. Are you up for the challenge at the end of this video - figure out the angle sum of the 7-pointed star. Some of you had problems parsing the first proof I published the day before yesterday. This second proof would actually have been the first proof, but I actually did not think that I could get it under a minute (the maximum length of videos on Instagram). The version of the video that I am posting here actually has two extra frames with the challenge accounting for an extra 5 seconds.Enjoy
If you dont like this one ... Mathologer 2 2021-12-21 | Quite a few of you will know this one. Still, for me at least the animation really adds to the enjoyment I get out of something that I've been familiar with for a zillion years :)
Pentagramath(s) [not evil :] (xpost from Instagram) Mathologer 2 2021-12-20 | Day seven of my Instagram adventure and still going strong. Such a nice break from my usual gruelling monster video projects. Takes me a little bit more than one hour to get one of these done, no scripting no separate video. This one also came together very nicely. See what you think :) Still looking for a bit more critical mass on the Instagram side of things. And so if you are prowling Instagram anyway ... instagram.com/the_real_mathologer
A cute proof with a very dramatic moment :) (xpost from Instagram) Mathologer 2 2021-12-19 | Here is the sixth chapter in my Instagram adventure. The "artist" in me went a bit wild in this one. Of course, this is just a special case of summing a geometric series. Still, I also love this sort of little gem. How about you? :)Enjoy. P.S.: If you use Instagram please drop by the Mathologer testing ground at instagram.com/the_real_mathologer/.
How do you best motivate the counterintuitive (-1) x (-1) = 1 ? (xpost from Instagram) Mathologer 2 2021-12-18 | Here is the fifth chapter in my Instagram adventure. How to best motivate/explain why a negative times a negative should be a positive (say if one of your kids asks you.) I also like this one:If there is a God, then ...0 = ((-1)+1) x (-1) = (-1)x(-1) + 1 x (-1) = (-1)x(-1) - 1Then shuffling the 1 to the left gives 1 = (-1)x(-1)Enjoy. P.S.: If you use Instagram please drop by the Mathologer testing ground at instagram.com/the_real_mathologer/. Also if you like the proof in this video, definitely check out this Mathologer video youtu.be/p-0SOWbzUYI were I showed this proof for the first time.
A cute visual proof with a great punchline (xpost from my Instagram) Mathologer 2 2021-12-17 | Here is the next chapter in my Instagram adventure. Enjoy. P.S.: If you use Instagram please drop by the Mathologer testing ground at instagram.com/the_real_mathologer/. Also if you like the proof in this video, definitely check out this Mathologer video youtu.be/p-0SOWbzUYI were I showed this proof for the first time.
My favorite proof of Viviani (xpost from my Instagram) Mathologer 2 2021-12-16 | I just started experimenting with putting short clips on Instagram. This is one of them. instagram.com/the_real_mathologer
Solution to the X+Y maths olympiad problem (winner in the description) Mathologer 2 2021-10-21 | Thank you to everybody who contributed a solution to this problem in the comments section of the main video (youtu.be/04x4ZdLpN-0) What I like about the solution that I present in this video is that it also gives a complete characterization of those special colorings of square grids on top of giving the answer to the problem. Lots of other nice approaches to this problem are possible. I'd just like to mention one more approach which I particularly like, not least because it uses a very cute trick that I also talk about in this video youtu.be/9JN5f7_3YmQ Basically you prove that you get different color corners in any 4-colouring of a 4x4 grid. Then you scale up the proof as indicated in this diagram imgur.com/a/affkdH7 So for a special coloring of the 8x8 grid you divide the grid into four 4x4s, and combine the corners of these four 4x4 into another 4x4. Now it's not terribly hard to see that this new 4x4 is also colored in a special way. Therefore its corners, which are also the corners of the 8x8 are have to be of different color. Nice, and in the next step divide a 16x16 into four 8x8s, etc. Some of you actually argued in this way. One little problem was that most of you who did argue this way thought that it was completely obvious that the new 4x4 carries a special coloring, which is not the case. Remember that you have to check that all 2x2s in this new 4x4 are colored differently. This is really obvious for most of these 2x2s but not for all of them. In particular for the 2x2 consisting of the squares 2,5,4,7 in my diagram you really have to argue separately that these are of different color (e.g. like I did in this video). A second problem is that this approach only captures square grids with sides that are a power of 2 and not all 4n x 4n grids. Anyway nice. Also you may have noticed while watching my proof that what we are really showing here is that in any special coloring of a 2n x 2m rectangular grid the four corners are of different color. Anyway, who gets the t-shirt? Bit tricky. The first submission was this: Joeeeee For the challenge: Same as Pascallian Triangles but for n=4. youtu.be/9JN5f7_3YmQ Where can I get my t-shirt? 😜 So that looks like Joeeeee is on the right track and that his arguments is supposed to be something like what I just outlined. A "bit" short on details though :) The next proof that was submitted and that I thought worked quite well was also along the same lines. This proof was by Tommaso Gianiroio. Again not quite complete, but complete enough to warrant a t-shirt :) Anyway, I am in a generous mood today and so happy to give a t-shirt to both Joeeeee an Tommaso. Please get in touch with me through a comment on this video and by e-mail.
