Random things that will (likely) surprise you

this video is sponsored by curiosity stream home to over 2,500 documentaries and nonfiction titles for curious minds okay do not blink that's probably the only cool card trick I can do but I can't tell you something interesting about playing cards I learned a few months ago in the standard deck there are four suits for the four seasons in a year there are 52 cards for the 52 weeks in a year and if you let a saw a value of one then all other card values follow from there up to 13 adding up all the cards in the deck gets you 364 add one joker and you get 365 the number of days in a year add the second joker and you get the number of days in a leap year that blew my mind the first time I heard it I don't know why I didn't learn it earlier in life but hopefully it surprise some of you as well and the rest this video is gonna be some more random things will be all over the place that will hopefully surprise you and these are things I learned over the past like year or so which just didn't make it into any other videos like this puzzle that was taken out of the is it possible video that I did a few months back imagine a circle with four points a B C and D on the perimeter then we'll put three points on the interior B C and D as shown but the second a will also go on the perimeter the question is can you connect a to a B to B C to C and D to D so the connections don't leave the circle and also don't intersect this probably wouldn't take many of you that long to figure out but I think it's the mathematical reasoning here that's really interesting now if you just think about connecting A to A this seems like it might be impossible because if you connect the a to itself like this then you have completely eliminated the ability to connect the D points together since to do so you'd have to have the lines intersect which isn't allowed if you instead connect the A's like this you still cannot connect the D points together if you do something like this now you can't connect the B points together so this isn't looking possible but let me ask this instead if I move the points around as shown can you now complete the same task asked before that answer is obviously yes a goes straight to a and everything else connects easily but the fact that this works allows us to see that the previous question is possible because if we just rearrange the points back to their original configuration while pushing the connections around as needed we get a configuration that satisfies everything we said and connects all the letters to their partners what we did was turned a more difficult problem into an easy one through a continuous transformation or homeomorphism that allowed us to get a solution so it didn't even require trial and error but rather just being a little clever now this next puzzle is a seemingly simple one that has to do with a rectangular region but we can really think of it as a one-dimensional line segment as well the question is can you place five points we'll label X 1 through X 5 such that X 1 is within that region so that's easy then X 1 and X 2 must be in different halves X 1 X 2 and X 3 have to be in different thirds X 1 through X 4 must be in different force and all the points must be in different fifths of that region okay at first I thought this was going to be super easy but it's not here let's just build up to it if we place X 1 here that just satisfies the first condition no problem now for the second point let's place it here since it's in a different half of the region then X 1 we've satisfied condition 2 but the third condition says that the first three points must be in different thirds and these two points already in the same third so that failed but let's just move them aside and put X 3 in the middle because order doesn't matter then here's the region split into force and we see there is an open spot 4 X 4 to go so that condition 4 is satisfied everything's in their own separate fourth but now we got to split it into fifths hope that x1 through x4 are in different fists while allowing room for x5 and yes in fact that does workout x5 just goes here and we're good this of course isn't the only solution but we can't just switch any of these that we want and have something that still works like if we swap X 2 and X 5 then that's fine for condition 5 but if we go back to condition 4 then now x1 through x4 are not all in different force so we can't maneuver as much as we'd want ok fine that wasn't too tough but what if we were to keep going well if we do this with 6 points or 7 or 10 and so on using that same pattern of rules well if you want to try this on your own try it with 10 points because you'll see it's definitely not easy but a more random question is can we keep going well this even be possible with 10 or 20 or a hundred points because definitely seems like it is it seems like we're just really clever we can arrange the number to satisfy all the conditions depending on how many points you have but we actually can't go on forever it's a seemingly endless puzzle but it does end and the last number we can work with is 17 if you have 17 points and have satisfied 17 conditions you can make it work but at 18 and anything after it becomes impossible here you can see a picture of a solution with 17 points but you won't find anything after that then if you've seen this year's math Avengers video the next few minutes is my part it's a very simple unsolved math problem that didn't make it into this video