Congruity is an advanced Sudoku puzzle by Andrewsarchus that combines four distinct rule systems: (1) Latin square rules requiring digits 1-9 once per row and column in the central 9x9 grid; (2) Index lines where the nth cell value indicates where n appears along the line; (3) Zipper lines where equidistant pairs from the center sum to the central cell value; and (4) Full rank rules where external cells indicate the rank of the 9-digit number formed by reading rows/columns, with ranks 1-36 being distinct. This puzzle requires solving multiple interconnected constraint systems simultaneously, making it one of the most challenging Sudoku variants available.
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If You Have A Brain The Size Of A Planet...
Added:Hello and welcome to tonight's edition of Cracking the Cryptic, where a couple of weeks ago on the channel, I mentioned a puzzle that was on my list of things to or puzzles to try. Um, but I knew it was going to be absolutely brutally difficult. So, I had to allow myself a very long window because I mean this could be a three-hour 4hour solve. Um, the puzzle on the screen is called Congruity and it's by Andrew Sarcus. Uh, we've had a lot of recommendations to try this, but apparently it is very hard. And I can show you some of the comments on Logic Masters. Hopefully, this will Yeah, that that looks good on the screen, doesn't it? Sorry, I'm I'm in this new software.
So, um, uh, if we if we just scan some of these comments. The Sedoku equivalent of mud wrestling, says Senator Gronk.
Many times I thought I had a firm grasp only to find things slip through my fingers. Um, Henry Pi James. Uh, oh, that's that's a technical comment on how the puzzle works. snooker fan who I always rate as not only a very good solver but a very good judge of a puzzle says incredibly clever, original and satisfying. Um so you can see everyone's saying is very very good. Um but this is what I'm going to attempt for you today.
Now, uh, the reason I have found myself with such a long window of time is that I was actually meant to be playing in a golf tournament today, a president's putter down at Ry, but the weather is so inclement that the competition's been deferred until tomorrow, and it may be deferred again, I think, because the forecast is for sort of um, well, 45 mph winds, rain, and um, and very, very cold temperatures tomorrow even. Um, but so I've got I've got I've got found myself at a loose end today because I've already pre-recorded the video for tonight. Um, because I thought I was going to be away, but now I'm not. So, I'm going to try this one instead. And I'm looking forward very much to having a go at it. Uh, even though I suspect this, if I can solve it at all, it'll be a very long battle. It's got it's got the full rank rule um going on in it.
Um, so it's a fivestar out of five difficulty puzzle. Um, and because I have no idea when this video or even if this video will ever see the light of day, I don't have any birthdays to tell you about. Uh, so I will make a short appeal. If you do enjoy our video content, please do consider subscribing to the channel. Um, we would be grateful. We we really really would love one day to to close in on the magical million subscriber mark. I and that would get us a gold plaque. We we've we've had a silver plaque now for many years. It sits over here in my office.
Um and um yeah, my secret ambition is a gold plaque. So you could make it happen. Subscribing of course is totally free to YouTube channels. Um so that if you enjoy the videos that we would enjoy that would be a very easy quid proquo.
Um anything else to mention? I don't think so. Let's just let's just kick into it, shall we? And I will read you the rules to congruity by Andrew Sarcus.
They are as follows. We have got Latin square. Place the digits 1 to nine once each into each row and column. Now, I think that must be in the middle of the grid here cuz there's there's this stuff outside the grid, which I think is an index line, and there's and there is a 9 by9 grid in the middle. It just hasn't got 3x3 boxes in it. So let me just highlight these cells quickly. So I think what we're doing is those cells have to exhibit the properties of a Latin square. You can't repeat a digit in any row or column of those yellow cells.
Now index lines turquoise.
Yeah, that is that is that really turquoise or is that turquoise? Hang on.
Oh, it says the zipper lines are lavender. that that I'm I'm pretty convinced is lavender. I've seen lavender in the garden and it's it's definitely that when it's flowering. So, this must be a turquoise line. Um the value in the nth cell along an index line starting from the diamond. So, it's definitely an index line indicates the position along the line when the value n appears. For example, if a diamond cell contains a three, oh, I tried to write three into that, the third cell along the line must have a one. Yeah. So, I've looked at these sorts of things before, but never and never with an index line this long. But the what what you're doing is you're Yeah. I mean, that's why it's called an index line. So whatever digit you put in the first position is indexing the position that the one appears on the line. So whatever digit we put in this position is indexing where the two appears on the line. Whatever digit we put in that position 1 2 3 4 5 6 7. So this would be saying where the seven appears on the line. So if that was an eight then we would write a seven into position eight. And you can then see they sort of index each other because then in position eight you've got a seven. So they often go in pairs. Um these index line thingies. Um now we've also got zipper lines though in this one. These are the lavender lines, the sort of purple lines.
Pairs of digits equidistant from the center of a zipper line have the same sum.
Um that's very interesting actually in the context of row one. Uh where the center lies in a cell the s hang on hang on there's more there's more to this rule though. Where the center lies in a cell the sum is the value in that cell.
The sum may be different for different lines.
Ah, okay. I think what's happened there is that that rule has been borrowed from puzzles where you could get an even even length zipper line. Imagine you had a zipper line that looked like that. That has no center.
Well, the center is actually right in the middle. It's along the edge. But if this was a zipper line, we'd be simply saying that those two cells, the sum of those two cells is the same as the sum of these two cells.
But actually, if you look at all the zipper lines we've got in this puzzle, they've all got circles in the middle of them. So, I don't I don't know this, but I I suspect that each of these lines is an odd length over one to it looks they look like they're all odd lengths, and that's uh so that's um what's going on. So, so what that means is that this cell is the sum of these two, and it's also the sum of those two.
Um, now this is going to be the tricky part. I think full rank a value in an external cell. So that's one of the cells around the edge of the grid represents the rank of the number formed by reading the digits in that row or column from that direction where one is the lowest such row or column and 36 is the highest. The 36 numbers are distinct. So there are no ties. I mean this is an incredible rule. So what that means is let's imagine that this column was was this.
Now if that was the case, what is the rank of this number?
Is it possible that there could be a lower number than this in this column for example or this this row from this direction? No. This is simply the lowest number you could possibly put as a nine cell number in Sudoku given you can't repeat digits. So this would be a one.
Similarly, this would be the highest number you could ever do because 987 you we can't do 988 or something like that. It just is impossible because of Sudoku. So, this would be 36. I don't know how we'd show that, but um because I can't write 36 into a cell. Uh can I do it like that or something? That would be the 36th rank number. And then oh and this is where it's going to get very complicated because now that's going to have some implication for this index line. Um so I mean it's it's going to be very difficult. That's what we can that's what we can immediately discern. But it's a beautiful looking a very resting looking grid, isn't it? Do have a go.
The way to play is to click the link under the video as usual. But now I get to play. Let's get cracking. Now let's double check that I've got the puzzle on the screen of that I'm recording on and that looks fine. So, no excuses.
Well, actually, I'm going I'm going to start then in the Latin square. I suppose I better call it that rather than the sodoku. So, because this row somewhere in it contains a nine. Now, if the nine is anywhere but the center, the say the nine was here, it would be being summed with this digit to give this central cell. Well, that central cell can't be bigger than nine. So, the nine in the row must go there. And what that's going to mean is that each of these sort of pairs along this zipper are going to be two digits um that sum to nine.
So these this these could be 36 or 45 or 27 or 18. We don't know. Um but that is how the top row is going to work right now.
Okay. Sorry, I'm just trying to think about what this means because a nine uh Okay, let's let's let's talk about some of the secrets of full rank. There are very few I I know very few secrets, but I do know one. And that is if we look along row one, if we look along column 9, if we look along row nine, and if we look along column one, in each of those Sedoku units, let's call them that, those nine cell pieces of the grid, there will be exactly 1 one, exactly 1 2, exactly 1 3, etc., etc., all the way through to exactly one nine.
Now the nine therefore because there are four uh on the perimeter. I mean you may say ah but you could put a nine in the corner. Yes, you could put a nine in the corner but it's still only once in the sedoku unit that is column 9 and only once in the sodoku unit that is row 9.
But the point is that there will be four such instances.
Nine will be once in row one, once in row nine, once in column row once in column 9, once in row nine and once in column one.
And those nines on the perimeter then are going to be responsible for the four biggest ranked numbers, aren't they?
