The Moessner Miracle. Why wasn't this discovered for over 2000 years?

[Music] welcome to another mythology video it's been an absolutely insane first half of the year in my part of the world busy busy busy luckily i can see the light at the end of the tunnel and so i'm looking forward to a lot more mythology action in the coming months anyway let's have some fun today have you heard of mercenar's miracle no not many people have which is a real shame well today's mission is to do something about this sorry state of affairs as usual we start with an easy warm-up and most of you will be very familiar with this one we begin by adding up the odd numbers there's 1 plus 3 is 4 plus 5 is 9 plus 7 is 16 plus 9 is 25 there's 36 so adding the odd numbers gives the squares a very pleasant surprise when you first encounter this famous mathematical gem right okay but how can we be sure that this pattern really continues forever and ever after well there's also a famous and beautiful visual proof that goes back to at least the time of pythagoras it may be due to pythagoras himself or one of the members of his fan club the pythagoreans proof time here we go okay odd numbers let's add them again one plus three is four plus five is nine all clear now right there there very nice very nice very nice people have been in love with this proof and some related proofs for thousands of years and for the longest time pretty much everybody was under the impression that there was nothing left to discover in this respect that is until about 70 years ago when alfred mersner the guy you've never heard of discovered a truly miraculous extension to our odds into squares result muslims discovery was first published in 1951 in the that's it over there just one page and not hard to make sense of if you happen to speak german so let's just get to it we'll start by reframing our classic odd numbers theorem in a special way start with all the positive integers now highlight every second number there and there now let's add all the numbers that are not highlighted that's just the odd numbers again right so 1 then we get 4 then 9 then 16. 25 36 there's the squares again now mercenary's idea was to start this process in a slightly different way instead of highlighting every second number he highlights every third number there there there there now he adds all the numbers that are not highlighted okay so one plus two is three skip the highlighted three so plus four is seven plus five is twelve skip the six plus seven is nineteen and so on there there there hmm okay but now what those new numbers don't look that special well we're not finished yet now we highlight every second among the new numbers highlighting it done and now we add up again one skip plus seven is eight skip plus 19 is 27 1 8 27 do you see the pattern yep those are the cubes there amazing isn't it how on earth do we end up generating the cubes this way also there is more much more and now that i've announced that there is much more can you guess what the more is can you see where we're heading i'll bet plenty of you have got it to begin highlighting every second integer and then adding up gives the squares then highlighting every third integer and adding up followed by a second step of adding and highlighting gives the cubes so what about skipping every fourth integer or every fifth integer let's be brave and go for every fifth integer add up all the grays highlight every fourth add up highlight every third and so on okay there there there yep there are the fifth powers and in general what mercenar discovered was that if we start by highlighting every nth integer we end up with the nth powers of the integers amazingly cool isn't it having fun so far [Music] okay this is mathologer so on was the proof right and we must be close after all moistness paper was just one page so it should be quick and will be done this could be our shortest ever mathologer video well it's not that simple that one page write-up of moistness miracle actually did not include a proof the first published proof was by the math professor oscar perron one of mostner's acquaintances i could not find a photo of mersna but there that's peron over there with his proof so plan b let's go through perron's proof okay first we'll have a quick look [Music] okay maybe not parons proof is pretty much impossible to explain here on mathology in a nice visual and accessible way how about a different proof nope plan c d and e are also doomed to fail a number of other algebraic proofs have appeared over the past 70 years and also don't lend themselves to mathologerization so plan f it is and plan f is to give up of course not mathologerals never give up when we cannot find a proof in the literature that can be nicely mythologized we often try to customize and prove things ourselves so let's give up on the algebra how about the visual proof for mercenaries theorem maybe something that generalizes to pythagorean gem well i gave it a go and it turns out there is actually something there the classic proof for odd numbers relies upon turning one square into the next larger one by adding an l-shaped shell like this right at that add larger and larger squares the pythagoreans have a special name for these very handy l shapes they call them nomons for what we have in mind the first thing to notice is that there are also no mounts for 3d cubes the 3d normans look like this okay there's the blue one there's the green one there's the red one from the other side this shell looks like a hexagon and the number of little cubies in this shell is 37 there that's the 37 up there great and what about the number of cubies in the other shells yep as we would hope 19 7 and 1. very promising it's also possible to spot the remaining gray numbers in those shells to see those numbers note that the brown dots marking the cubies can be separated into hexagons like this now let's count there's the one let's highlight two dots on the smallest hexagon then there are four dots remaining in this hexagon here are five dots on the next larger hexagon and there are seven dots remaining in this hexagon and so on pretty obvious that this pattern will continue forever now to prove that moistness method always gives cubes we can argue visually like this the shells stack