3rd proof that the coefficients of (x+2)^n count the numbers of vertices, edges, etc. of an n-D cube Mathologer 2 2021-09-14 | A nice animation that I put together which complements the last video on the main channel on hypercubes, Euler's formula in higher dimensions, visualising hypercubes, iron man :) and so onyoutu.be/E2l95ttmJOgMusic: FloatingBranch by MutedEnjoy!burkard
Intro to MagicTile (Part 2 of Can you solve the Klein Bottle Rubiks Cube) Mathologer 2 2016-10-08 | Following on from the video about Klein bottle Rubik’s cubes and other topological twisty puzzles on Mathologer youtu.be/DvZnh7-nslo in this second part I talk about the MagicTile interface, show you how to design and record algorithms as macro moves, as well as talk you through a complete solution of one of the easy Harlequin edge-turning puzzles (featuring the all-time simplest three-piece cycle algorithm as well as some cute parity problems)Also check out the following videos for more background information. "A simple trick to design your own solutions to Rubik’s cubes": youtu.be/-NL76uQOpI0(for an introduction to designing your own algorithms for solving twisty puzzles)A mirror paradox, Klein bottles and Rubik's cubes: youtu.be/4XN0V4xHaoQ(An introduction to what Klein bottles are all about and a bit of fun with putting Rubik’s cubes INTO Klein bottles.)Cracking the 4D Rubik's Cube with simple 3D tricks: youtu.be/yhPH1369OWc(Your next challenge after the the Klein Bottle Rubik's cube. Another hall of fame awaits.)
Cracking the 4D Rubiks Cube with simple 3D tricks Part 2: Magic Cube 4D Mathologer 2 2016-06-17 | Following on from part 1 youtu.be/yhPH1369OWcon the main Mathologer channel, this video is a hands-on introduction to the 4D Rubik's cube simulator Magic Cube 4D, with a special focus on building macros for solving this 4D puzzle (for all those of you who would like to give solving the 4D Rubik's cube a go.)You can download my full set of macros that I put together using the simple methods described in both videos from here:http://www.qedcat.com/misc/burkard_macros4.logHaving said that I'd like to encourage you all to find and program your own macros. After all, the more work you do in this respect by yourself the closer you will be to being able to say that you solved the 4d Rubik's cube "on your own". I've also included the macros used in Roice Nelson's published solution (http://superliminal.com/cube/solution/solution.htm ) at the end of my macro file. To understand what most of these algorithms do, you will have to study Roice's write-up. Having said that the algorithm labelled 2nd 4-color series is a very nice example of an elegant short algorithm that twists just one of the corners. Well, worth studying in slow motion. Similarly, the 3rd 3-color algorithm is used for twisting edges (without worrying that it also mucks up some corners). Enjoy!Burkard Polster
Footnotes to the video A simple trick to crack all Rubiks cubes on the main channel Mathologer 2 2016-01-15 | Check out the main video before your watch this one: Here again is the description for the video on the main Mathologer channel. The vast majority of people who tackle the Rubik's cube never succeed in solving it without looking up somebody else's solution. In this video the Mathologer reveals a simple insight that will enable all those of you who can solve the first layer to design your own full solution for the Rubik's cube, as well as for many other highly symmetric twisty puzzles. For more details about this really very fundamental idea behind many twisty puzzle solutions have a look at this article by the Mathologer from a couple of years ago http://www.qedcat.com/rubiks_cubeGoogling "commutator, Rubik's cube" will also produce links to a lot of very good articles on this topic. The Rubik's cube animations in this video were produced using the program CubeTwister by Werner Randelshofer: http://www.randelshofer.ch/cubetwister/Enjoy!Burkard Polster, Giuseppe Geracitano, Karl and Lara
e to the pi i = -1 paradox Mathologer 2 2015-12-24 | A nice puzzle that did not quite make the cut on our main channel. Enjoy!Burkard & Giuseppe
Intro to juggling three balls Mathologer 2 2015-12-05 | The Mathologer explains how you go about learning to juggle three balls. This complements a video on the mathematics of juggling that is part on the main Mathologer channel.