Because the nine will always be, you know, wherever eight is here. Doesn't matter whether this eight is followed by nine, it's still going to be a smaller number than whatever this number is. So, so the rank of this number going down here is massive. It is one of the four biggest numbers in this puzzle or yeah in as as a rank. So So this is a 36 length line is it?
That's what the instructions seem to imply.
Yeah. Yeah, that's right. So there. So there's nine there. 18 27 36. So the biggest rank number which would be the 9876543 2 1 if it exists. It's in the puzzle is 36.
So this number which is telling us the rank of this is is at least a 33.
I don't know how to pencil mark that.
Um, maybe I'll put a Okay, maybe we'll put a 10's digit in the corner and then 3 4 5 6.
So, what that's saying is that the rank of this number is 33 34 35 or 36.
And then that is putting 1 2 3 4 five on this on the index line going round the grid. That is putting a five in 33. So 33 is going to be four back.
So there. So there is a five in one of those cells.
Yes. So it might actually be helpful to divide this line into sort of four slugs of four.
Um because the nines in the perimeter wherever they live are always going to relate or the nine. Yeah, the nine.
Let's say that was a nine.
So that's 1 2 3 1 2 3 4 5 6 7 8 9 10 11 12 13 14. That's 15.
So that would be saying write a 15 into one of those cells.
But the yeah there's a there's a sort of conccommatant implication isn't there because whichever one of these is five for example is saying you put the f the fifth ranked digit which is very low digit.
So say that was five.
Yeah that's going to say that's going to be a two isn't it?
That's weird. So, but because because the fifth the lowest number is one, the fifth lowest number will begin with a two because the first four lowest numbers will all begin with one and then then we're going to get the twos after that.
Yeah. Okay. So, that that that I think is what we should do. I'm going to just highlight using my um my color scheme.
I have got nine colors in my color scheme, so I might as well use them. Uh let's make those red, those yellow, and those are going to be blue. Right?
So the these effectively are where the nines get indexed. These are where the eight and when I say where the nines get indexed, I mean the nines in the grid. The nines in the grid get indexed into here.
The eights in the in the grid get indexed into there. So this is a nine. So it get indexed over here.
And the thing that's being indexed over here is a five. So that means there is a two in one of these.
Can that be a two?
If that's a two, you've got to be double one there.
Oh, this is very confusing actually. If if that is double one, this would be a this would be an index of one. So that that can't work.
Yeah, that can't be a two. I mean that's quite even that's quite complicated but I don't think it can be because the only way you can make the zipper work is with double one. But now this one is saying that this is actually the lowest ranked puzzle um digit in the whole puzzle. So it must begin with a one and that's going to break the Latin square rule.
We've got two ones in column one there.
So that is not a two. So that is not a five.
Unfortunately, I think this being a two on the zipper is a very different thing, isn't it? That's probably fine.
That nine is a nonsense.
That was me hypothesizing.
Wow.
I mean, where is the where is, you know, when you look at a puzzle, I'm always trying to think about where's the weak point in it. Where's the bit where we can get stuck into the logic?
Actually, I better just check my phone in case that's about tomorrow. Uh, no, that's No, no, it's it's all right.
That's not it's not a message telling me that tomorrow's also cancelled. Um, yeah. Where's the weak point in this puzzle?
It feels like the weak point or it feels like you we're being told the most amount of information about row one except that knowing that these two digits sum to nine.
Yeah, imagine they were eight and one.
If this was an eight, this would be indexing a six. A six would be being thrown over there. Are we okay? Oh, hang on, hang on, hang on. I'm confused about that.
I think I've No. No. Okay.
Well, am I wrong about that then?
Sorry. Let me just think this through because I think I think I am I don't think that's possible.
Okay. Well, this this is very interesting if this is right.
Yeah. None of these digits can throw an index into this column because this cell has already thrown a two into this column and an eight here.
Yeah, these purples outside the grid effectively equate to an index of two inside the grid. Ah, this is very hard to explain. But what I'm seeing here, and I hope this is right, is that the eight is saying, okay, so this number is going to equate to one of these yellow uh that's where this digit is going to be thrown.
But we know that having been thrown down here, this is this this digit here is going to pop a two into one of those cells where it can't go by sudoku. So what we're actually learning here is that the eight in row one is not in those three cells.
So I am certainly going to make a note of that because that eight I think eight can be in the corner because if eight is in the corner what that's doing is throwing three which is it's throwing a three into one of these cells. In fact it's in fact it's Oh hang on. No that doesn't work either.
This is bizarre. It's absolutely bizarre. It's quite weird and quite lovely in a way because if this is an eight, what I was going to say, and it was true, is that an eight here is throwing a three into one of these cells. And that's fine, it looks like. But it's not just throwing one because it's in the corner. It's associated with two ranks.
So, in fact, uh, nine and 10.
So if this was an eight, two of these digits would be the digit nine and the digit 10. And that would put two threes into that sequence. Well, I can't put two threes into that sequence. I'll break Sudoku or break Latin square rules. So that is not an eight.
That might mean that we were meant to start with the corners actually because I'm now I'm going to come back to that thought.
I'm going to come back to that thought.
I'm going to just focus on the eight first because now this is really weird, but we've now worked out the eight in row one is definitely underneath the green digits.
Now what that means I think is that there is a one in one of those cells because let let's go through the logic and check that we're right. Let's say the eight was here.
This would be the index the full rank the full rank of this number because it begins with eight. It's massive is going to be 29 30 31 or 32 and that digit would be a three. That's in the third position along the line.
So that would be saying one of these yellow digits is a three, which of course means that that is the third lowest number in the puzzle. So it must begin with a one. Yeah. So that we have we are sort of starting to understand it now. The fact eight is in one of these means that one by zipper logic is in one of those.
And that means oh what does that mean? Why why do I find it easier to think about eight going down here than the one being one being in one of those?
So that's that's putting a two or a three in one of these. I think actually look we can do the trick with one again here.
If one goes there, then the digits 9 and 10 are in those cells. And that and that that's putting two threes into row one.
That doesn't work.
So So that means if this can't be one, this can't be eight. And now one of these being a one means one of these, which are positions 6, 7, and 8 along the line. So a six, seven or an eight is getting thrown into hang on. So So if this was a one, that would be a seven.
So it depends how Hang on. No. Am I wrong about this? Hang on. I'm getting confused. I think I'm getting myself confused.
Oh, this is so confusing. I'm so I'm losing the will here. Oh my goodness.
Oh, I thought I had it and now it's gone. It's My brain is like popping.
Hang on. Let's Come on. Just blank your brain. It's like the Etcher sketch. I have to go and now re reboot. Um, oh gosh. Right. So, let's go through the logic carefully. If this is a one, this must be a very low number because this is one of the smallest digits. So, what we're doing is we're throwing a seven cuz this is So, we're saying put a seven into positions 1 2 3 or four. And that's going to put a two into one of those cells, right?
Because these are definitely underneath.
These are definitely associated with the blue number, aren't they? So, we're now saying one of these digits, there's a two in one of those, which means there's a seven in one of those. Oh, seven in one of these.
But can we play the same game here and get the seven out of the corner? That's the question. So, a seven here, that's going to be a going down there.
So, we'd be putting nine and 10 into the red cells, which would h that works, doesn't it? That's really weird. Right.
If that's a seven, nine and 10 are getting thrown outside the grid into the red cells. Now, this therefore couldn't be this couldn't be a nine or a 10 because that's putting a three there.
And then whichever one of these was a nine or a three, we'll put a second three in the row. But what would happen if exactly this cell was the three and these were the nine and the 10? That would work.
Yeah. Okay. So what I'm now I think we have to do is to look at the corners more carefully because what I'm realizing is that this digit here has to be associated with a color that bends around a corner so that we Oh, it could which could be itself of course.
Yeah, that could be a three, couldn't it?
Oh, that's that's cunning. Yeah, if this was a three in the corner, nine and 10.
Yeah, if that's Let me just think this through. So, if that's a three, that could just be indexing itself or they could be indexing each other and that would still put a three here. So, that can be a three.
That's we So, that could be three.
It could it can't be a four because that would be associated with two threes in in these positions. It could be a five.
Maybe it could be a seven. We've looked at that because that's red. Can't be anything else. So that corner digit is 3, five or seven. But I think that must apply there as well and there. But I'm going to come back to that in a moment cuz I want to think about row one column one. Now by the in the zipper line that digit is 6 4 or two which equates to just a million things outside the I mean a six here would be throwing a one into one of those cells which would be throwing a one into one of those cells. A four would be throwing a one into one of these which would be throwing a one into one of those. and a two.
