In Region Sum Line Sudoku, the fundamental mathematical principle is that each row, column, and 3x3 box must sum to 45 (the sum of digits 1-9), and when box borders divide blue lines into segments, each segment must have the same sum. For a line divided into three segments, each segment must sum to 15 (45 ÷ 3). This constraint allows solvers to deduce digit placements by analyzing segment sums, box sums, and row/column constraints. The perimeter of a Sudoku grid can be calculated as 4 × 45 = 180, but corner cells are counted twice, so the actual perimeter value is 180 minus the four corner values. Solving strategies include: (1) identifying segment sums through box and row constraints, (2) using parity and divisibility by 3 to limit possible digit combinations, (3) analyzing wing cells (edge cells) to determine their values, and (4) systematically eliminating impossible digit combinations through logical deduction.
This Sudoku Has Many Lines!!追加:
Hello and welcome to Friday's edition of Cracking the Cryptic second edition I should say cuz there was a crossword video earlier on but this is not a crossword video. This is the Sudoku video and we're going to be doing another puzzle by the great Nikolas Du Hayle incredibly prolific constructor and just a wonderful constructor frankly. Every single one of Nikolas' puzzles I've done on the channel I've absolutely loved and that we continue to be sort of deluged by recommendations because Nikolas produces so many puzzles.
This one is called co-terminus and again this is it's just region sunlines so don't worry if you don't know what they are I will explain all when we do the rules and but you can see basically the entire actually that cell isn't in there but basically the entire grid is covered in region sunlines.
So this is what we're going to be working with it's got three stars out of five difficulty and apparently it is wonderful so I'm looking forward to having a go at it. Anyway we'll do the rules together in a moment or two's time.
What can I tell you about before that?
Let me think.
Actually there isn't too much is there?
Oh.
There is one thing though.
Um on the blueprint stream for this coming week is on Sunday so it's in two days time. It is not Monday because we are doing a slot as part of the cerebral showcase which is a big event over on Steam and they've asked us to do a blueprint stream as part of that so because that what the cerebral showcase I think it runs until basically the time we'll finish streaming on Sunday. So, we're going to be the final slot in it.
Um so, I know some of you have been following um that series where we've been going into the Blueprint's Mansion basically every week for over a year. Um so, please remember it's not going to be on Monday this coming week. It will be Sunday night at 10:00 p.m. instead. Um other than that, over on Patreon, we've got all our bonus content over there.
So, please do think about um supporting us. We've got a brand new Sudoku competition starting on the 1st of June, which is which is only what, a few days away. Um and I've got a birthday to do today as well.
Oh, Mr. I don't know your first name. Mr. Fleming. Mr. Fleming is a maths teacher and is turning 25 today. And Mr. Fleming, I have to say, um you are the first teacher that we have had um we've had requests for a birthday shout-out from two two of your students, but it wasn't the same email. This is independent emails that we've had from two of your students. So, from Wilson and from Brecon. Um and apparently you're an amazing and confident person.
And they really wanted you to get a shout-out on the channel. So, I think that's fantastic. And Mr. Fleming, I I'm sorry I don't know your first name.
Hopefully someone will enlighten me at some point. Um but thank you for watching the channel and um yeah, thanks for being obviously uh a bit of an inspiration to the kids that you teach. Uh I hope there is some chocolate cake in store today.
Uh but that's all the news. Shall we have a go at Coterminous by Nicholas du Hayle. The rules are not going to take long to read, but they are as follows.
We have got normal Sudoku rules applying. So, we have to put the digits one to nine once each in every row, every column, and every 3 by 3 box.
And then the box borders divide each blue line into segments with the same sum.
Um so, let's look at that one. So, those two, you can see that this blue line here, the 3 by 3 box borders cut this line into segments.
And each of these segments has the same sum. We don't know what it is. We're going to have to work that out. Um but that is all they are all the rules. They are all the rules. 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.
I mean, I can see one thing here straight away.
It might be to do with the perimeter, but the thing the thing I'm seeing straight away is is column column one and column nine.
Because there is a Sudoku secret we can avail ourselves um something I share with only my very favorite people. But if you're watching this video, you're definitely one of my favorite people.
