A Breakthrough In Sudoku Technique

[Music] hello and welcome to sunday's edition of cracking the cryptic where we've got a classic sudoku for you today and a very unusual way that we're going to be presenting this one because i've been advised to start this video by putting this puzzle into a computer solver and then showing you that the computer solver can't solve it without using basically bifurcation and then my challenge is going to be to solve it obviously using some sort of logical technique which sounds incredibly daunting um and i have no clue how i'm going to do this but we shall see and but don't worry if i if i do mess it up i have a couple of other puzzles on the back burner uh which i will i will try and record instead so you may never be seeing this video um and it's called steering wheel the puzzle by the way by sudoku explorer who's publishing puzzles on logic masters germany i know for the last few months uh often to absolutely rave reviews so this should be a very interesting challenge i'm certainly looking forward to trying it um but before we kick off a couple of things to mention some of you have been asking when i'm going to be live streaming my attempts to solve a puzzle game called the witness again um well my current plan is tuesday night it does look like there is a window for tuesday night so probably around 10 o'clock in the evening uk time uh that's when i'm going to try and do that so uh keep an eye out if you're interested in that sort of thing and the other thing we wanted to mention is that we aren't that far away now from the extraordinary total of 400 000 subscribers on youtube these are numbers that we could not have dreamt about um but we were trying to think about what we should do if we get to 400 000 by way of a thank you um and mark suggested we might do a video where you guys can ask us questions and then we'll answer them you can ask us anything you like um within reason um and um uh yeah so if you do want to ask us a question send us an email to crackingthecryptic gmail.com and um if you aren't subscribed and you do enjoy the channel please consider subscribing and trying to get us up not just up towards that total it will make us very very happy if we ever get there um so yeah that's just something to mention now with all that said let's get on to steering wheel um i guess this does actually resemble a sort of wheel doesn't it um by sudoku explorer and i'll read you the rules here we go normal sudoku rules apply there i've done it um so nothing more complicated than that now i put it into a solver i put it into andrew stewart's solver which is basically the only solver i know about online so apologies if there are other ones i should be using um and you can see down the right hand side it's basically got a whole load of very uh well progressively more complicated techniques many of which i have no clue what they are 3d medusa for example doesn't ring any bells with me at all exact i do know from a shy puzzle actually that i covered on the channel when i say no i sort of i sort of backed into it um i do know what empty rectangles are but these things down here these are the things i've been told to turn off digit forcing change initial forcing chains self-forcing chains basically forcing chains are what i consider to be bifurcation or good lyfisms so we've turned off good luffisms on the basis that i don't like to solve puzzles using gudlifisms and i'm going to click grader okay so this is telling us that this puzzle is very hard i don't know what that means to be honest i'm not surprised it's very hard but it yeah so it's very hard that's what we know and now i'm going to try and make the computer solve it um so let's try and do this i don't i think i have to click the next step button um repeatedly until it gets stuck it's not getting stuck at the moment so okay still not got stuck still doing things oh it's doing some more things come on get stuck whoa whoa whoa okay well it looks like it's very nearly stuck it's gone all the way to the bottom something called a pattern overlay method i've got no idea what that is um [Music] right and now it is stuck okay so there we go i think that the point of this is just to show you that even with an absolute multitude of techniques that you probably only spot if you're a computer you can't solve the puzzle without guessing except that's what i'm being asked to do which seems a bit mean but anyway let's let's have a look at this in some ways actually it's interesting to me because i can almost um disregard some of the things i would normally do when i solve um when i solve classic sudokus so let's just have a look around the grid and see what we can do twos have got to be in one of those squares i've only got two twos in the grid threes have got okay so the this is the wheel this is the steering wheel straight away that's really cool actually look at the uh disposition of ones they lock a one into those twos are locked into those threes are locked into those and fours are locked into those now to me and i admit i am a bit strange that sort of feels like it's completing the steering wheel so i have no idea what that means in terms of solving but it did seem quite interesting fives uh no the computer did get a digit i should have made note of what that digit was um oh dear dear dear sorry about this i am not uh sevens look we can put a seven in one of those squares seven in one of these squares seven in one of these squares ah there we go there we go we've got the digit the computer got sevens sevens in this square here that's got to be a seven by sudoku now seven has got to be in one of those two squares eights we have got three eights eight maybe the most sort of profligate digit in the grid eights have got to be in one of these two squares eights have got to be in one of those two squares eights have got to be in one of these