A two we've already looked at. A two here would be a two here would be throwing the one into one of these cells. So it would just mean that the the the one digit coming down here would be the very lowest of the low.
That's very very annoying. Ah but from there as well it's got to work doesn't it?
So, if that was a six, this one, that's where that's throwing a 36 into one of those. So, that's putting a nine into one of those. Well, nine could go in the in the bulb of that zipper.
Couldn't go there by Sudoku.
Yeah. Okay.
Okay.
I tell this is a weird puzzle. I actually felt as I was starting it like I understood it better than I do now. That must be Does that sound weird?
No, but I think I've got I got myself in a real pickle trying to think about these ones in row one. For some reason, I find that harder to think about than the eights. I don't know why. Um yeah. Okay. But what I do think is true is that these corners which are ploning two digits, two of the same digit somewhere, they have to also yeah I mean imagine that this was a four. Let's just I don't think it can be. So I just want to reconvince myself of that because we know these two digits are associated with the digit five in the sense that they are 12 16 that's 18 and 19. So we're throwing an 18 and a 19 into some of those cells. And when we put say an 18 there and a 19 there, that's going to put a four there and a four there.
No, it's not. It's going to put a five there and a five there because that's associated with the fives, but it is going to put two of them in the same column. So, that is not a four. That also has to be three, five, or seven.
And so does that. by the same logic but unfortunately I don't think there's any mathematical relationship between is that true I don't I don't see anything do they I I suppose if that's a seven that will be a three then. But if that's a five, that would be a three.
That probably be a seven. Then maybe they will have to be different.
I don't even see how to use that though.
They actually Oh, they No, they don't even have to be different, do they? What if one of them went there and one of them went there?
That's weird, isn't it? So, imagine that this was five actually. Oh, well.
Okay, you can't quite Okay, I I couldn't do those two if this was a five because that would put a three on this zipper circle.
So, I think if that's a five that that is a three in the corner.
I'm not sure it's the same with seven.
If that's seven and it plunked, sorry, if this is five and it plunked seven into that one and that one, that would be a seven. And that might be okay with that being three or five.
Oh dear. Well, I don't know what to do then.
It's not this thing in the grid, is it?
What's that? That's another That's another index line. One, two, three, four, five, six, seven, eight, nine. Okay. So that's right. So I mean one of the things about index lines is that the digits on them are always distinct because imagine you put I don't know two twos on the line.
Then that would be saying write a one into position two and this would be saying write a four into position two.
You can't do both those things at the same time and obey Sudoku rules. So this line will contain the digits 1 to 9.
And actually I I now think this line is not relevant. I think this is a disambiguating line, believe it or not.
It's going to be to do with trying to fix the numbers once we've got a handle on the perimeter.
So I'm not looking at that again. At least not for now. Um, so I have no idea what to look at now.
Wow. Uh, what do we think is the most likely thing to be restricted?
That digit has to be a small indexer that I can see cuz that digit can't be more than nine, can it? So this is being summed with. So if that was as small as it could be, that could be a one. That cell is not bigger than eight. And that that that is important because that's going to be putting a three. Ah, yeah. Okay.
Okay. Okay. Let's think about this, right? 1 2 3 4 5 6 7 8. I know that's the most ludicrous pencil mark I've ever done in my life. But it it is far more potent than anything we could do in the I mean, if I if I did that there, it would be absolute nonsense. Yeah. But it's not such nonsense here because this being only because because what we're doing there is we are sending the digit 9 10 we're sending the digit 11 into an early position.
Maybe that one as well. No. No. Cuz that would be nine. We can't send it there because that would be a nine and then this would be at least a one and that would be a double-digit number. Yeah. So we are sending So this is this is telling us where three goes in the first row of the puzzle. We are sending an 11.
Well, we're not sending it into position five cuz that's got Okay, so I've improved a lot in life.
Ah, I'm not sending it into position one. That can't be a three.
Oh. Oh, no, no, no. Right. This is great. This is great. It's bizarre, but it's great. Right. So, what we've just learned is that this digit is 2 3 4 6 7 or 8, which means throw the 11 in the perimeter into one of those three or one of those three cells.
And that's telling us to throw a three into one of those three or one of those three cells. So, this digit is not a three, which means that digit is not a six.
and three and six are in these cells in row one.
So now this uh so that's uh I don't know do I ah bobbins that was nearly that was nearly intelligent and then it's not. So what's that digit then?
It depends. It's a It's a one or a two, isn't it?
Yeah. If this is a three, say, then we're putting 11 in here, but that says put a three back here, which is going to equate to a one. So, this digit is a one or a two.
I don't know what that means. That's just infuriating.
Um, is this in any way? Is this obvious?
Wow.
I don't think it is.
Not to me at least.
It's bizarre actually. Low digits outside the grid are much more valuable.
Yeah, let's do that one as well.
That also can't be a great big digit, can it?
So again, if this could be a nine, which it looks like it can be the value of this well the minimum value of that digit is two actually can't be a one because of what we did earlier. So the maximum value of this is seven.
So that digit is another ludicrous pencil mark.
Okay, it's not five because five we know is associated with a nine. And this would be throwing an eight into that position.
Right. And the position we don't have an eight here. So we're not throwing that into position one. Oh, in fact we hang on. We are throwing an eight. This is throwing an eight. Oh, I didn't realize this. I mean, how are you meant to realize this? I don't know. But I write this digit. We know the eight in row one is in one of those three cells. And this is throwing the eight into row one because it's indexing a massive number.
Whatever that is 30 uh what is it? 28 29 30. It's throwing a 32 into one of those positions roughly. And that's going to put an eight in in in row one. And we know the eight in row one is here in those three. So that is that is actually a 2, three or a four.
And one of these is a 32.
The thing is that that coming down has not really helped me has it.
So now the minimum value of this is four if that was two and that was two but it could be four 5 6 7 8 or nine and that puts no pressure on this bother.
Okay, I'm floundering a bit here, aren't I?
Um, I don't think it can be this line. This line's far less constrained than that one is.
This can have repeats on it, but I don't I don't think I mean this this has to be a relatively small number but it c this can be absolutely a double-digit number. Um because well how could we do it? We could have an 8 n pair there perhaps and that would so the maximum value of this is 17.
I mean, I know that doesn't sound sensible to think about because okay, that can be a number from one. It can't be one. I suppose it could be that's a minimum of three.
Uh, and that's a one. The minimum value of four. I don't think it can be. No, no, it can't be four because that would put 28s into row one.
In fact, ah ah hang on, hang on. Right, that's important. I've just realized something because I've got an eight in row one already.
If this is anything projecting an eight into row one, that's terrible news. That's terrible news cuz that would put two eights in row one. We know this thing is throwing the eight into row one. So this must be throwing the eight into a different sodoku unit.
And because this can't be bigger than 17, it's going to be something in this column.
So 9 10 11 12 13 14 15 16 17.
So I think it's throwing an eight.
looking at our pencil marks into one of those cells, which means that Yeah, but it isn't 17, is it? Actually, it's not 17.
Oh, this is confusing to explain but it isn't because imagine it was 17 which is associated with this position.
So we're saying we're going to write 17 in there which is obviously going to make these two in fact it can't be 17 for many reasons. I've just seen a different reason why it's not 17. That would require this to be an 8 N pair and this to be an 8 N pair by the zipper logic. That's not going to work. So, and actually I was going to get rid of 16 as well for the same reason. 16 can only be made with seven N pairs. So, this would be 7 9 the two tips and these would be 7 9 and we'd have three cells in column one all from 7 and 9. So, that is a maximum of 15 now.
So, it is now putting an 8 uh 12 13 14 15 into one of those positions.
It's quite difficult to put it there, isn't it? That would make that a nine.
I don't know. But 8 in column 9 is in one of these positions.
So this number is um 9 10 11. So it's a 12 as a minimum but it could be 13 14 or 15.
Now what pressure does that put on this digit though? Because I think there is some Yeah. See, I mean, there's a lot there's a lot because depending on where the eight goes, that's putting a three or a four into this position.
So now, this is really clever. for this line by the way. I mean the only way you even begin to study it is because you get so stuck but but having these two digits sort of related to one another because of the indexing is really putting pressure on this because because this is massive now isn't it?