I did before see for most unsolved math problems you need a degree in mathematics just to understand the problem here's an example called the Hodge conjecture which comes with a million-dollar prize by the way but as you can see there are very few words and non mathematician would understand so let's see an example of a much simpler question what we have here is a break a rectangular prism and I've labeled all the side lengths a B and C and also label all the diagonals XY and Z or the diagonals of the faces now what makes this brick in a l'heure brick is if all the values are integers however finding integers that makes this work is not easy for example if we were to try like 3 4 & 5 for the side lengths well that's fine here 3 & 4 have a diagonal of 5 that works that's an integer but 3 & 5 use a non integer diagonal so that doesn't work now this is not the unsolved part though so the next question would be what's an example of an Euler brick what's the smallest Euler brick that exists give that some thought but not really because it's gonna take you a while just tightening these first few integers is not easy but I memorized them for this video here we have them at the first Euler brick so the side lengths a B and C are 44 117 and 240 respectively and then if you do the Pythagorean theorem with each of those pairs you get these numbers like 44 and 117 give you 125 do it with 44 and 240 get 244 and these two give you 267 so we can keep going from there but again this is the first whaler break now the equations that represent this are called IO fantine equations which are just polynomial equations that only have integer solutions and in this case those look like this so we have three Pythagorean theorems and any integer solutions to these equations yield an Euler brick now the unsolved part of this comes in when we add a fourth bag one that goes from the bottom left corner to the top right I'll draw that in and I'm just going to call that D if you can find an a B and C such that everything here is an integer you have found a perfect Euler brick however the question of does a perfect Euler brick exist is still unsolved that's the part we don't know let's actually eight perfect Euler brick is unsolved when we add in that fourth equation which would be a squared plus B squared plus C squared equals d square the diagonal then we don't know if there are integer solutions that satisfy these four equations however through the use of computers and just number theory we have found out a lot about the properties of these numbers if an oil rig does exist for example just from the Wikipedia page we know if a perfect oil and brick exists the smallest edge must be greater than 5 times 10 to the 11th we know one edge to face diagonals and the long diagonal would be odd we have a bunch of divisibility rules and so on that's about it though solving just those three Diophantine equations is not meant for a quick video like this so it wasn't much I even could go into depth on but I love unsolved problems and this was an easy one to talk about and just because we're on this topic here's another simple unsolved problem this here is a minimal Sudoku that means it only has one solution but if you remove any number you now have a Sudoku with more than one solution meaning it's not a proper Sudoku anymore they very recently confirmed that 17 what you see here is the minimum number of clues you can have for a minimal Sudoku anything less and you'll have more than one solution what we don't know is the maximum number of clues for a minimal puzzle however we believe it's 40 and only two of those puzzles have ever been discovered when you're seeing now compare that to the over 2000 minimal puzzles with 39 clues that we have found so if we remove one number from this Sudoku then we'll have a puzzle with multiple solutions however we think if you have 41 or more clues then yeah you have one solution but remove a number and you'll still only have one solution so it's not minimal we still aren't sure yet though there are several other unsolved problems that are easy to understand but I might save that for a future video if you want to continue learning more random cool things though you can head on over to curiosity stream the sponsor of today's video curiosity stream host thousands of documentaries and nonfiction titles put on by industry leaders that you will very likely enjoy if you're a fan of this channel they have documentaries like the hulking pair if you want to learn more about one of the greatest mistakes made by Stephen Hawking in a search for a deeper understanding behind the physics of black holes or for those who like the real-world applications of mathematics I recommend the secret life of chaos which will take you through several examples of how chaos shows up in reality and even math itself curiosity stream is available in a variety of streaming services including Roku Android Xbox one Amazon fire Apple TV and more and it only comes up to $2.99 per month plus if you go to curiosity stream comm slash major prep or click the link below you'll get your first month membership completely free so no rest and just giving it a try but with that I'm gonna end that video there of course big thanking my supporters on patreon social media links to follow me are down below and I will see you guys next time

Channel: Zach Star

Views: 2,180,907

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Id: 3ARVIkbbf3o

Length: 11min 51sec (711 seconds)

Published: Fri Jan 03 2020