So, I will certainly share my secret with you. The secret of Sudoku is that every row, every column, and every box of a Sudoku, because of the rules of Sudoku, contains the digits one to nine once each. Now, that's not the secret, but it means that this column, for example, contains the digits one to nine. Now, if you sum the digits one to nine, this is the secret, you get 45.
But we're told that this line Well, well, we're told that let's see let's just divide it a little bit. Um we're told that this segment sums to the same as this segment, sums to the same as this segment.
So, if this if these indigo cells sum to X, these green cells would also sum to X, and these purple cells would sum to X. So, the line sums to 3X.
So, 3X is equal to 45. So, x is equal to Mr. Fleming could tell us 15.
So, that means that each one of those segments is 15. The same is obviously going is true in column nine as well. Um I'm not not going to keep the coloring in cuz I'm not sure it serves much purpose, but so, we just know they've got lots of 15 segments here.
Now, what I'm wondering about is whether it's to do with the perimeter of the Sudoku.
Now, the perimeter of any Sudoku using the secret we can define the perimeter as being four lots of the four it consists of row one, row nine, column one, and column nine if we were to include these cells twice.
Um because obviously this cell here is in column one and it's in row one. Same is true for this. Same is true for this.
Same is true for this.
So, if we were to view the perimeter as being defined in sort of whole rows and whole columns, it would be four lots of 45, which would be 180.
But, in in using the number 180, we would overstate the value of the perimeter by these corners because they would have been included twice. So, we'd have to deduct out the values of the four corners to arrive at the value of the actual perimeter.
Now, let me just think about that for a second. So, I know that's 45. I know that's 45. I know that is Mhm.
Let me I don't I don't think I do know how to do that.
I can do something different to that.
Um, I can Well, if we look at we look at row row one. It's the same is going to be true of row nine, but let's look at row one. We can We've got a line here.
And I've got no idea what this line sums to.
But what I do know is it sums to a number that is a multiple of three.
Because if this is X, that's X, and that's X. So, that line sums to 3X.
Now, we can we can think about the whole row the whole of row one sums to 45, which is also divisible by three.
So, if this line if this indigo line is divisible by three, and the whole line the whole of row one is divisible by three, then these two green cells, the sum of those digits must also be divisible by three.
But unfortunately, there are there are quite a lot of options, aren't they?
There's a minimal option, which would be one and two.
And then this line would sum to 42.
Which would be 14 for each segment.
But the the maximum size of a segment here could be 15. It couldn't be 18 because we can't make two digits sum to 18, but we could certainly make two digit digits sum to 15. They were seven, eight, or six, nine. And then this this line would sum to My phone is buzzing at me. Um, this line would sum to 30, and each segment would sum to 10.
Um, okay.
So that's So the So each segment on this line sums to a minimum of 10 and a maximum of 14.
I don't think we can do that, um, either.
Uh, sorry. I can't quite see what to do here. I don't think I don't think it it can be that complicated.
Because it's a three-star puzzle.
It's not going to be It's not going to be something Oh, hang on. There's another cell there that's not I didn't see that. Oh, so there are Hang on. There are There are several cells that are not on blue lines. I thought there was only this one.
No, there are Okay, I think there are two.
So those two cells are not on blue lines.
So we can't We know they're 15.
I'm wondering if it's some sort of parity or division by three, uh, idea.
Because I mean that line that if this is X, that's X. So that line is sums to 2X. So that is an even line.
That is an even line. That is an even line. So these cells definitely sum to an even number.
But the problem is that a number that's divisible by three could be even or odd.
For example, 9 and 18.
I'm not getting this yet. Sorry. Um Is it going to be Mhm, well, okay, I can I can limit the I actually know the value of that domino.
Right. Okay, so maybe that's a better strategy rather than me trying to do something holistic.
Maybe what we do is work with our 15 segments.
Let me just think about this one for a moment.
I think that's the same as that one actually. So, okay, so now I'm switching. I'm looking at the box.
So, that box we know from the secret sums to 45, but I know these three cells sum to 15, so I know those cells sum to 30.
But those cells are the exact same sum as their equivalent segments in box two, aren't they? So, those must sum to 30, which means these must sum to 15, which means they are either 6 9 or 7 8.
Um, let's try that the other way round then. So, those sum to 15.
Oh, no, that doesn't work. I was about to say. So, so these sum to 30, and then I was going to try and do some trick in this box, but this one sticks out in the wrong direction.