Cuz even if that's 12, which is the least it can be, because that's so low, that has to be at least eight.
Yeah, that's true. But more more more relevantly, even if this is nine and this is four, this can't be more than 13 now. So it is 12 or 13, which might Oh, no. It still means there's optionality.
It is 12 or 13 which is this one or this one.
So if it's Oh, I can't quite visualize it.
It's very forced either way.
That's that's certainly true. The only way it can be 13.
13 is very specific. It's probably right. Therefore, because 13, the only way to make these numbers add up to 13 is 4 and 9.
And what that will do, this will be a 13. That is position 29.
So you'd write 29 into position 13, which is this one.
29, which plunks an eight here, I think.
And then those two digits would have to sum to eight. These two digits would have to sum to eight.
And if that is a four, that would put a two in the corner as well, which is interesting.
Oh my goodness. That doesn't work then.
I mean, this is barbarically hard, but it's really clever. It's probably therefore what we're meant to do, right? Look at this. So if you do go 13 here, the point really is not about that digit after all. The point is about this digit, believe it or not.
Because if you put 13 here, this becomes a four. And that makes this digit in row one, column one a two by Sudoku or Latin square, which makes this digit a seven.
But remember what's that seven doing?
that is throwing a three because it's indexing positions nine and 10 into red.
Two threes into red, in fact. So, the only way you can put two threes in red is by putting three in the corner. And we might we normally like putting three in the corner, but in this instance is deadly because it says that's a three.
And now, how do we make these two cells add up to 13 if this is a three? You can't do it. You can't write 10 into this cell.
So that is not well so this is not 13. I don't quite know whether it means more than that but it it definitely means this is 12 which definitely means I don't know how we do the sum here then but that's throwing what I need to do as well if I actually work out any of these ranks and I do know that this is a 12 now I need to somehow you know notify myself that that is correct it's Done. So, how am I going to do that? Do I I'm going to circle it maybe. Does that work? No, it's not great. Can I circle it in gray? It's a bit easier to see.
Actually, the gray isn't that easy to see. Maybe I'll circle it in green. No.
Uh, red.
But then red. I'm not Maybe I would see the red, actually.
Yeah, I might do red. Okay. So that is a correct index. But I get two correct indexes as a result because I have to throw 29 into position 12 which is that one. So that is a 29 which we know equates to an 8. That's huge actually. That's going to give me a nine here. Um because in fact it gives me a one here as well. That's got to be a one now because I can't make this index more than nine because it's in a sodoku.
So, that's really cool as well cuz that being 29 is the lowest of the eight ranks, isn't it? And then I've got an eight followed by a one from that direction. So, that makes total sense for that to be the lowest of the eight ranks. You would expect that with an 81.
Now, this being a nine is throwing a four. Oh.
Oh, botheration. That's almost brilliant. I don't think it's quite good enough. But but but it's saying that this this rank is a third is the same as this. It's a 30 345 or six.
Uh and so what that's doing is writing four into one of those cells. Well, that that's good. No, that's good for a different reason. I thought it was going to be good cuz I was hoping it was going to resolve row 1, column 1. It doesn't resolve that, but it does resolve this.
So, that's now got to be a three because we've got to get because it can't be a four. It's a three. And now we've got to get to 12. So, that's a nine.
So, this is not a three in the corner.
So, hang on. Hang on. That might mean something else. These are now five and seven. Does that mean that's a three? It might do. Um, let me just mle this over. Let me just get this straight in my head. So, we're now Yeah, that makes sense as well. That being 39, cuz that's position 12. In fact, maybe that was a way. If that had been 49 and that had been 13, you'd be saying that the all of the rank four numbers would have had to begin 49, which is probably impossible for some weird reason.
Um, okay. I Well, this must tell me something.
So, position 12, which is this one.
I get a 39 here.
Oh, I tell you what it does do. I've got to make the other part of the zipper add up to 12. So, that can't be a one anymore. Now, where did that one come from? Oh, that was that was whichever one of these is eight being thrown down here. So, we now know that whichever one of these is 8 is associated with 28 29 30 31 or 32.
I don't think we can use that though.
It's not going to help us.
But this is a five or a seven now, isn't it? Cuz we're saying that these two cells add up to 12. And therefore this digit is throwing.
What is that doing?
Five or seven.
So if this was seven, it would be indexing itself.
And now that would be throwing that would then be five.
Throwing an eight. That looks like it could be an eight. Oh, actually that can't be an eight.
It's quite interesting.
It would do a lot actually would do a lot if that's a seven in the corner.
I think this becomes five. I think that puts eight here.
I think that this is a three five pair.
Then adding up to that eight.
But I think all of those things are possible.
Oh, right. Let's look at that corner digit. Now, that can't be five, can it?
If it's five in the corner down here, both of these corner digits are seven.
And that means I think that's a problem.
because this being a seven definitely requires that to be a three. In fact, that can't be a three. It simply can't be a three anymore. So that Oh, this is massive. That Let's forget what I've just said. Let's just ask the question.
Can that cell be a seven anymore? And I've just knocked out three from this position. So, it can't.
If that's seven, we're throwing 9 and 10 into these positions. So, it's putting two threes into red. The only way you can put two threes in the into red and not break Sodoku is if you put the three in the corner, which you can't now do.
So, that is a five. And that means that's a four.
Right now, that must be important.
So, this is not a five. That's Sedoku that's telling us that. I'm not sure that can be, but we'll come back to that in a moment.
This has got to be it now.
How do we do how do we use this?
So, five.
Yeah, this is great, isn't it? This is great because now what we're saying is that we're throwing nine and 10 into into these purple cells.
Well, that's putting three into two of these purple cells.
And I can't put three there because of this short little zipper that it won't add up. So that is not a three. So I'm now putting three into two of those cells. Well, the only way I can do that without breaking is by putting three in the corner. And oh, it doesn't know. It doesn't know. It doesn't think I have. I have. That's three in the corner. That's three in the spotlight. Losing its relation. I have whatever it says I have. Um, right.
I mean, this is going to be impossible to pencil mark now, but we're saying that those are nine and 10. So, these are 9 and 10. How do I do that? 9 is 9 and 10. I mean that I mean I know it looks bonkers, but that that I think it means something to me. Now similarly these positions are 16 17 18 and 19. So we're putting 18 and 19 into those cells. So that's a one in the 10's position. That's an 18 and a 19 now.
So, what does that mean?
I don't know yet.
Um, I'm sure one thing we ought to be able to do when I got this four, wasn't didn't I have a four in one of those positions?
That's this one, isn't it?
Yeah. This one was throwing a four into one of these, which is now going into the very top rank. So, that is 36 and it gets a red ring. This is its complement, which is 12 13. That is a 13.
It's not got a five in it anymore.
So, the five is is in one of those two.
And that's this one, isn't it? Depending on what Well, this isn't 36 anymore because that's 36.
And it's not 33 because that can't be This can't be a two.
So, it's not 33. That's 34 or 35, which is one of these. One of these is a two.
Also, this cell is now associated. That's putting a one in one of those, isn't it?
That's big. That's going to do that digit.
Let's just check that I'm right about that. This is a rank which is 13 14 15 or 16. So I'm putting a one into one of these, not 13. So 14 15 or 16.
So one of these is a one, which means this cell is a two. And if that cell's a two, that's not a 2, three, or a four.
because that would be Let me just think that through. I think that's true.
I'm so bad at this.
It's throwing a three.
Yeah. For this to be a one, I h I have to be indexing into those cells, don't I? And I'm not now because of this one. I'm indexing into one of these with a three. So one of these is a three. So one of these is So one of these is a six.
Six, seven, or eight.
But I don't think I know which one.
But I I mean slightly that's not a one actually by Sudoku. So that means this one isn't an eight.
And ah no doesn't quite work.
Oh that can't be a seven because if that's a seven that's a two to add up to my nine down here. So this is six or eight. So this is one or three.
Now you'd think I could do that, wouldn't you? You'd be wrong.
Um, I I also have the feeling as we're doing this that it's incredibly easy to miss an implication because they all have such weird knock-on effects.
Um, yeah. Uh, so I don't know what to do now.
It's yeah it's very very difficult to know to know even remotely again where the pressure is where where have I I've done a bit I've got some digits but I don't feel I've got very much pressure on the grid but I must have somehow It's very difficult to put nine in the bottom row. It's really difficult because of this nine here. Can't put nine halfway along a zipper cuz the circle can't be more than nine.