two squares eights have got to be in one of those two squares ah eights have got to be in one of these two squares as well oh sorry no i thought that might be more helpful but no i've only got two nines in the grid basically we've only got two of an awful lot of numbers nines have to be locked into one of these two squares so okay now ordinarily if i was solving a puzzle and i did this and got to this point what i would start to do i was about to say i'd be looking for weak rows and columns i rows and columns and boxes where there are digits that are you know this square for example that looks mildly weak to me and i say mildly weak because although i see that this this this cell here sees four different numbers it sees the seven and the nine in the box the eight and five in the row and the two and four in the column so it has to be a one three or a six it's not it's it's only relatively weak it's pretty pathetic it's not really restricted is it and this is what i would do normally in a puzzle is i would try and find weak weak cells on the basis of boxes rows and columns but we know because the computer failed the computer would pick up you know anything like this instantly so i sort of feel like this is a fool's errand um and indeed that my second port of call on hard hard classics would then be to try and find digits that are that can only appear twice in a row column or well really in a row or a column i'm now trying to see an example of that just to see if i can find a digit that can only appear uh i'm not doing very well the problem is there are so few digits it's actually or so few instances of multiple digits that it's not that surprising but i can't find very good examples here um no i still haven't got one sorry about this i wondered whether i could do something with the one and the three here because there's a one and a three in these two squares so if we think about where one and three go in row five it's either here here or here so that's no use it's nearly useful but not quite um wow wow so yeah i mean the reason for looking for things that can only appear twice in a row column or box is that that's that's the basis for so many of the basic techniques like um x-wings empty rectangles um skyscrapers all of which you know that they're patterns i'm relatively familiar with and can sometimes handle in difficult puzzles but here we know none of that none of that is there because the computer would have found that so i've got to be looking for something the computer doesn't understand which basically means geometry doesn't it so there is going to be some geometric trick we can perform on this puzzle that computers really haven't yet been programmed to understand um [Music] probably to do with the wheel somehow let me think about the wheel what is this wheel of ones twos threes and fours meant to be telling us um i don't know i'm afraid i i suspected this is what might happen um because i think this is going to be you know it's almost a game of guess guess the geometric trick so the standard the most basic geometric trick of course is the fistomaphel ring which is which tells us that these cells the purple cells are equal to the green cells which is an extraordinary thing if you've never seen it before but it is true but that's ah no no no okay i have got i think i do see what this is i do see what this is that is that is that is actually reasonable to be very fair that is reasonable you can see this because if you are familiar with geometry tricks if you try you'll look at the phistomophel ring as i did then now what's the what's the basic expansion of the vista mobile ring if we stretch the fistumaphel ring where do we go next with that well it there's a technique that basically looks at these squares and if this is if this is relevant i will explain it so worry not there is an equivalence between these squares now that does look interesting because i can see all these digits are high and all these digits are apart from that one are low three one two three four five six seven eight one yeah that's beautiful that is actually beautiful okay fair enough fair enough so this is definitely at least it's definitely worth explaining so i will now explain that right i realize i'm being completely ineloquent and i apologize for that so let me try and correct my eloquence and bring bring this to some sort of into some sort of reasonable order the purple squares whatever digits are in the purple squares are identical to whatever digits are in the green squares and if you've not seen that before that may sound like complete magic and it is in a way it's very but it's very understandable once you've seen it a couple of times so let me try and explain why this is the case we'll highlight those squares now these squares clearly if we look at box one that's definitely if we correctly complete box one it will contain all of the digits from one to nine once each so this is one set of the digits one to nine this is the second set of the digits one to nine this is a third set of the digits one to nine and this is a fourth set of the digits one to nine so the purple squares contain four sets of the digits one to nine in some order now let us um consider what that column contains that complete column well obviously that's a complete set of the digits one to nine it's just one set of the digits one to nine so the green cells i've highlighted are one set of the digits one to nine now now i've highlighted two sets of the digits one to nine in green now if i highlight row one and row nine as well i've now highlighted four sets of the digits one to nine in green if i count the corner green cells twice why do i count them twice well it's because that this cell here is definitely in column one and it's definitely in row one so the green cells if i count the corner cells twice are exactly equal to four sets of the digits one to nine