That's a very annoying um little dog leg triomino there.
Okay, let's not do that then.
Let's try this one. So, these Yeah. So, these ones sum to 30. Same logic, these must be 15, the whole box is 45, so these sum to 30, so these sum to 30, which is their equivalent segments in box five, so these sum to 15. So, these are 6 9 or 7 8. So, that's they're different now.
Which means the rest of this column is all low digits. Well, relatively low, ones, twos, threes, fours, fives.
Ah.
Right. Look here.
That segment is sums to 15. So, these cells sum to 30. So, these cells sum to 30. Whoopsie.
Those ones there.
So, this sums to 15.
Now.
Now.
Um These sum to 30.
Okay, and those sum to 15.
Which means those three cells sum to 15.
Which I don't see how to do that. Um Oh dear.
Uh okay.
But I don't think we can do anything with this. Let me just think about that.
So, I'm saying that that that's That's 45 minus that. So, these are 30.
Ah, those are 15. Okay. Okay, hang on.
Yeah, so I see. Now, that that does do something. Yeah. So, these add up to 30.
But these, remember they added up to 15.
So, we can cancel those out, and they add up to 15.
So, now No, they they don't add up to 15. Those two add up to whatever that one is.
So, this one That one's 15.
That one is 30.
But, I don't know what that one is.
Let's say that was six. If that was six, 30, these would be 24.
But, that would be relative Actually, hang on. Hang on. Do you have to Okay. No, no. I I don't really know what I'm doing here. I'm trying to mess about. Um Th- These two, let's assume they could be double six. I They might not be able to, but let's just just say that's clearly the minimum they could be. So, the minimum value of those would be 12, which means the maximum value of this is 18. That's too big. That's definitely too big because um the maximum they could be really would be an eight nine pair, which is only 17.
So, if this was six seven on the other hand, I'm sure Mr. Fleming has heard that a lot in his classroom recently.
Um they would be 13. This could be 17.
Oh, no. Ah, tell you what, that can't be 17 for a different reason.
Because if this was 17, then this should be 17 as well, and that would also have to be an eight nine pair.
So, that So, in fact Ah, hang on. Hang on. Hang on. I've had a different idea.
Or did we know this already? The maximum size of that is 15.
Cuz that could be 15 and then that could be 15.
Oh, no, but then that would be 15. That would be 45. Oh, we did this work earlier. 15 wasn't possible. It was 10 to 14, wasn't it?
So that That can't be That can't be 10. There's no way this can be 10.
And we can see that very simply.
If this was 10, these have to add up to 20.
And they can't add up to 20 because even if these are double nine, they'll only add up to 18. So 10 is not a valid option for this.
11 would make this 19. That's still too big. So it's 12 It's 12 would work, wouldn't it? Because if this is 12, that could be 18 and that could be double nine, I think.
So Okay, so the segments in row one either sum to 12, 13, or 14. So it is getting more restricted. And that means that the edges of row one either sum to nine if these add up to 36 from three 12s.
3 * 13 is 39. These would add up to six.
And 3 * 14 is 42 and these would add up to three.
So the So these two digits here, the wings, if we can call them that, of row one, are not huge.
Neither of them can be a nine.
Because the maximum sum of those two digits is nine.
Right. Okay. Now I'm going to look at that cell because that can't be six.
And that's going to push up this one, which is going to put even more pressure on this.
Can that be seven?
No. Right, here you go. Here we go.
Let's look at this digit. That can't be six, because then these would be a 1 2 3 triple, and how would we make this domino add up to six? It couldn't be 1 5, it couldn't be 2 4.
If this is seven, this triple has to be 1 2 4.
How would we make a domino add up to seven if you can't use 1 2 or 4? You couldn't be 1 2, couldn't be 1 6, couldn't be 2 5, couldn't be 3 4. So, that's not six or seven. Now, if this was eight, then then it's 24 overall. This couldn't be 1 2 5, because then there would be no way of making this equal to 1 7 2 6 or 3 5. So, if this is eight, eight, it would have to be 1 3 4 and a 2 6 pair.
If this is nine, nine will be easier, because we can only knock three pairs out with if this is 1 3 5 or 1 2 6 or 2 3 4, you'll knock out three of the four possible ways of the dominoes adding up to nine, but you won't knock out all of them.