But I think I could put nine here. I think what digit would that be associated with six?
So that would be saying write a six into one of those. That looks fine. Although I say that there probably are some things I can't write in there. Now I've got this 57 pair down here.
What's that digit?
That that Oh yeah. What is this digit?
That's a good question.
That digit.
It's not it's not one. It's not two.
It's not three. Four. Five. It's 6 8 9.
It's not nine. It's 6 or 8 only. So, it's even.
So, that means this digit is throwing a nine either into one of the six positions. Oh, that makes sense though, doesn't it?
There does have to be a nine cuz we just said the nine is in one of those two positions in row.
Yeah, but I suppose that six could be up there if that's a two because he can't use it, I don't think. Or this is eight.
Uh if it's eight, it's very difficult because the only position for nine now in the eights, which are these would be that one.
But that might be fine. So this is eight. This is nine.
Okay. So, we can't really do that then.
Oh, this isn't two anymore because two is no longer able to be an eight since we got rid of a one here. I mean, this is barmy. So, this Oh, this could be double three. No, it can't actually. Oh, in fact, that can't be three.
Oh, that's huge. That's it. There you go. If this is three, whether this is six or eight, this digits impossible cuz it would either have to be three or five to add up to six or eight. So, that's got to be four. And now, now that's great. That's absolutely what we wanted because we now know that being four is throwing 31.
No. 32 into that position.
So that is a correct index. That is a correct index. And that 32 puts an eight in the grid there.
Now that 32 puts a one in the grid here.
So that is now a 37 pair.
This is now a 2 six pair.
This one we know means this digit must be 1 2 3 or four.
But it's putting a two in in the first row. You can't put a two in position one. You can't put a two in position four. You can put a two in these positions. So this is one two three four five six. So there is a six in one of those equating to the two in one of these.
Don't really want to pencil mark that. I think I just leave that like this and hope that I understand what it means.
So, this can't be eight anymore because that would be a four and that would clash.
So, that's got to be a six and that's got to be a two. This could be huge, couldn't it? Let's hope so.
Now I've got I mean every time we do this I have to go through this exercise again of trying to understand what it means.
So this is saying we're putting a nine into one of these four cells. Well, you can't put it there and you can't put it there. So the nine is definitely in one of those two cells.
Um so one of these two cells is 33. One of those two is 33.
Which means 20 21 22 or 24. This is 22 or 24.
It's not going to be easy to do this even if you had the whole perimeter cuz all of these ranks have to be fulfilled correctly, i.e. in the perfect ordering.
But this two means this is the five, doesn't it? That's actually a win for the goodies. So we are writing the number 34 into that position.
And that is a correct rank. And that is a correct rank.
Unfortunately, it doesn't take us forward, but it's still true. Now, sorry for the pregnant pause. I need to think.
Um, I need to think. Come on, think, Simon.
Maybe I pencil mark that digit. Do I?
So from a sedoku perspective, that's eight or nine.
So this is a massive digit.
Yeah, it could be indexing itself. It could be a 35 and then that just would be a nine. And if it's not 35, it's it's it's indexing the eight into here. Ah, this is a four. This is a one, is it? Yeah, that's got to be a one.
Didn't put it in, but that's true. So, now this is eight or nine. Oh, bother.
So, these either index themselves or they they swap their indexes.
Okay. But this one, this is putting eight into fives or sevens.
So, one of Oh, if eight was there, that would have to be nine.
Jeepers. I mean, this is so confusing.
Um, do I know hasn't hasn't six gone now?
Isn't that six?
Hang on. That might be important.
Um, this is if that's six, it's saying write 9 10. Write 11 into that cell, which would make this a three. That's wrong.
That's got to be eight.
That's good. So, this is actually a one.
I've got three ones on that diagonal.
But the key surely is that we're now writing 11 into position 8, which is that one. So that's a correct index.
That's a correct index. And that's a three, which means that's a six, that's a two, and that's a seven. And I've done the top row. Wow. Wow.
Now, obviously, we're going to have to go through this and work out what all of this means, but that that feels like a real win for the goodies. I've got a 71 here, so that's probably a low rank within the sevens.
Um, so we're writing a two into the sevens, aren't we? That's what that means. Uh because seven equates this rank is one of the red ranks which are 24 25 26.
We can't write them in there. So it's it's 25 or 26.
So, it's saying there's a two in one of these.
I know. Oh, I know which one. That's great. That's great. Okay, I know where it is by Sudoku. So, this is 28 27 26.
That is finished.
This one is therefore uh what's that? 1 2 3 4 5 6 that's seven.
and they sort of swap with each other and throw their digits in the other directions.
Now this digit is now a three because we now know that the it's sort of swapping with these two to create this one two pair.
So this digit is now six and they can swap with each other.
Okay, I don't think that's wrong. It felt right.
Um, now this digit is throwing a one into this part of the world. That can't one can't go everywhere. Actually, it really can't. Oh, that's good. That's good. We're going to get this. Yeah, because what this is doing this rank here is 21 22 23 or 24.
But obviously that's going to this is throwing a two into the outside of the grid which is the second lowest digit then going up which is going to begin with a one. You can't begin with a one here. These two can't add up to one. So we're not indexing that by sudoku. We're not doing those two. So the one is definitely being indexed here.
Well, no, it's actually a two that's being indexed there cuz it's the second position on the line. But this is therefore the one and this is 20 21 22 23. So that is a 23.
That is correct. This is correct.
Now, can you imagine setting this? Uh, no is the answer. Um, I can see now that's got a bit bigger.
That's at least a four. So, that's at least a seven. Let's put that in. Seven, eight, or nine.
Does that help us?
I suppose if that's seven, eight or nine. I know this digits is four, five or six, don't I? I might as well mark it.
And that being 7, 8, or 9 means we're throwing what is it going to be a five?
Well, could that be the five? Maybe. I mean, there is a five in one of those.
But maybe that could be thrown by this digit. I'm not sure.
Yeah, that's that's horrible. Actually, if that was a five, that's throwing seven into the purples twice. But that could be a seven. And that could be a seven.
That's Yeah, that's that's tremendously terrible, isn't it?
That is tremendously terrible.
So if, so let me just think that through again. So if this is seven then it's seven and if it's five that's seven. So seven in this row is in one of those two cells definitely. So it's not that digit.
Yeah we can probably get a bit of a handle on this. Uh it can't be eight actually.
Yeah, let's just think about this this zipper in the bottom row, the big zipper because primacy this could be a four and this C central cell could be a five. I don't think that works. The reason I don't think that works is if we look at row eight, I can't put three in any of those cells.
So let's fill them with the minimum the other digits could be it's almost difficult to put a one on the line but one two say can't use three four five so there is a five at least in one of those cells that's being added with another digit to create this digit so this digit is at least six so it is 6 7 or 9 by sudoku it can't be eight now I've just said it can't be seven actually and the re I'm just going to recapitulate that logic to check I agree with myself.
If this is five, I've got to put two sevens are in in in these cells. So they would have to go there. So this would be a seven. And if this is seven, it's obviously seven. So that is definitely not seven. So it's six or eight. Now if this is six, this is five. And if this is nine, this is eight.
Which is that a problem?
Yeah, that is a problem. This can't be six.
If that's six, this is five and this is seven. So 6, five, and seven have all gone. But if this is 65, this is a four. And four + 3 is seven, which I've just put there.
Can't work. So this is nine. That's really cool. So that's nine. That's eight. Now that's going to No, this is going to do things. So go. That's a seven. Now this is a four. So this is a five. That's do I know that's doing something. This bottom row is finished.
I haven't put six into it.
That's that could be just such an enormous deduction actually because now this five remember is putting two sevens into the purple cell. So that has to be seven and we can we can work this out. Now, we might not be able to work out the order, but the fact that this is five means we're throwing uh what are these? 27 and 28. These are 27 and 28. I don't know the order, but that's what they are.
And these are 16 17 18 17 and 20 into those cells. That's horrible. 17 and 20. So it's that I mean it's a horrible pencil mark. Ah, but I know the order here cuz 52 is less than 53.
So this one must have a lower This one has the lower rank. That's got to be 17, doesn't it? So that's 17.
This is 20.
