One Of THE Great Arrow Sudokus

[Music] thank you hello and welcome to today's edition of cracking the cryptic where we're going to be taking a look at a puzzle that has been very very heavily recommended to us it's called The Vow of three worlds and I just realized I shouldn't have turned the webcam on before I actually understood what that means the vow of three worlds I don't know I don't know you'll have to let me know in the comments how this puzzle relates to the bow of three worlds or indeed what the vow of three worlds is but it's by playmaker 6174 one of the sort of stalwarts and dorians of the um of the Discord server a phenomenal phenomenal solver and Setter and this puzzle has been praised the nines by people like Vista mafel so I know it's going to be brilliant um and it's just a normal Arrow Sudoku today so if you if you have a go at this and enjoy it there are lots more like this in our Arrow Sudoku app which I certainly commend to you is full of handcrafted puzzles by the Great and the good of the stoku community um and definitely worth checking out I don't have any news for you today so I suppose just an appeal if you like the videos do click like um if you really like them click subscribe we'd appreciate it and um and that's all I've got to say so why don't we have a look at the rules of this puzzle they are as follows normal Sudoku rule Supply so that means we've got to put the digits one to nine once each in every row in every column add in every three by three box and then digits along an arrow sum up to the digit in that arrows Circle so that's telling us that I'm trying to see that there aren't very many long arrows I think that's the longest Arrow so those three digits there you sum them up so imagine that was a one two and three then you go one plus two plus three equals six so you write six into the circle so it's a very simple rule which leads to some absolutely beautiful geometric considerations often so do have a go the way to play is to click the link under the video what do I mean by that some of you actually asked that basically go to the video description just under this video and you'll find a link to this puzzle click the link and you'll be able to play the puzzle in exactly the same software that I'm playing in now um on whichever device takes your fantasy indeed but now I get to play Let's get cracking and while I'm wondering if these are the three worlds referred to in the title because this this thing well this box looks very suspicious with its sort of arrangement of it's very densely packed let's put it like that now I mean the simplest thing I can see about this Arrangement is to have a look at those six cells there now those six cells there have to contain different digits because they're all in the same box of the Sudoku and the minimum I could make those six digits add up to is 21 because one plus two plus three plus four plus five plus six is twenty one it's good to know your triangular numbers in Sudoku and that means that these three the scissorgy of planets here the uh maybe this is the three worlds these three worlds have to add up to at least 21. now that in and of itself doesn't feel like it's quite good enough because what I'm not going to do is put four five six seven eight nine into these cells because obviously they're they're each individually now of at least a four if I try and put three in there they'd have to be double nine as if in order to get these up to 21. so okay maybe I'll just put that one in there so that's got to be at least six because that must be at least one plus two plus three um hang on a minute okay so I'm just wondering I don't know I was about well okay okay here's something tiny but I'm gonna say it anyway nine in this box can't go on a two celero because if you put nine on an arrow you're going to have to put 10 at least into a circle so nine in the Box must be in one of those three cells which means that this cell is not a nine okay all right we'll let that that means let's do the same trick then in box one where does nine go well by Sudoku it's not in those squares and it can't be on a two cell Arrow or these would have to be ten at least so nine is in one of two places now in box one oh yeah yeah this is going okay all right so this might be how we're meant to start because now where does nine go in box four Again by Sudoku it's not in those cells can't go on an arrow so it's in one of those two positions can't go on a two cell Arrow oh look it can go there look that's just a one cell Arrow so if that was nine that would be nine that would be nine uh no that looks okay actually all right what about box seven Nine's in one of three positions oh no I'm wrong sorry sorry that's another one cell Arrow completely missing the point of these one cell arrows all that all that's basically here to do is to tell us those digits are the same as each other so that's not actually I think I think there's something else going on up here because this this is where all the action is isn't it I have a feeling I meant to under this is probably a 789 triple um and there's some way I've got to appreciate that if that was 789 that would be six that would be one two three that would be a four five pair and that would probably make that all quite sensible those three digits have to add up to an odd number and that is because of a secret that I only tell my favorite people now if you're watching this video if you're still with me after six minutes you'll definitely you're definitely one of my favorite people and let me tell you what the secret of Sudoku is the secret is that any complete box of a Sudoku indeed