which if we remember back is exactly what we said the purple cells were equal to so what i can do is i can remove cells that have two colors like this one if i remove this cell from purple and i remove it from green it's still obviously true to say the purple and the green cells are the same sets of digits and i can put i can do that for all the cells that are let's i'll leave the corners to last because we have to think more carefully about the corners all of those squares can come out of both purple and green now if we think about the corners if i just cancelled out these corners and didn't do anything else that wouldn't be quite correct because we have to think of the corners effectively as double green so i can remove i can remove i can remove them a purple and a green from the corners but then i have to reinstate the green that was sort of underneath them because the greens counted for twice and we end up with this situation where we should find we've got 16 purple cells and we do and we should have 16 green cells there's 10 13 16 and we do and we know that the purple and the green contain the same set of digits and this is the geometry trick that i think is interesting because if we look at the purples we have got eight high digits in purple therefore i've got to put those eight high digits in purple in green or where can they go i've got eight low digits in green so that that all of the rest of the cells in green that are not the low digits must be high digits in order for the geometry of this sudoku grid to work so those squares there are six seven eight and nine in some order i don't know where they go yet apart from we know this one by the rules of basic sudoku and of course all of these squares in purple it works exactly the same way in reverse we know that there are low digits in green that have to therefore appear in purple and the only place all these low digits can go is in these little two by twos so they are one two three four in some order and now this is what i'm hoping the computer didn't know and does it matter that we know this let's try and tidy up our pencil marks and see if we can spot what this means um six and eight here so i can remove six and eight from this one seven and nine here so i can remove nine from this one six and eight here that's not doing anything nine and seven here so this is not nine um now the other thing we might want to think about here is five because five as a result of all this uh jiggery pokery we've basically removed five as a possibility from every single colored cell in the grid so here yeah okay so now we can ask where five goes in box three it's got to be in one of those two squares now this is going to go around the grid now where does five go in box one it's got to go in one of those squares where does five go in books seven oh no hang on no where does five go in box seven it goes there by sudoku that is that now this digit is a hard one beautiful digit because this digit we couldn't get before i see yeah the simpler way of thinking about this digit look is look at this five once this square can't be a five you've got to put a five in one of those three squares and now 5 by sudoku is here and we didn't know that we couldn't have got that without being able to eliminate 5 from this cell and the only reason we could rule 5 out from this cell is because of the mad geometry trick so i'm now feeling like i might be able to solve this puzzle if this keeps going so we've now got um [Music] now we've got what have we got here have we got anything very helpful right i tell you one thing that's strange and that's these ones twos threes and fours in the corner in the corner boxes don't actually see any other ones twos threes and fours because we never get we never got given any ones twos threes and fours in any any of boxes one three seven and nine uh oh goodness man i'm so sorry i'm not i'm just not quite spotting i have a horrible feeling there are going to be sort of three or four just digits we can write into the grid here that can't be a seven and i'm just not seeing where they are oh goodness me okay uh maybe maybe we do what i don't know ah yeah ah right no let's come back to my 1-3 pair because i've now got a one-three pair which i didn't have before this one and three here we had to put in row five and they could have gone in that square that square and that square well they can't go here anymore so that that is now a one-three pair so this one three pair here is actually it yields a hidden pair in row five now this must be important come on let's get let's get something from this so i've i'm now wondering whether i've got sort of low high coloring to do in column three i just need one more cell that can only be one two that can only be i know the problem is this could be a high did can that be a could only be seven if it's if it's a high digit but it if it's a low digit it could be three or four so i think that's three four or seven i might be wrong about that but i think it sees six eight nine it sees one two in the row and it sees five but oh hang on now i've got a seven pencil marked into those squares oh yeah which is fair enough so that can't be seven right let's check this digit again one and two are ruled out three and four we're saying is possible five is not possible six is not possible seven is not possible eight and nine not possible this is three or four right so now i've got a quad yes i have got a quadruple in column three on the digits one two three and four so those two squares oh that would have been a simpler way of seeing this this is a six seven pair yes ah i see so this six here what we could have said if we'd been sensible is that there's a six in one of those three squares and therefore where do we put a six in box um in box four it's got to be in one of those two squares which where it joins its friend the seven to make a pair now