together to form real 3d cubes so to show that moistness process produces integer cubes we just have to show that the gray numbers in the second row are indeed the numbers of cubies in the different shells but having prepared our hexagons earlier that's now easy there's the one to get the next shell take a copy of the one and add two and four to to get the 19 take a copy of the seven and add five and seven to it and so on and all this amounts to a visual proof that mercenaries miracle really produces all cubes very pretty isn't it and as i said as far as i know we're dealing with a brand new proof here very cool but of course there's more to moissan's miracle than just the cubes how would you go about trying to prove that we're always getting the fifth powers up there well if you're a five-dimensional creature that's a no-brainer just start by considering 5d shells of 5d cubes in fact when you do this you'll find that these numbers here are just the numbers of cubies in these 5d shells promising but now what also 5d cubes are a bit hard for us mere 3d creatures to visualize so if we want a visual proof that works in our 3d world for all powers we need a different approach luckily there is a totally different but still super nice proof i'll show you this proof at the end of the video but before that there are some other beautiful merson-like discoveries that i really must show you [Music] okay here's a new trick highlight the one skip one and highlight again now skip 2 and highlight skip 3 and highlight skip 4 and so on now do the mersenal summing with as many steps as you need for the larger gaps do those numbers we end up with look familiar hmm looks like the factorials right there 1 times 2 times 3 times 4 is equal to 24 that's 4 factorial that's very cool but there is more have a look at this interesting huh one plus two turns into one times two one plus two plus three turns into one times two times three and so on plus turns into times and this is not a coincidence have another look at the cubes so here 1 times 3 turns into 1 to the power of 3 2 times 3 turns into 2 to the power of 3 and so on so what's going on here in general the question is given a sequence of highlighted integers what sequence will be produced by mercenaries method and how are the two sequences related here's one of the absolutely crazy answers to this question pick a few non-negative integers a b c d however many you want one of them has to be positive then make up a highlighted sequence from these numbers like this there first second third fourth and so on those are the numbers that get highlighted now hold on to your hats the sequence generated is this so again sums have turned into products and multiples into powers feels familiar doesn't it it's a logarithm exponential kind of thing here are some examples of this remarkable relationship in action if a is non-zero and b c d e are all zero we get this that's merchner's original result a is equal to three gives the cubes a equal to five gives the fifth powers and so on here's another example make a b c d etc all equal then we get okay and we get well if we choose a to be equal to 1 then what we get are the factorials again and you probably guessed it there's more for example given two input sequences those two with these output sequences then the sum of the two input sequences gives the product of the two output sequences whoa as well if we avoid some predictable hiccups the output of the different sequence will be the quotient sequence and there's plenty more but that's probably enough new stuff for today i'll include some links in the description of this video for those of you who want to delve a bit deeper challenge for you if the input sequence is the sequence of squares what is the simple formula for the output sequence [Music] to finish off let me show you a methodologization of a very nice graph theoretic proof of mercenary's original ends power miracle this proof was published by the mathematician karel post in 1990 a real treat for you mathematical gourmets of course we're getting into the real thing here so this is a good time to grab a cup of coffee and buckle your mathematical seat belts ready well then here we go we first show that we get all fifth powers by using mercenaries method all other instances of mercen's miracle can then be proved in the same way to start with have a look at the triangle of numbers on the left does this look familiar no maybe tilt your computer's 45 degrees no wait bad idea i'll help you still does not drink a bell how about if i add the string of ones on the right now you've got it right this is just the tip of pascal's triangle which i'm sure you've all seen before clearly something is going on there so let's rotate back and add the top row of ones to moisten the setup like this that looks very nice and that extra row actually meshes in seamlessly with the original scheme just watch let's get to mercenary so start adding 1 plus 1 is 2 plus 1 is 3 plus 1 is 4 plus 1 is 5 6 and so on so we have seamlessly slotted pascal's triangle into our mersnotable but how does this help well pascal's triangle is special in many ways but for us what's important is the way it grows the left and right sides of pascal's triangle consist of ones then any two side-by-side numbers had to give the number below so starting with the ones we get this one plus one is two one plus two is three two plus one is three okay three plus one is four and so on here's a related property of pascal's triangle replace every number by a circle make the top circle blue and connect every circle to the two circles directly below like this turn any one of the green circles into an orange circle now how many different journeys following the arrows are there from the blue circle to the orange circle you know this well it turns out that the number of journeys from the blue top to the orange circle is just the pascal number in that orange circle let's check that this is true for the example there here is one journey there's a second one and here is a third one so that one's fine and why does this work in general well for trips down the sides of pascal's triangle it's all obvious there's just one way to keep going down the edge and therefore we also have ones all the way down now what about an orange circle in the middle somewhere well to get to this orange circle we have to travel via one of the two circles directly above it the pink or the purple guys there but that means however many ways there are to travel to the pink guy and similarly the purple guy the sum of these would be the number of ways to get to the orange circle and of course that's exactly how the pascal numbers are generated so the numbers of travel paths start the same way and build the same way as the pascal numbers and so must always equal the pascal numbers easy peasy what about the other triangles how are they built well let's have a look at the second triangle turn so it's still a pascalish triangle right the numbers on the left edge aren't all ones anymore but the numbers in the middle are all produced by the same addition rule there 6 plus 1 equals 7. 