any complete row or any complete column must because of the rules of Sudoku contain the digits one to nine once each so if those digits are the digits one to nine once each we know what the sum of all those digits is and it's 45 and that's the secret now the second way of thinking about that secret is that 45 is odd but those cells there must sum to an even number because of the nature of an arrow and what do I mean by that well let's imagine that these two digits here sum up to X well we then know we've got to put X in the circle so we know that the sum of the arrow and the circle together could be expressed as 2x now 2x is an even number um hopefully that's obvious because any two times any number the result of that must be an even number because you can divide it by two and get an integer so if this is even and this is even those six cells are even 45 is odd so these three cells are odd now then we can say though ah right so when we learned something about this Domino here because if these are odd and they are and we've just said that an arrow plus its Circle adds to an even number then we can conclude that those seven cells are odd in total because we're adding an odd number to an even number now if their audit if these seven are odd and we've got to retain oddness in the row overall because the row adds up to an odd number 45. these must have even parity so those two digits sum up to an even number ah okay so that means that these digits all together sum up well we already knew they summed up to an even number oh no this is no I don't sorry I'm not I'm not understanding why that's important if this was you see this would be important if I could prove this was a 789 triple is then knowing they were even would force them to add up to 16. so there is there's something about this total here that we've got to think about ah got it right okay beautiful absolutely beautiful and in fact this line of thinking is important because right I've seen this trick before by the way what I'm about to show you here where does that digit go in the in in the yellow cells so by Sudoku let's just make that purple for a moment purple has to live somewhere in box two where does it live and the answer is I have not got a Scooby-Doo but I can say it is in yellow because it's not in those cells because of Sudoku well therefore we can we can think about what these two cells sum to in a different way because purple can't go on its own Arrow not that that's really important in what I'm about to tell you but it can't because if purple did go there this would have to be a zero so purple is in one of these cells let's just pick one let's just pick that one so that's purple but now if we think about the nature of these three arrows and what they sum to we know they sum to these six cells but we know that this one and that digit are the same number so these two sum up to what the yellow digits sum up to and the yellow digits are all different and the Triangular number for 5 is 15 and that means these cells sum up to at least 15 but they don't sum up to 15 because 15 is odd there is a knowledge bomb for you so they add up to 16. so I haven't proved that this I haven't proved this is a 789 triple but I have now proved that because these two digits have to add up to at least 15. they have to add up to 16 because they have to add up to an even number and they can't add up to 18 because that would require them to be double nine so that is so this is not seven or nine that's really clever actually isn't it that's beautiful setting and ah now I better just remove the purple there as well because I don't know where purple go I know purple's in one of those cells but I don't know exactly where it goes so there's an okay so now we know there's the nine is definitely in this Domino this digit is at least a five now because remember these cells added up to 21 at least because there are six different numbers well we've only got 16 here so that digit's got to be 5 6 or 8. um which is probably in some way important so if that hang on I'm just trying to work out what this means [Music] um um how do I how am I meant to this then if that if this is eight I've got a 16 32 these have to add up to 13 in order to make the row add up to um 45 overall 13 here she looks very easy if this is double six I've got 28 here these have to add up to 17. that's also looking fairly straightforward isn't it so hmm okay okay so the all right let's try to do some stuff on this box again though those are up to 16. so those add up to 16. that means together they add up to 32 which means these cells add up to 13. which is oh so there's a parity thing going on oh that's strange ah okay but this is very important isn't it so because these three digits here let me let me make those a different color for a moment let's make them blue these digits add up to an odd number but this is even so these two digits sum up to an odd number in order to make sure that blue overall is odd and there's only one option so that's five I think that was probably a better way of seeing that but I couldn't I didn't appreciate what it was and now this is huge as those digits add up to 21 so I know what yellow is because that must be one two three four five six mark would be very proud of my pencil marking there that is an eight it's an eight arrow without a five on it this is one three four those squares are two and something two and six I want to say um now I'm gonna get rid of my one two three four five six because that seems excessive um now that's got to uh that's got to be one two three or four to add up to five so now we can do some maths on this box can't