And those ranks are correct. And well, they're going to have equivalents over here. Uh I can work work it out now. uh because 28 is going into position 17 and 27 is going into position 20.
So actually this number reading in this direction is going to be bigger than that number going up although they share the second digit. I mean that's it's going to get very confusing that but but it's still great news. Now, now this must be important because this digit now we know from our earlier escapades is throwing a six. Ah, hang on.
So, we Oh, I see. Because now we know this is 24, don't we? The nine. The nine could have been there before, but it's turned out to be here. So, this is 24.
That is a correct rank which is going into which means this is 33 to make that work. So we can do that.
So in terms of our nines now how many nines have we got? We've got 33 34 36 and we're going to have 35 in either this position or this position.
That's the only nine we've got left to place.
Now that six is a weird creature because that's in its own region. So that's got to index itself.
Uh which is 20 21. That's a 22. I think now this one doesn't index itself.
That's throwing a six which I haven't put into column one into some of those position some of these positions.
So uh 12 13 so 14 15 or 16. So it's like this one.
So uh 14 15 or 16. And these digits actually we might as well pencil mark them centrally. We've got 1, six, and four.
That's not the one at the top. Look, because of this. So that is four or six.
Um, now the one was related to this. No, this is four.
It's uh that one that's not 14 anymore is what that's telling us. That's 15 or 16. So that we can put a one into one of those. It's really hard to pencil mark this. Well, and I fully admit that I'm probably doing a terrible job of it. Um now, oh, we can do the ordering of the 18 and 19 here because the 53 is bigger, isn't it? than the 52 coming down. So that's got to be 19. That's got to be 18.
And that's going to have a conccomative effect for this because we're now throwing the digit nine into position 18, which is this one. So that is a nine on its own. That is a correct index.
That is a correct index. That is a correct index. And once I make this 10, which I thought I could do like that, that's a correct index. So we now know that this number, which is 374, we know this number's bigger than it. So it can't go 371 because that wouldn't be correct. So it must that that's the only position left for one. And now that one was 15, I think.
So let's make sure we don't misanalyze that. So that becomes 15. That becomes one. That is the lowest digit. The lowest rank in the whole puzzle is this one. Um I've only got four more perimeter cells to do.
That's actually a little bit terrifying, isn't it? because that's probably horrendous in terms of working out all of these ranks.
I haven't I haven't even pencil marked this one.
Ah, but this is swapping with this one cuz this is throwing a seven. Yeah, these two are swappers.
Um, right. So, that is that's position 25.
So, we're throwing a 25 into this position.
And this is throwing it back in the other direction. 29 a 30, I think. So, that is a 30. Or have I done this the wrong way round?
No, I I think that's correct. I think that that feels right.
I mean, is it it must be completing this line. We know everything on the line adds up to nine now, don't we? So, I'm just, if you forgive me for this, I'm going to pencil mark. Maybe I No. Do I pencil mark the whole row? I've got to put an eight. Actually, the eight could go there.
None of those can be a one.
I don't know. Let Let's Oh, hang on.
Where's No, it doesn't work.
I am going to have to do the whole row.
I I apologize for this. This is just absolutely catastrophically terrible, but I've got I feel like I've got no option. Um because it's so complicated.
Right. I'm going to just whittle these down as best I can. Oh, that one's terrible. That's can't be one.
That can't be eight. What about that one? Can't be two or eight now. So, how does it work? These two are meant to What is going on with my phone?
Hang on. I do need to just read that.
What's that mean?
Hang on. I just need to check what's going on. It's something to do with tomorrow.
No.
Wow. Okay. Okay, the president's party has been cancelled. Um, okay. But I I don't know what that means, but it means I can just carry on with the puzzle. Um, okay. I get your head in the game, Simon. I was trying to do logic and then I've been totally discombobulated.
Right.
I mean, I suppose these ones might be No, not they're not brilliant, are they?
That one can't be seven.
So this one can't be two because it's meant to add up with nine with something. So this digit here is four, five, or one.
And this digit the opposite which would add up to seven with it. I mean it's so silly. 7 4 5 3 1. It's not one. So that's not eight.
Is it really that one and that one have to add up to nine together? So 8 1 4 5 uh two doesn't work for some reason. Why doesn't that work? No, two doesn't work cuz that's going to need a seven opposite. And six doesn't work cuz that's going to need a three opposite. So we've whittleled that down to 145.
Wow. Okay, that's very very distressing.
Oh, I know what it might be. You know, I've got a 29 number which is a seven.
So there is a bigger two number than 29. So So 8 wherever that is has to be followed by nine.
Beautiful. There. This is a bigger number by rank than a number that begins 29. So that Oh, in fact, look at this.
That's got to not only has that got to be nine. Oh no, this is great. This is great. The eight number, which is the biggest of the two numbers, begins 291.
So, this can't begin 294 or 295. That's a one. And that's going to give me another digit on this zipper down here.
Um, uh, that's an eight.
Okay.
So, that's an eight.
We still got to be a bit careful. I don't know how I'm going to do this.
I've got to I've got to keep track of the fact that that number has to be a bit less than that number somehow.
What's the 29 number?
Have I got that anywhere? Oh, 29.
The 29 number is there. That That is 81.
So that doesn't put pressure on that.
I'm desperate to find out more about this putter situation. Right. That's a four five. So that's a 45 now.
So they're a four five pair. Okay, that's great. So four, five, four, five, four, five. That's become two or six. So this has become three or seven.
Um, which is presumably important in some way. This is bizarre actually because now we're getting to the point where I'm literally going to have to start tracking these numbers.
And that does not feel like a very easy thing to do. Ah, hang on. And I've got an 11 which begins 3 91 now.
So I know that position 12 for threes has to be 3 9. Oh, and it is. And it's bigger. That's That's distressing, isn't it?
26 begins 71.
So 26 that's really is isn't that very surprising.
It's not very surprising. It means that 25 needs to begin 71 which it does.
27 and 28 just need to beat that. 27 is there. 28 is there. Yes.
Okay. So, so that digit can't be enough, is it?
It's not sedoku, is it?
How do we say that? How many how many ones have I put in the actual grid? I've put seven in the actual grid. I know I've got to put one on that thing.
That might be important. Where's Okay, one. Where's one on this thing? Oh, yeah. Where? Where? Oh, for goodness. I mean, oh no, I don't know. I don't think I know. But it is very restricted on this thing because there's got to be a one on it. It's got to contain all the digits from 1 to nine.
I'm just going to double check the one is in one of those two. So, it could be indexing itself or it's there and it's throwing and it's throwing an eight into this position and that doesn't work because I've got eight in the column already.
That's lovely. So, that's a one, which actually doesn't give me a second number, but it's going to give me the last one in the grid cuz now I think I've got eight ones in the grid. I've not put a one in column four.
So, that goes there. Oh, and that's been available forever. Just didn't spot it.
Never mind.
Okay, so 33 now begins 91. That's not surprising.
That's a nine. So, this is an eight.
That's good. That's going to allow me to complete some indexing in the column 9.
How long have I been going? 95 minutes.
Well, I mean, that's frankly I I don't know whether I think I've I must have broken the back of it.
I'm bit worried about how long it's going to take me to tidy up all of these ranks.
But we I mean the progress has been very stuttering, hasn't it? But let's think about this.
So that is throwing the number 35 into this position.
That is what it is doing. We can red outline this. And this is throwing the number 28 29 30 31 into this position.
Okay.
And 35. Can that begin with 93? That seems a bit low.
Where is 34?
Oh, it's here.
So, that digit is at least a three.
So if that was 93, that would have to be a three as well.
And 36, which is here, well, that can't be a two, is what we could detect from that because that's at least a three.
So this is at least a four. I mean, I might as well. I know it's ridiculous, but I'm going to actually pencil mark that.
Where's nine on my funny line? It's not there, and it's not there. And it's not there.
So, nine on the funny line is in one of two places. Either indexing itself or throwing a three into this position.
So this digit is three or nine. That's what that means.
Okay.
That's not a six apparently. See six in the row. I just didn't spot that.
So I've now done all of the perimeter.
Why can't I do these two?
Is there some reason for that?
Uh, probably one of them is indexing themselves.
Oh, and so it depends what this one is.
So that's not 15 anymore.
Oh, so I don't I don't think I do know.
I probably know for some reason I don't understand.
So if this was 14, where's the 15th rank four number there?
48.
Where's the 13th rank four number there? 4 six.