But, but we did know, didn't we, that these two digits sum to 15. I'm sure that that is something I had in my brain.
Yes. Yeah, that's correct. Because this is 15, that was 15, so they add up to 15. So, this digit is coming down. Now, if this digit comes down, what's it doing to the value of this? It's pushing that up, because the maximum value of these two cells now is 9 7, which is 16. Yeah, and that's done it.
So, nine seven here is the max, which means the minimum of this is 14, but that's also the maximum, because if this was more than 14, we worked out that this line would add up to 45 on its own without including the wing cells, and the whole row, remember, adds up to 45.
So, so we found the sort of equilibrium.
It's not that easy, this. So, that's seven, that's nine, this adds up to 14, this adds up to 14. So, they those digits are five, nine, and six, eight in some order.
And the whole of this line in row one adds up to 42, which means the wing cells add up to three, so they're a one two pair, and this is three four seven just by Sudoku.
And this is an eight.
Because they have to add up to 15. Now, that's not eight, so that's not six.
Um Do I know what that is? That Okay, that they were they added up to 15, didn't they?
That seems that seems sensible.
Uh no, that might be wrong. That might be wrong. Hang on.
No, they did. They did they did. From that's from this box.
That's 45 on the box minus the 15 on that segment, so those three added up to 30, those three segments, which means these three add up to 30, which means these do add up to 15. So, that's a six.
This is a seven eight pair.
So, this is not a seven.
And those digits are quite restricted now. 1 2 5 8 I want to say.
Now.
Uh So that digit has to be Actually, I'm not sure.
I was Ah, no, okay. It's not a 1 2 pair.
It's not a 1 2 pair because it was a 1 2 pair, this would be a 3, and this 7 can't couldn't work. In fact, maybe I should pencil out this first cuz it can't be 1 6. So it's either 2 5 or 3 4.
2 5 or 3 4.
Oh, hang on my I was thinking 2 5 or 3 4 and I'm not looking at those digits.
Right. Okay, but this can't be a 1 2 pair.
It can't be a 1 5 pair.
Now, it could be a 1 8 pair.
Probably, if this could be a 9.
Okay. So So 9 is possible here.
If there's a 2 in this sequence, it would have to It couldn't be 2 8, could it? So it would have to be 2 5 adding up to 7.
And it must have Okay, it must have a 1 or a 2 in it.
Because if it It can't be a 5 8 cuz I can't write 13 into there.
So this has got one of one and two in it. This has got one of one and two in it.
>> I think this has to have eight in it.
If this didn't have eight in it, this would be a five and a Well, the maximum sum this could have would be seven.
5 + 2 And then the but those you'd have to include one and two in a three cell sum adding to seven. And if this added up to six, it would have to include one, two, three and the corner would break. So, this must have eight in it.
Which means this doesn't have eight in it. Which means this can't add up to nine. So, this adds up to seven. It is 2 5.
This is now 3 4.
This seven can go in big digits.
This adds up to nine.
Um What does this mean? Seven.
No, it doesn't. Oh, I was hoping I was going to get my seven in my green over here, but I don't think I do get that.
Um Uh what How does this add up to nine then? It can't be 1 8. It can't be 3 6.
It can't be 4 5. That is 2 7.
Right, that's good. So, I get one in the corner here, two in the corner here.
These digits add up to Oh, they add up to 14. So, they're just they're just whatever this one isn't.
So, this is either 5 9 or 6 8. It's just the opposite of that. And it's going That one's going to be the same as that one in row one.
This can't be nine, so this can't be five, and this can't be six, so that can't be eight.
Uh okay. Where is one in row three?
It can't be here.
Because remember this segment adds up to 15. So, if this 1 + 2 is only three, this would have to be a 12. So, one is one is there.
That's quite interesting because now this domino adds up to eight because we know we've got a one and an eight on this side of the balancing of the scales and a one plus these two digits on the the other side of the scales. So, this has to be 3 5 now cuz it can't be 2 6 or 1 7, which means that the 14 is 6 8, which means these are 5 9, this is 6 8, and I know the order.
And now we might get somewhere. Four has to go here, so this has to be nine by Sudoku. 2 4 and 9 do add up to 15.