we because we've got 30. so now these are a seven nine how there was okay there's some way I was meant to appreciate this instantly and I didn't at all but I can now tell you that's a seven nine pair it's really beautiful okay so so once we work out these squares at up to 13 then by maths we know those squares add up to 45 minus 13 which is 32. but we know that basically these circles must add up to half of 32 because of the algebra of the arrows half of 32 is 16 so if these have to add up to 16 there is only one way of doing that with two different Sudoku numbers so that's a seven nine pair which means that the seven arrow is not 2 5 and it's not 1 6 so there is a 3 4 pair in one of these and a 1 8 pair adding up to nine so this is from one three four and eight now do we know the order um not not yet I don't think um okay and what's this doing to this box here answer is I don't know although it looks like we've got very similar geometry here haven't we that digit is in green right so that's eight I think let me let me run by you who am I thinking here I'm not sure if this is right but I'm going to say it out loud and then we can we can debate it together so that digit lives somewhere within the boundary of green because it's definitely not in those white squares so whatever this is is in green somewhere and it's not on its own Arrow so let's just make it there now that means that this Arrow well too lot this whatever this number is it's basically it can be expressed as two times because I'm not explaining this very well but the these match off obviously if we're thinking about what is the what is the sum of those and now I'm really not explaining this well I can see that these green digits are double what this is because this has two arrows going into green plus this that is a way of expressing green two times this plus this but we've just said that this is in green here so these cancel out which means that the green the cells remaining are two times this number now the minimum I can make five different digits is one two three four and five which is fifteen fifteen divided by two is seven and a half which I cannot put into a Sudoku cell so I'm looking for a number that is bigger than seven and a half and is not nine so that is eight and now uh well okay so then we know that the five digits that are adding up to 16 double eight are one two three four and six that's the only way of making five digits add up to add up to 16. I don't know which one is orange that's that's the thing that I've got to be careful about here so in within the panoply of green we've we can describe the panoply of green as one two three four six plus Orange um one two three one two three four six plus Orange okay but can I say some I think Orange make a similar logic to up here Orange has to be at least five I want to say because the minimum I can make green is 21 triangular number for six I know I've got 16 here from the double eight so this is at least five and it's not nine or eight so it's either five six or seven here but if it's six haven't I got two six ah yeah okay yes it's not six because if it's six I've got to put two sixes into this because I've got to put 16 in five digits plus six and the only way of doing the 16 plus the fact that in five digits is with a six as well so that won't work so this is odd uh this is right okay so maybe this is some sort of parity trick again if this is five this is one two three four five six and that is seven nine foreign then these digits are one two three four six seven one two three one two three four six seven so these are five nine right ah right so this is a five seven nine triple and that's not five by Sudoku and oh for goodness sake right I've just seen something else what's that upward strut of the eight arrow going to be it's beautiful again it's not one seven and it's not two six that's a three five pair and that does it that's done it now now because there's a five up here that can't be five so this is seven nine and that's five which oh so orange was over here there the four and these squares are one two four and six and the four oh this is it it's just just beautiful it really is and it's like one beautiful deduction after another because now I can't put four on an eight arrow because then it would have to be double four so there's no four on the a Tarot which means there is a four on the five arrow which means the five arrow is one four it means that eight arrow is two six foreign and okay pregnant pause when I try and work out what that means this means that digits under pressure so that digit is a maximum of six and if it's not six it's four and it's not four I can tell you because if it's four indeed or if it's anything lower than 4 the puzzle's broken because you can't put the same digits on those struts wait well actually you could if though if they were allowed in there but by Sudoku if those are the same digits where do you put those those digits in this row well they've got to go down here but if this is four this is a 1 3 pair and one and three are not there if this is three they're a one two pair and that you can't put that there either so in fact that is six but that's still interesting because now that version of six is different from this version of six running out of good colors here um actually maybe I'll recycle my yellow so there we go and we can see that these are different because if they were the same we'd have to have two digits down there that sum to six which we clearly don't have so these digits oh in fact look they say oh this is it that's right so that I've got a 5 here so that's got to be 2 4. that's got to