So whatever whichever one of these is four, it's got at least a six in its second position.
I don't know how to use that one. Okay, I I'm going to have to do this in some sort of logical way. I think I'm going to have to start using colors or something to indicate the lines for each type of digit. So like the ones we could the ones are going to be like this. I'll use red for all the ones. And we we'll try and put them in the right order. So we'll start off with numbers that we've got a bit.
So twos maybe 2 91 2 91 there. So I've got to extend that to a fourth position.
So the low see there two nines. So the lowest rank two twos aren't where it's at. Twos are not where it's at. Let's try Let's try threes. That's the top rank three. 3 94 or five.
I've got a 3 91 which is the second highest rank. What about um the three Oh, they're both down here.
Ah, there. That's where we look for our next deduction because I've got rank nine and rank 10. And I know that rank nine, which is lower, begins 3746.
So if this was a four, the rank 10 number would begin 3741. And that's lower and the wrong way round. So that's got to be a six. And that's got to be a four. And that that's going to allow us to finish the the outside. That's therefore 16.
Uh throwing this digit is now 21.
which also of course gets a red line.
And this one once I've worked it out is uh 12. That's 14.
It's 14. Okay, that's the whole perimeter done. That's all the ranks. In theory, it should be completely straightforward now, shouldn't it? Uh no. Um so let's go.
So actually are the threes now done? No, the threes are not done. But these threes are irrelevant now because we've discriminated completely and fairly between the 9 and the 10. And we know that the 11 begins 3 91. So it's totally not dependent on these anymore.
So all we've got to do Yeah, it it's very interesting.
No jokes. No jokes about me. Look, if this was an eight, this Oh, hang on, hang on, hang on.
No, I'm wrong. I was looking at 2 91.
Not and that that's a very different thing from 3 91. So, actually, I can't do anything. 3 91 will always be less than 3 945. So, the threes are done. The three the threes so far as the ranking is concerned I can't do anything more with them. So we take out their blueification and we move on to a different number.
Perhaps that number should be four.
So let's try the fours instead.
Now what do we know about fours? We know the lowest ranking four number is going to be 13 which is here. So 4 six. So every other four number has at least a six in its second position. And that's a four number.
So that number here is at least a six.
So that's got to be six or eight.
And if it was eight, would that be okay?
It probably could be okay. We would have to go 482.
It's quite tricky, but it's probably all right.
And 16. This just has to be at least a two, which it has to be anyway.
Bother. Okay. So, maybe it's not the fours then that we have to think about.
Should we try fives?
Let's try fives. So fives, we've got these two are sorted.
So it's only where the 19 and 20 are relevant. Where's 19 for the fives?
Sorry, n 20. Where's 20 for the fives?
It's down here.
Oh.
Oh no. Hang on. I've got to be careful here.
Golly gosh, that's really amazing actually. 17 begins 52891 and 18 begins 52894.
So they are but they are distinct already.
18 19 is definitely bigger than and 20 is definitely bigger. Yeah, that no. So the the fives are irrelevant. We can't do anything with fives. So we should tick these off. So threes and fives I can't do anything more with in the perimeter.
Ones, twos, fours I still have jobs to do. Let's try. Six looks horrible. I hardly know any I'm not doing six. I come back to six cuz I hardly know anything about them. Sevens. What about sevens?
I have got a couple of seven ones.
So I probably ought to check sevens. So 71 there, seven something there, 71 there, and seven something there. So let's try the sevens.
So the lowest seven is going to be 25 which is where's 25? I can't see it now here. And it begins 71. Then we go 26. We just have to. So there is an implication for that. And then we go 27.
And then we go 28. So we just got to make sure this digit isn't bigger than that one, which is very unlikely to be.
I can't do Sudoku, can I? It could be something as simple as Sudoku or it's this line somehow.
Eight. Let's should we check eight on that line?
I think eight can go in so many different positions though.
Or the 349 line.
Is that going to help me? What's the 33 N line? Let's look at nines just in case they are relevant. So, let's use pink.
So, 9 9.
So, the 33 begins 91, which is of no use nor ornament. The 35 might begin 93 or might begin 97 and the 36 we don't know.
Okay. Uh so I did so I don't know about sevens. I don't know about nines. So, it's the only eights. These eights are going to have to do some work here or I have or I am in real trouble. Eights. Eights. 8.
So, 29 unsurprisingly, is 81.
30 is 84 at least. That's a bit bigger.
So, 31, this digit here has to be at least a four.
So, let's put the options in. And then it's at least four. So it's four, five, or 7.
And that has an conccommatant or a knock-on effect on 23. Look. So 23 now is at least 64. So 24 here, this has to be bigger than it has to be at least four. Well, it actually has to be bigger than that number as well. Oh, maybe sixes are coming into their own then.
Yeah. But what I'm seeing there with six is that whatever this number is, there is an absolute relationship between this number and this number that this has to be lower. So this has to be at least five and it's not six.
So that is five, seven or eight I'm going to claim.
Now that's in position two. So a two on this line.
Please be ruled out of some of these positions.
Position five. 1 2 3 4. Can that be a two? Of course it can. Position seven.
So position seven can't be a two actually. Position eight.
Ah that can be a two.
Bother.
Okay. So that is five or eight.
and then 32 which is this eight number has to be bigger than that eight number.
So again there's a relationship whatever this number is which could be a four as low as a four that has to beat it. So that's got to be at least five. So it's 5 six or seven in fact therefore that can't be seven can it? Cuz if this was seven, this would be an 87 number. This would need to be a bigger number and eight and nine have gone and seven would have gone. So it can't work. So that that is now four or five which doesn't put any pressure on this. But it is true.
So this is 64 65. This is 65 or 68.
Okay.
It's probably It's probably some sort of pencil marking extravaganza. 3 4 3 4 578.
Yeah, I'm going to pencil mark this column. 3 4 5 78. And tick off what I can.
This digit can't be four.
That's not good enough, is it?
Where's the third? We don't know. Of course, we don't know.
Um, okay. So, let's go back to 21 then. So, 21, which is the lowest six number, it's second position is at least a two.
And then the 226 number, which is here.
Ah, hang on.
So, 22 begins six and then a four or a five and 23 begins six and then a four or a five.
So the only way this can be five is if this is also five and then that would definitely be eight and that would definitely be two.
It's something to do with this, isn't it? There's a real congregation of digits here. Ah, hang on.
Yeah. So, what happens if that's four?
Or what happens if that's four?
If that's four, that has to be four.
This has to be five. And that's the wrong way round. That's it. This is the key digit. That one there. Let me just highlight that cell.
It's quite I mean it's it's incredibly complicated, but let's try and work this through. If this is four, then we know that the 23rd 6th number begins 64, which means this must begin 64 in order to be less than it. But that's going to make the 30th number for begin 85 because of the zipper 85. But the 31st number begins 84.
That's not right. That's the wrong way round. So this must be five.
This is now eight.
Okay. Let's be very sure that we don't overstep our marks here because this this is doing a lot of different things.
This eight is plunking a two into this position on this zipper.
So the twos and the eights are sort of friends.
We've got 23 is now 65.
So 22 is is under less pressure than it was. 31 is 85. So 30 could still be 85 or 84.
Uh, it it did things, but it didn't quite crack it open as much as I'd hoped. So, 3 4 6 7 into the rest of row two.
So, this is now.
And I've I've extended that line quite a long way, haven't I? I've got Oh, this was eight. This was the top rank two number 291 and then 347.
That's nearly good, but I don't think it's good enough. And 32 is 86, which is always bigger.
That's I'm sorry. This is probably obvious to everybody, but it's far from obvious to me.
So I mean another thought here which is a a mad thought but I might have to do it is do I have to pencil mark this thing? I mean I know all the digits on it are different. Yeah.
Oh nearly.
Okay. I was going to say where's two in row five?
two is in one of those cells in row five, but maybe where's two in column five is also relevant.
Can that be a two? That can't be a two.
That would be double one. So two is in one of those two cells in column five.
Um, the two wing cells of this tiny zipper, which I've paid no attention to at all, neither of those could be ones by Sudoku. So, they're at least twos. I don't know whether they can both be twos. They probably can, but that means this digit in the middle is at least a four. So, this is four, five. It's not 6, seven. It's four, five, or eight, actually.
Is that somehow I mean it is going to be important at some stage. I just don't know if it's now one. The first rank number is here, isn't it?