8 7 is lovely there. We get some more work done.
And Okay, so seven in column nine is down at the bottom, which means nine in column nine must be in this sequence. How do we know that? Well, obviously we if we were to put the seven and the nine together, they would already add add up to 16. And that that segment is only meant to add up to 15 altogether. So, the nine goes in this segment along with two digits that add up to six. It's not 1 5, so that's got to be 2 4 9. And now this has to be 3 5 7.
Just to complete the Sudoku and all the math seems to be okay there.
So, these digits now 1 3 5 6.
Uh which Let's check the maths on that.
That adds up to 15. They add up to 15.
So, so far we're balanced. Ah, now this is an eight. That was restricted in the row.
Cuz we couldn't use 1 2 5 here or this domino would break. So, this is 134. You can totally see why Nicholas is so popular. The He's just a brilliant constructor. This is now a nine.
By Sudoku and that digit's a free five look. So, these add up to 10 and they're not 28946 or 19. So, they are 37.
And now the bottom of the column has to have 168.
Now, can we do better?
Don't know.
I know there's an eight in one of these three cells. That's just Sudoku doing its work. There's a two in one of those.
Now.
Um I think I better look at these then.
It's a 2569.
We can't put nine. Oh, Maverick's flying over. Can't put nine on this cuz we're adding up to a single digit total.
In fact, one of those has to be a two we can immediately see. 5 + 6 would force us to write 11 in here which won't work.
So, this is either seven or eight into this cell.
Which is okay. So, it's the sort of It's the opposite of this digit. There's a Oh, well, there's a seven eight pair in row six which makes this a three.
This is seven. This is seven. This is an eight. So, this is an eight which tells us that now this is 26.
It's beautiful. Now, that's not got six on it. This is a four. This is a two.
This is now definitely a 14 and those are 1 95.
No, that's not right. 194. I was thinking the maths on that felt wobbly.
Oh, one. I can put the one in.
So, this is 4 9.
That's 5 9. So, we can do the 9. We can do the 4.
I don't know whether I'm just being lucky here, but it just seems incredibly smooth so far.
Well, although it did take me a while to get going, didn't it? But since we've got going, we just we've been very fortunate.
Ah, probably shouldn't have said that because now I can see that I'm about to get stuck, aren't I? Do I know what that is?
I know Okay, I know they add up to 15.
So, I know these add up to 15. Those add up to 15. That's 30. So, these add up to 15 to get us to 45.
Mhm, not sure. Maybe we should pencil mark column two.
2 3 5 7 are the options.
And column three, 3 4 5 9.
But those two Well, we don't know, do we? We don't know what's going on in the bottom row.
Because Well, we know these two have to add up to a number that's zero mod three, i.e. divisible by three.
Because this this thing in the bottom row is in three segments. So, that's divisible by three, and the whole row is divisible by three.
But I can see loads of ways of that working. 8 7 5 1 6 3 Um Let's try Um maybe this digit.
If that was It can't be two, actually.
So, you can't write double one here.
Ah, hang on there. 15 Oh, no.
I was thinking maybe we have to have a nine in here.
But it But if that was seven, we would be fine, wouldn't we? Then we only need eight more. So, 735 would work. Oh, no, that wouldn't work there, though.
Hang on, then. We might need a nine on this line. Let's think that through again. I think I better think it out again.
Uh job I'm getting possibly. I wonder who my boss will be. I wonder if he'll talk to me.
What bonuses he'll make to me. I'll start at eight to finish late at overtime and overrate. I think I'll better think it out again. I think I got that slightly wrong. Um anyway, let's think again, Fagin.
Um Yeah, how do you make that add up to 15 without nine? 3 4 5 isn't enough. That's only 12. So, we're going to need the seven. Which means these have to add up to eight. Which means I think looking at the options, they have to be a 3 5 pair, which would break this digit. So, there is a nine in here.
Um which means there is no seven, because that would take you that would blow the total. And we need two digits on here that add up to six. And looking at the options where we haven't got one available, so we can't use 1 5. We can't use double three. So, we are going to be using 2 4.
So, this sequence So, the two is necessary. We've got to put two there.
And this is 4 9.
Which somehow Uh that one is not four, right? So, this is three or five.
So, the Oh, so the right. That's kind of right. That must do something to this row.