be one five now I can see that one is in this Domino by Sudoku where there's one going row six is the way to see that there's two is in this Domino where does two go in row six foreign this is one something one three now so the place I'm thinking I've got to look now which is probably about to be proved totally wrong is this digit because that immediately gets reflected there and at this digit can't be one two four five or six so it's three seven eight or nine which means that is three oh okay that's interesting a little it can't be three so this is a big big digit it's seven eight or nine so it it five whatever this is finds a friend up here and goes in one of those three cells but I think also all the digits seven eight and nine are available in those three cells so we don't learn anything more the other the corollary of that is that these two cells have to go in one of those cells so I wonder if we've got some sort of um that's not a six Arrow not that anyone was probably thinking about it being a six hour but I've just seen it's not a six Arrow because that's a two four and that's a one five so there's no way of making it though that Domino add up to six but unfortunately I think it could still be something like a three if it was a three no actually it can't be a three because then I would have too many low digits in this column same is true of four if this is four I've got um five digit in fact I've got more than that I've got six digits in the comma column actually that can't be five can it ah this is actually important I think so I I flippantly said this couldn't be six but it because we've got three digits in the column already that are five or lower how could this be five even it can't be because then there would be six digits in the column that was selected from five or lower that's that necessitates a repeated digit so this not being able to be five or lower or six itself means that is massive as well it's a seven eight or a nine so so hang on come on brain what does that mean well okay it means that digit see this digit here whatever this is is not that digit because at that digit gets reflected into this one's column so this digit goes over here and is therefore in one of those two cells um let's let's recycle orange to indicate that so Orange is in those cells are orange is over here oh dear okay so I was I was anticipating a bit more a bit more a bit more resolution from this but I'm I have definitely not got it so what is it I'm missing here um I have got a actually maybe it's this column because I have got five definite digits there so this column needs one one two four six I think into those squares it doesn't seem terribly appetizing to me that's two six so these have definitely got one and four in them which okay which might matter I have no idea why it's me okay what am I missing here so five five there is a five in one of those cells um four here that that this arrow is adding up to it well I suppose it can't be five that Arrow there because it can't have one on it it's a minimum of two and three so that's at least five but it can't be five so this digit is relatively high it's at least six but I can't see I suppose oh no okay it's not six is it because I've got I've got because I've got one and four in these cells if I try and make this six I can't use one five and I can't use 2 4 on the Arrow so that's not six so this is set it might be the same as this one this is seven eight or nine now and whatever this is has to go in one of two places in box in box five um well the other thing about this Stitch actually let me just highlight this digit for a moment I'm going to recycle purple making double use of lots of colors today but if this is whatever this is it can't go in those cells and it's too big to go in those and it can't go on its own Arrow without being accompanied by a zero so this digit goes in one of two positions in box seven so let's label that up in case that's important now how do we make this work for us foreign there must be a way what that way is is a mystery garlic golly gosh um okay sorry I've got nothing here I'm just I feel like I've totally hit a brick wall can I can I do I don't know where I don't even know where to look actually I can see I've seemed to have lots of low digits in rows four and five I'm wondering if I can do something with that don't think I've quite got have I got no I don't think I have got a s uh um I haven't got a quintuple but I very nearly have in both of these rows actually very nearly having both of those rows what what is it I'm meant to appreciate here this is really strange is this seven nine what did I say about this seven nine one of them was one of them is three four whichever one of these is seven sees three four on its Arrow and whichever one is nine sees one eight on its arrow and that's going to pair up with this eight so whichever one of these is one eight is gonna have a friend here but I don't see a problem there um okay I've got a 2-6 pair in this column is that somehow important doubt it two six pair here is there some sort of digit that's forced down here that's going to be more more relevant than I think it's going to be or indeed in this column one twos threes fours fives sixes sevens no I think everything's catered For there and therefore everything's going to be catered For All there as well because there's the same options basically but there's two six pair is that somehow relevant um so that okay that Arrow cell there is not one or two so it's not three is not three then it wouldn't be able to be four so it would have to be at least five if it was five that digit would be at