that means that square is not a six so six seven eight nine um so these are ones threes eights and nines so this square is a three or a nine i think i don't see what this means oh good grief um so maybe maybe maybe we're wrong maybe the computer is still going to win the battle here because i feel sure that what we did this is an amazing it's an amazing thing and i can't remember see i'm sure i must have seen geometry applied in a classic before like this but it really is very beautiful but what i'm not sure about is what to do with this how do we force how do we force more deductions so i've got to put yeah okay that's that's something i hadn't really thought about if you look at the purple squares i've got two sixes i've got two sevens i've got two eights and i've got two nines so i've got two of every high digit that i have to put in the greens so are there any are any of these digits difficult so for example i've got to put two sixes and two eights is there some way of saying they have to be different digits then or maybe they have to be the same digits don't think so um sixes oh good grief i don't know i've got a bad feeling i'm not appreciating some some little facet of this that would help me to solve it it's one three pair is bugging me as well maybe this row i've got to put two four six eight nine so that square there is a two four or a 6 2 4 8 9 here that can't be an eight look this has got to be just six or nine i could have um there's a y wing that would have allowed me to remove a nine from this square but that you can see is completely useless this six or eight here if that's a six this square's a nine if it's an eight this squares are nine so these two squares we know at least one of them is a nine and they're both looking at that square nine eight ah right okay that's interesting i've got something okay that is weird nine eight six yeah that is weird okay well that square cannot be a nine ah that's going to give me a nine here right look at this square so the question i was asking i was thinking about nines basically uh off the back of this sort of well this failed y wing pointing at this square now this square can't be a nine because if it's a 9 that square has to be a 9 as well and that means we couldn't put a 9 in this box now how do we see that well we see it just by unwinding the y wing so if this is a 9 that's 8 that's six and that's nine that's just what we were looking at with this square so you but you can't put two nines in those two squares because if you do look where our nines are pencil marked in box nine there would be nowhere for a nine so that's not nine and that's interesting because now i get a 6 8 pair here which is beautiful but more beautiful than that even is that now the 9 in this column has to be in one of those squares and that means this this must be a 9 and we have another digit so this is probably the second hardest one digit in christendom um [Music] and that means that's a nine just by sudoku which means that's not a nine now we have to hope that that is the end of the jiggery pokery and we can we can we can solve it from here so now i can see those ah this is interesting those two squares here are a 2 4 pair look they've got to be two low digits and they they are two and four which means those squares are not two and four so now we know what these three squares are these are a five six eight triple and that is not six and this is not eight five six eight and that gives me a seven here by sudoku which gives me a nine here now that's a seven if i trust my pencil marks and i do um this is an eight oh come on this eight is giving me a six over here which is giving me an eight five six up here that's not six anymore plea yeah now now where does eight go there are four eights looking into this box and whenever you get four of the same digit looking into a box you can fill in that digit we must have almost done all the eights well yes look we can put six in here and that means that square's an eight and now oh no we can't do all the eights look we're going to get left with a little x swing of eights there rats um [Music] okay well let's right these two squares now are a 3 4 pair which means those two squares are a 1 2 pair and that means we know these two squares have to have a 7 in them so that's a seven that's a five now we know these two squares i suppose these two squares have got to be oh this is so strange it's really beautiful so we keep getting low pairs going around the steering wheel as we turn that way and that's it's not resolving but it's it's affecting all of the low digits that go in the purples look so now those two squares are a one two pair which means those two squares are a three four pair and um nine now that's interesting where does nine go in box two this goes in those squares so that's an eight nine pair three four five and six to place your rotten thing that's a three or a four these ones are three four five or six i think that's not three that's not four i've just got a quadruple there that's nothing it's not helpful um [Music] what about what about i don't know what about sevens if we do oh hang on i've got two sevens here that's not gonna work is it how did i do that that's bad surely i'm meant to type in five there let's just delete that yes okay i will give myself a bit of a break there i was just looking at that triple thinking it had to be five six eight and where was the five in it well it should have been here i misclicked it but i didn't use this digit for anything so i'm fair i think we're still good to go sevens no set so the sevens are in an x swing in those squares the eights are in the next swing and those squares what about nines so the nines have to be in one of these two cells and we don't actually know very much about nines in box five sixes six has to be in one of these two cells this is not giving up two two four and nine into