Cuz the maximum that can be is 12. Now, 36.
Um Ah, I'm sorry if you can see this. Hang on.
So, Okay, let let me just think through the options for those purple cells.
I think there were three options. Eight is two mod three.
So, that needs to go with a one mod three digit. The only one available there is seven. So, seven eight is a combination that obviously add up to zero mod three, 15.
And that would make the rest of the row add up to 10 Well, 30. Divide that by three, 10. So, we'd have 10 here, which could be 3 7.
3 7.
What were we saying? Oh, no, no. 3 7 won't work cuz that we start from the premise that's 8 7.
Yes, there's no way to make that add up to 10. So, it's not It's actually not 8 7. We've managed to get rid of that.
Well, go I've got rid of eight from here. I'm reluctant to take the seven out. Now, I can take the seven out cuz seven is one mod three, and there is not two mod three digit there. So, it is fair to do that. Now, if this was six, this would have to be three to make these two work. So, they would add up to nine, which means these add up to 36, which means they Each domino adds up to 12.
Or that's 12, that's 12, and that would be 12 as a triple.
But that could be 5 7.
This would have to be 4 8 cuz it couldn't be 3 9 cuz there would be 6 3 in the wings.
Mhm, might work. So, 6 3 1 5 is the alternative. 1 5 is 6, which means the rest of the row adds up to 39, which means 13. No, you can't make that add up to 13. No way. Right, so there is only one way for this to work is with six in the corner, and you've guessed it, that's three in the corner. That's three in the spotlight losing its religion. Um so, 6 3 So, these add up to 36. So, each segment adds up to 12. So, this has to be 7 5.
So, that's five, that's three, that's three.
This has to add up to 12. It's not 7 5, it's not 3 9. So, that's 8 4. That's two by Sudoku. Now, it's just unwinding our pencil marks.
That's four, that's three, that's three, that's six.
This column This digit is one, while that digit is 1 5 or 9 by Sudoku, but it's definitely not one cuz it's the sum of those two. So, that's five or nine, which means the one in the column goes here. That's going to unwind more, you know. Five and the nine therefore get resolved. That becomes a nine, which is the least profitable thing we wanted, but it's still a thing.
Four is on this line.
This is a 1 8 pair. That Oh, no, that's not that can't be right. I must have made a mistake.
That can't be right, can it?
Cuz I can't have a 1 8 pair. Oh, no, it's not 1 8. That's a relief.
Cuz I didn't see how I could have a 1 8 there and a 1 8 there. I didn't think there would be anything about the logic of the puzzle to resolve it, but I realized this this 8 pencil mark is an anachronism. When I put this 8 in, I could have removed it.
So, this is a 1 6 pair actually. That's that's actually something of a relief.
So, this Oh, so this line adds up to nine.
So, that must be a 4 5 pair.
So, this is a seven. This is a five.
This digit is not seven or nine anymore.
The way that this domino adds up to nine is not 4 5. It's not 2 7.
So, it's either 3 6 or 1 8.
So, this is a six or an eight, and this is a one or a three, and that's a 1 3 4 triple in the column, which makes this a five. Good grief.
And that we can finish this column now.
Well, we can at least write the two into it. It's got to go there. Now, what are these digits? One and something. One and nine actually.
And they do add up to 12 with the two.
So, that makes sense.
If this was six or eight, those had to add up to 15, didn't they?
So, this has to be odd. So, that's got to be a seven.
Which makes this a seven.
Now, how do we finish it?
I mean that might be presumptuous, but it does feel like we're close, doesn't it?
Working this out now.
They added up to 15, so that is an eight.
So that's eight, that's one.
That eight sees that cell, which forces it to be six. So six, three, six, one, one, eight can all go in. Six, two can go in. Two, five can go in. Five, nine can go in. Good grief, we're there, aren't we? Well, if I haven't made a racket, we are we are we are there.
Um this is a three. Three, four, four, one. What a beautiful puzzle again.
He is so talented, this constructor.
It's just beau- I hope you enjoyed I did enjoy it. Um just super talented. Really cool idea, beautifully executed.
Quite mathsy, so hopefully Mr. Fleming enjoyed it. Mr. Fleming on your birthday.
Um let me know in the comments how you got on with the puzzle. 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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