least two so this would be at least seven which is what it is so that's absolutely of no surprise to man or beast okay so I'm now completely and utterly stuck now okay so okay let's think about other things let's try this box I've got one two three four and five in it so these squares are six seven eight and nine that's not eight whatever that digit is has to be in one of those two squares oh oh ah okay I've got a six seven eight nine quadruple apparently in in box or what column four which I've not appreciated was on the cards so one of those is six which means neither of these is six so this has come right down now this is seven or nine only come on oh wow okay let's keep going on this column because now I've actually got five six seven eight nine approximately placed in this column which means these are really low digits and that one is getting reflected there and it can't be one or four so this is two or three oh for goodness sake oh that is so pretty oh my goodness me this is so pretty because look that two or three rules out two or three on that Arrow that's that's absolutely beautiful and that bounces back down there therefore I make this a two so now I've got a 2 3 pair here oh I've got it well I've got something I've got something really cool here that I think there's quite a lot going on here but one thing I've noticed straight away I'm going to color to show you this um I probably can get rid of my coloring on my six Arrow now because I just wanted to record these were different and I've used that that logic already but watch this so that digit I'll make red that digit is different from Red obviously so this is red but look at this cell now that cell sees both yellow and red so that cell is no longer able to be three which means it's at least five which I don't know maybe that doesn't do as much as I was thinking well it's got to be five six no it's not six I've got two and six in the columns it's five seven or eight it can't be nine because it's on an arrow so that is actually really restricted five seven or eight now that digit right okay so that digit is yellow because what would happen if this was not yellow if this was not yellow then it can't be red and it can't be one or four so it's at least five and five plus five is ten which is greater than anything we can put in that Circle so the only way this exists is if it's yellow and that gives me a 2 3 pair in this car now that gives me a 2 here because now those can't be two which means this is one four and six which means that is two that means that is six which means that is two that is six and this is two three which means these are not three now if these are not three I've got a one four eight triple here but let me just think about this three three was on the seven Arrow so if you can't put three here you can't put four here and I've got one four and eight here so this is a seven or a nine I think which doesn't prevent it from being purple so okay this is not eight anymore because if that was eight it would need to add a 1 to it to make nine and that can't be eight then so if this is seven this is two five if this is eight 3 5 is 9 it's 7 2. bobbins uh eight in this column is up here so that means that yeah this is this matters okay so if we look at that triple one four eight the eight is at the top now on the arrows here the eight goes with the one so now I think I have to a limit because the eight is in this Domino the one can't also be within this Domino or I can't make the nine Arrow work so that becomes a 4 8 pair which means that is a one that's very strange and pretty so now these are not eight so this is a one three pair this is a 4 8 pair oh look that's a one four pair so that four gives me that digit so that's a four that's a two that's a one that's a five which means that's a five that's a three good grief that is a three by Sudoku that's an orange ah look that could have been that but now it can't be so we now know that is orange which means this is seven eight or nine oh boy that's no it's actually not really we we already would have been able to work that out that digit is seven eight or nine so this I see so the six in this row is now identified identifiably there why couldn't that be a six because for some reason I think the 6-0 because the 6 in this column okay well that seems fair so that's a six and that's not a one anymore that means that's not a three bikers of this three oh I don't know I don't know I'm loving this though this is so interesting every little every little deduction leads to some little uh you know just a a wand wafting of magic okay come on what digit have I not put in this box or pencil mod I've not pencil marked an 8 in this box you know so the eight is in one of those two cells now be lovely to know whether it was this digit that's eight um but how do we do that something's got to disambiguate all of these all of these arrows as well that's going to have to be something over there I think foreign I've got a 2 here and a 2 here so there's a 2 on that Arrow that's great that's great because this 2 here can't go with nine on the Arrow because that would lead to a 2 7 Arrow I mean that's got no nothing no way of being made so that's got to be seven that's got to be a two five arrow which means that this is nine hopefully there'll be a way left to make nine here three and six make nine so that's a three six Arrow which means this is a three and this is a one so now I've got well that's not seven and this being nine means those two are not nine so this is now seven or eight so if that's nine that would determine that this was seven or eight oh I tell you what that's