16 plus 7 equals 23 and so on and now that we know what we know it's not hard to check that not only do the numbers in the first triangle count the number of journeys from the blue circle but then indeed all the numbers you see here count the numbers of journeys in this bigger arrow diagram either take my word for it or convince yourself that this is true really easy anyway as an example to get to this circle label 6 there we have to pass by one of these two circles and of course five plus one and six works another one down there one plus 31 is 32 and another one 32 plus 211 is 243 and so on super duper nifty now comes the key trick reverse all the arrows now all the reverse parts are heading to the blue target circle at the top so what do the numbers now indicate they tell us the number of journeys from that circle to the target circle at the top okay the stage is all set remember we're trying to show that most summing method result in successive fifth powers at the bottom circles of the triangles so what we have to prove is that from here there's exactly one journey to the blue target that's pretty obvious right then we have to show that from here there are 32 journeys to the target that from here there are 243 and so on okay let's start from the red circle then it's completely obvious that there's only one way to get from the red circle to any of the circles above right let's move the red circle to the right okay how about the number of journeys from this red circle to the blue target circle let's count them from scratch well first going straight up it's all once again what about this circle here well there's only one arrow leading to it and it originates in a one this means that the circle in questions also has to be a one in other words there's only one journey from red to this new circle and obviously the same is true for all these circles now one plus one is two the same for all these circles all right two plus two is four and so the next diagonal is all fours next is eight 16 32 and there we have it the 32 we were hunting for and a very nice power of two pattern in the triangle on the left to get there you can sense how this is working right let's start the next card then because of the repeating pattern our count again starts like this nice huh arguing as before we get this transfer of numbers now let's just fill in the rest 243 in the blue circle as expected great but why we'll get to the explanation in a moment promise let's just do one more count first there as before we get this transfer of numbers and so on time for some pattern spotting for this we'll scan things from the right to the left watch [Music] powers wherever you look let's go again there the powers of one turn into powers of two the powers of two turn into powers of three and so on very very pretty how all the pieces fall into place so that has to be it right if we can prove this pattern that in general the powers of n on the right side of a triangle are converted into the powers of n plus one on the left side of the triangle then we will have proved mercenaries miracle all clear hope so okay to figure out what's going on let's see how the powers of three on the right combine into the powers of four on the left let's focus on this entry here there obviously the only powers of three on the right that can contribute to this entry are the bottom four so with just a bit of algebra autopilot we get this what are those coefficients in front of the powers of three one three three one they're very familiar right well for example the three in front of 3 to the power of 1 is equal to the three different journeys from the 3 to the power of 1 circle to the orange and similarly with the other coefficients in the sum what this means is that those coefficients 1 3 3 1 are just the numbers in the fourth row of pascal's triangle and that means that our sum amounts to an instance of the binomial formula one of those distant things from high school and the encapsulation of pascal's triangle remember now and setting a is equal to 1 and b is equal to 3 you get our power of 4 sprouting from our powers of three nice and this all easily generalizes to produce all the other powers of four on the left and that's how karel post proved moistness miracle a very pretty and ingenious proof don't you think made my day the first time i encountered it well and that's it for today until next time [Music] do [Music] [Music] [Music] you

Channel: Mathologer

Views: 409,960

Rating: 4.9552913 out of 5

Keywords:

Id: rGlpyFHfMgI

Length: 25min 38sec (1538 seconds)

Published: Sat Jul 17 2021

Reddit Comments

[u/andororand]

Insane. Can anybody explain why this method turns addition into multiplication?

[u/uxnuxn]

C++ version of factorials generation using Moessner Miracle:

https://gist.github.com/uxn/1af19ead0fe5980fe631748bad1775d3

[u/columbus8myhw]

At 20:53, look at the ratio of each number with the one below it :)

[u/xThomas]

!RemindMe 2 months, watch this again

[u/kochede]

Is this logarithmic/exponential transformation by Pascal triangle between addition and multiplication - something well known and trivial? Or is it something new? Sounds interesting!

[u/EnergyIsQuantized]

I am stupid. What's the Moessner's summation method? When generating the cubes we're skipping every third number in the first row, then every second number in the second row. But in the factorial generation we are skipping numbers in the same order in every row.