this oh that's no use two four or nine this square ah you rotten thing bobbins this is one three or nine that's only okay that can only be one or nine because it's c3 so look at it's almost like this this column has been designed to be difficult if we look at the digits we've not placed the one can go in two positions the two can go in two positions the three can go in two positions the four can go in two positions and you've guessed it the nine can go in two positions everything can go in two positions okay okay one um right okay there is something maybe one two uh maybe may hang on a minute maybe not um that's three that's my fault wow okay that's that's horrible that is horribly tricky um [Music] so is it likely i'm going to find anything better than this this is the question i'm now wrestling with about whether i'm even prepared to show you what i've just found there because that is very close to chaining in my opinion um is there another way of thinking about this so i found a way i think of making this square have to be a one um [Music] and the way i got that was noting that if this square is a three you get a one here and if this square is a four you get a one here so one of these two squares at least has to be a one and therefore this has to be a one but it's a bit indirect because it's very easy to see if this is a three this is a one it's less easy to see if this is a four this is a one because you have to go via the medium of the two there to get me the one here so it's a it's a step removed um it's a sort of if this then that type analysis uh and i got that by i was trying to find bent triples um or y wings in the grid which this most certainly isn't um [Music] but it's sort of an indirect y-wing it probably has a name but i don't know what the name is it's an indirect y-wing it's like a finned y-wing by virtue of that square i think i'd well there may be something easier and everybody in the comments will no doubt point out what i've what i've missed here but i think the point is that one of these two squares has to be a one and there may be a better way of showing that but that means this square can't be a one so that's going to have to be a three and let's see if that does damage if i fear it no i was about to say i fear it might not but actually i think it might be okay because look this three yeah this three is actually doing a lot of work in box four but it's also result revolving or resolving i should say some stuff up there and that's going to go around the steering wheel aha there we go three one one two and more to the point that's a nine and that's a one and now this is a nine which means that can't be a nine it's the only place for nine in the box i think now there's a nine in one of these two squares this two i think the low digits are almost going to be filled in in the puzzle aren't they they are this is so the whole perimeter is now being done um [Music] okay but so what are those digits we can place one look in this box and these two digits have got to be two and six and we can do that using the two at the top which means those squares have to be five and eight in some order okay and there's an eight here so that's not too tricky we can do the eight we can do the five we can do the eight at the top we can do the nine we can get the nine in box five and with our nines i think are now fully done which is a relief um now this five is removing itself from this square so this square has got to be three four or six and it's these three and four beautiful so that's a six that gives me a three four pair here which means that square is going to be a five that square is going to be a six it should only be i was about to say something that's not right is it so this is two four and five into those squares and that's not four and that's not two but the five could go in either place okay but that square therefore has got to be a three and something's either wrong or i've not finished off this cannot be correct what am i missing how is my scanning gone this badly wrong i'm going crazy that was a six here good grief simon get a grip now this square's a two or a five what and this four out okay that's why so that's two that's five that's two that's four that's five and this is four and this is three and this is four and i think yeah we have done it wow my goodness me that was a very very difficult um and i feel what do i feel i feel it's it's a it's a beautiful puzzle but it's devilishly tricky um and i'm not sure i feel like i cheated a bit to be honest because i knew at the start i had to look for geometry and i wouldn't have looked for geometry until i'd spent an awful long time on the puzzle so from that perspective i was lucky and once you have to look for geometry i think this is very very findable to be fair because it's it's a relatively understood extension of the beautiful original fistum of l ring and it's something we've seen on the channel relatively often but after that i think i found it very hard to understand i got a digit down here didn't i and i got a 1-3 pair that was useful but after that i really did struggle to understand quite how to use the logic inherent in the sets to really blast through the puzzle so i'm going to read the comments with interest but this does go to show you that the computer might the computer couldn't do this puzzle we saw that at the start the computer couldn't do that without bifurcation we could do it um with with a bit of human ingenuity and well done to sudoku explorer for uh explaining to us at least that human beings we're still we're still the bee's knees sometimes and thanks so much for watching we'll be back later with another edition of cracking the cryptic [Music]

Channel: Cracking The Cryptic

Views: 523,847

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Id: Ui1hrp7rovw

Length: 39min 56sec (2396 seconds)

Published: Sun Aug 15 2021