not seven anymore because seven went with two on the Arrow to make nine that was the only way of getting to nine so so yeah this this cell is now five always because we're adding at least two to a seven and therefore that once you take nine out of here that's not gonna work so that's five doesn't seem to do anything but it does mean does it mean anything I really don't that's so mean isn't it I mean that's just rotten right these digits are seven eight nine I think I think that was already known it's literally got nothing knowing this is five unless there's something it's doing over there that I'm not appreciating it doesn't look like there is but we might have to do some more coloring you know I can get rid of my coloring of green that's clear green has clearly been well thought about and I was just thinking about sevens and eights actually um although looking at it that doesn't well the this was the seven eight pair I saw but then I can't seem to extend that anywhere else so maybe that's not what we're meant to do let's check I think we've got to check the bottom of these columns because we have effectively placed five digits in both of these columns so I'm going to check this one first I need four five eight and nine down here and that can't be five and these two seem to be replete with anything they want to be so there's definitely a four and a five down here because this is only eight or nine that's not five again that's not nine you can't put nine ah hang on that's not nine because you can't put nine on an arrow that's two cells large can that be eight only if only if this is nine we'd have to go eight one nine now why does that not work I bet you that doesn't work for some reason eight one nine that will make this nine and this nine and this nine don't know maybe it does work oh all right let's try this column um one six seven nine one six seven nine now we can't put nine there still um in fact ah this is this this is just a one isn't it because if it was six or seven even if we had four to it we got a double digit number so that's just a one okay that's weird that gives me remarkably little but it does mean though that ah okay no it gives me the six in the column that can't be six because it would reflect here and Clash so that's not six which means this is the only place six can go in the column which means that's not six and this digit is five or nine only and I think it can still be both really are you really is it really true I don't know I think it is I think it really is true that's but this is okay this is seven or nine so this is seven or nine now ah so look it's really close to being a sort of triple in box nine or indeed in column seven ah I've got a seven nine pair here so that's an eight that's not the digit I wanted that to be to get this to be a triple but that gives me oh this is known this is nice though actually because that 8 gives me a four here which makes this a five which doesn't resolve this digit but it does mean that's a five by Sudoku this is not able to be for so I've got I've got this 789 triple the eight of which is at the bottom so that can't be an eight which means this is the eight in box number seven good grief five seven nine pair this knocks out four from that that digit so that uh we're still this is still not resolved I don't think so I've got seven nine patterns all over the place I've got I've got a whole variety of things going on these are not eights so that's not eight anymore so I've now got a seven nine pair here which means this is a that's good that's I can see that's useful because that means that's nine that's seven that's nine that's nine by repetition that's seven that's eight but more importantly that eight means that's seven which makes this two this two this three this three by coloring three in this row gets placed which means three in the top row gets placed okay has this done it this might have done it that's seven yes that seven gives me nine nine eight which gives me seven here nine here nine here seven here now that's one or six and can't be six so that's the six that's the four that's the one that's the one by Sudoku that's the one and the four these two squares are two and seven no points um is it this nine yeah look that Nine's gonna do some work so that 9 gives me nine seven two five uh okay we might have to do a little bit more in order to figure this out this eight arrow is not one seven or three five so that's two six and we know the order which gives me the two four here this squares are five no okay it must be these two squares then so these two squares are six and eight oh no it's fine it's fine I've realized what I've missed once I know this is nine I know this is one and eight and that's going to that's going to sort out the rest of this row so one eight three four four one three six six eight few and then these two squares are four and seven which I can do what a brilliant puzzle start to finish that is just it's just exceptional setting the vow of three worlds I still haven't got a clue what that means but every little deduction is it's well they were surprising which is really clever and interesting which is really clever and just as a whole it's just a sequence of really really nice deductions very very smart setting this one so playmaker 6174 I don't know how many Powers you've already taken but you need to take another one my friend uh loved it absolutely loved it and we'll be back later with another edition of cracking the cryptic foreign [Music]

Channel: Cracking The Cryptic

Views: 47,600

Keywords:

Id: PvDpp6w35_Q

Length: 49min 47sec (2987 seconds)

Published: Sun Mar 26 2023