So if that was six, this would have to be No, if that's two, this would have to be a two.
I don't know.
27. If that was a two, it would begin 762.
And then 28, wherever that is, which I can't now find. Oh, it's there.
No, that's not there. That's nonsense.
20. Where's 20? Oh, 28's here.
Oh, these share. Okay. Yeah, this this shares digit. This digit. So, all you have to make sure is that that's well, if it is eight, then it would go to the fourth position as the tiebreaker.
It could get very it could very quickly force this to be an eight. If you can get rid of twos in a couple of positions in column.
Ah. Oh, that's it. Oh, I focused on the low digits and I'm so stupid. I mean, it's I mean, you're all going to be cross with me and I'm cross with me.
I've got an eight here. So, why don't I do the high digits on the blue line and note that in column five, where is eight going? Well, the answer is it's not on the blue line again.
So, it's in one of those two. And if I put it there, that would have to be a nine. And it definitely can't be. So, that is eight. And I didn't have to get rid of twos on the wings, but I got I got I did it a different way. Now, if that's eight, what does that mean?
I don't know. I'm not sure. Um, it's really actually that's very that's a little bit distressing. I I thought that would do a mountain of work for me.
Where's No, it's not a good question. I was going to say where's two in row seven, but I think it's got three options, I think.
And if two was there, this would be six.
which is which is all very well isn't it?
Okay, this digit let's look at this digit this cannot be it could be two it could be three it could be four could be five it can't be six it can't be seven so it's 2 3 4 or five which means it's complement here is six 5 4 or three now can we get see I can't get rid of anything that is so unbelievably unfair I claim it's unfair. Okay. But now now we ask now we do ask where two is in this column because that does seem to now be resolved. So the two does have to go there.
And I mean how can that not do anything?
Oh maybe it does something to the ranks.
Yeah. Okay. 128 is the second biggest digit. So the first big biggest digit must begin one two. That knocks two out of this cell. Ah, it gives me a six here and that's on the zipper. So that becomes a three. That becomes a seven.
This is not Oh, this that's naked now.
That's a four for some reason.
I don't know how long that's been available, but that's that's terrific.
That that I feel better about that. At least I got something done.
I suspect this sip has more of a role to play. Um, but let's not get carried away. Nine. Can I do I have got quite a few nines in the grid.
Can we do better with nines?
Nine is obviously a very potent digit so far as all sorts of things are concerned.
I don't know actually. H 27. That can't be a Oh, that obviously can't be a nine. You silly man.
Can that be a nine?
25719.
Where's 26?
Uh, I don't know. Up up up at the top there. So, that would have to be bigger than 719.
Oh, but it could be cuz that could be a nine on the end of its thingy. Oh, bother.
Okay. But we did just work out we worked out that this was 76, didn't we?
But that but it's sharing a digit. So that's less relevant. Maybe this three matters then.
So I've got from I've got 2 913 going up.
21937 is what I've got. So this this if this was three that would have to be seven. I mean yeah that okay that that can't be three because this line would is going 2 91378.
So if this went 2 1 2 2 2 913 we'd have to beat 78 with the next two digits and the best we could do would be 76.
So that digit is more and that's affecting the one clue going down which now begins 1 four.
So position two begins one two. Position one begins one two. Position four needs to beat one.
Whereas position four it does beat it already.
But if that was 17 that might then it might matter.
Okay. So, we can't do that, I don't think.
Can we say anything sensible about these positions? I mean, I'm going to pencil mark them just because I'm getting to the point where I'm a bit desperate. So, they are three, four, five, six, seven.
Now, that's not four. That's not six.
And that's not four or seven actually.
So that's three, five or six. One, two, three, four. That's the fifth position.
So the five is going either in position three, that one. Position five, or this one. So the five is not going in position seven, which is that one. I wonder if that's how we have to do this.
We have to whittle away.
Yeah. If this was three, it would be say right a seven there. That can't work.
So, this is now right down to six or seven.
It's indexing the seven. So, it's either indexing itself or it's putting a seven there.
Well, okay. But that means the seven is in one of those two positions. So, it's not above it. I mean, it it might be that this is what we have to do. Now, this is position six.
Now, can we put a six in position three?
Yes. Can we put it in position four?
Yes. Can we put it in position five?
Yes. And can we put it in position seven? Yes, apparently. Bobbins.
Ah, but but where's four on this line?
That four is quite cool, isn't it? I hadn't seen this.
Four is placed. I hadn't seen that.
Okay. So, four goes there. Now, that's five.
That's Oh, that's 36. That's the biggest number begins 95. So this doesn't begin 96. That's definitely smaller.
Okay. And that four 1 2 Hang on. That four is in position six, isn't it? 1 2 3 4 5 6. So we now need to write six into position four. 2 3 4. So that's a six.
So this is not six. We can't repeat six on the line. So seven indexes itself.
That becomes an eight. That's that's the 14th number 48. That seems unlikely. Is that really true?
Where are the fours around the perime?
Oh yeah, look. Oh, but 15 is 482.
And 14 has got to be lower than this. So that has to be two. And that's placing two on this zipper down here. And that being on the zipper makes this a six.
And now that's a four. So that's a three. That's a five. That is Don't tell me it's in position five. Of course it is. Why did I expect anything different?
This is three or nine.
And either it does mean we've got a 39 on the zipper, which is still great. I'm very happy with that.
And this. Oh, you rotten digit. You rotten naughty digit. That's doesn't work for the row. Wow.
Okay. Well, that's I'm still delighted.
That's still enormous progress compared to where we were a couple of minutes ago.
So, 33 has got to be less than 937 and is.
And 35 has to be bigger than 937. Oo, and it's got a three in its second position. So that digit is at least a seven. And it is therefore a seven.
So that is not a seven.
This digit is four or five. So the six in the column is now placed. Six 7 four yeah five four by my sedoku I suppose five here this 937 this last digit here has got to be bigger it's got it's got to be bigger than 93745 937 so it would be even if this is fourh that's a shame Um, but we we're still we're still in the game. We're still in it. 3 4 9 into the row into this row here. Yeah, that this digit's doable.
That's a four.
So now it's a three nine. But oh, hang on. Where's nine in the row? Nine.
That's got to go there. That's got to be three.
So column seven.
3, five, six.
That's naked.
63. I mean, the fact that the Sudoku is working is something of a It's extraordinary, isn't it? That's a seven in this column. What do we need?
Three 38. I haven't put eight in. So, eight's got to go there. So, this is a three or a nine.
I've got a 39 deadly pattern. So that so something about the ranks is going to have to fix that. In this row I've not put four in.
In this column I've not put something in. Five is it? Yeah. I mean this is fantastic because I think the stoku has actually worked. So, it is only a matter now of working out which one of these ranks is determining the order.
Right there it is. So, I'm going to look at this digit here. Now, the but by by the fact that this is a sort of deadly pattern, I know those two digits are the same digit. These two here, they are definitely the same digit.
If this is nine, then that's nine. So let let's look at the rank here. We've got the four rank or this one. This four and this three is what I'm comparing. 1 7 something six.
Oh no, that's not it. That's a four is a different digit to seven. You absolute muppet. So it's not that. Sorry. Uh hopefully something hopefully it will be something. It's going to be some sort of logic like that, isn't it? Um, probably where it's got to be. But where is it? Is it the two four?
I really like the idea. It was those two. I have to say apparently it isn't.
Um.
Oh dear. Okay.
14. Is it this one? 482 something.
So, where is where's 15 or 13?
No, they're way over there.
Oh, but 15 is 4826.
So, this So, 14 can't be 4829. That's that's that's very clear. That would be bigger than 15. So that's three.
So it's a bit simpler actually than I'd realized. And but that get rid of that there. I think that's the solution. I mean, it felt logical, didn't it? It's the most incredible puzzle.
Wow. Oh, I'm delighted. I'm so thrilled about that. I'm thrilled to get through it. How long does it take? 2 hours 10 minutes. 2 hours 11. Well, I mean, I do think that was re really, really quite difficult. It was very difficult to start. I couldn't get my head around it at all, but I have managed to wade my way through congruity. What an incredible mind you must have, Andrew Sarcus, to set that. That's just it's a beautiful idea, but you've got to be barmy to set that. Absolutely bonkers.
Let me know in the comments how you got on with it. I enjoy the comments, especially when they're kind. And we'll be back later with another edition of Cracking the Cryptic.
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