American Math Teacher's New Sudoku Idea

Added: 2026-06-02

[00:00:14]Hello and welcome to Tuesday's edition of Cracking the Cryptic, where if yesterday was hot, today is even hotter.

[00:00:21]And I've just had to go out and do a load of chainsawing. This is just terrible. Um, but but I have returned indoors.

[00:00:31]Um, and it it's not cooler in my office at all. I've got the fan on today. There might be a tiny background noise compared to usual, but I I really couldn't um I don't think I could manage to record without the fan being blasted on me. So, I apologize if there is a tiny bit of background noise that's a bit different to usual. But we're going to be having a look today at balanced x sums, which is a brand new puzzle by the American math teacher Rocky Row. Rocky Row, who first featured on Cracking the Cryptic with a puzzle, I want to say in about August 2020, so almost Rocky's sixyear anniversary of first appearing on the channel. Um, and well, I think this is a new puzzle, but it's we've already had lots of emails saying we have to have a go at this. Um, apparently it's got a little bit of sort of arithmetic in it. Nothing monstrous.

[00:01:23]Um, but the the feedback that we've been receiving says if you if you like a tiny bit of arithmetic, this is an absolutely gorgeous puzzle. Um, and it's got three stars out of five for difficulty. Um, so we'll read the rules. You can see it's a it's a very sparsely populated grid today with just some arrows out. So, can I click? No, I can't click on the arrows. Um, but just some arrows outside. Um anyway, we'll read the rules of this one in a moment or two's time.

[00:01:50]I've got some things to mention before that. Let me start by saying many thanks to those of you who joined us last night when we streamed Blueprints again for the 46th time. I want to say um and uh I won't spoil what happened. Uh I do think it's probably worth watching if you if you have been following our journey and you missed the stream. So I will try and remember to put a link to to the VOD um on the screen now. And um and a reminder if you are following this series, next week's edition is on a different day. We are streaming on Sunday night at 10 p.m.

[00:02:28]Not Monday night at 10 p.m. And the reason for that is that Steam have asked us to be part of what they call the cerebral showcase, which is where um some streamers basically uh stream, you know, games that involve puzzles uh over a whole weekend. Um, so we are going to be closing the show, if that's the right way to describe it, at 10:00 on Sunday night. Um, so, um, if, as I say, if you do enjoy the blueprint streams, um, please don't be disappointed. Don't be disappointed and and make sure you you note it Sunday night this coming week.

[00:03:07]Um, now, what else can I tell you about?

[00:03:09]I would like to mention the winner of May's competition. Mark has used his random number generator and tells me that the winner of May, drum roll please, is Michael Straniieri.

[00:03:26]So, Michael, I hope you're watching this. Um, I will be writing to you to see whether or not you might be interested in coming onto the channel and solving a puzzle and hopefully appearing in a video. Um, very very well done. Um, very well done to all of you.

[00:03:42]There were hundreds and hundreds of you who managed to solve all of May's competition. I can tease um June's competition obviously with the the release in the US of all the classified files related to alien life on Earth. Um we've been uh going through those files and we've noticed some interesting interesting puzzleike uh content. So, um there's going to be an alien discovery type theme um based off those uh definitely the real US files um in coming on the 1st of June. So, do get involved. Make sure make sure you're uh a patron by then and uh that that competition will go live at 400 p.m. on the 1st of June. Um now, what else? I would like to say thank you to Preston.

[00:04:33]Actually, have I got Preston's um let me just see if I can find it here.

[00:04:39]Preston's email, which was very kind.

[00:04:41]Let me just I have I'm not very well prepared for this. Um but Preston wrote to us from across the pond and to say that I think I think Preston has now been watching the channel daily for 27% of his life, which is quite a lot. Um uh even going back to the dueling dragons video that was Icy Fruit, wasn't it? Which, um well, I'm I'm not sure I can say anything about the video, but I do remember the puzzle.

[00:05:11]That was an extraordinary puzzle by the by the rarely seen Icy Fruit. Um Icy Fruit, if you ever watch the channel and you're watching this one, I'd love to see another puzzle by you, my friend.

[00:05:24]Um, but anyway, Preston Preston just wrote a kind email saying that the channel provides um a good laugh occasionally at our expense and some mental exercise and um yeah, it was it was one of those emails that made me smile. So, I wanted to shout you out, Preston. Thank you for that. I have no birthdays to do today, which is most peculiar uh on a Tuesday. Um, and I think that therefore is all the news, which means that we can have a go at doing some solving. Shall we have a go? A balanced x sums by Rocky rower. What do we have to do? It looks like normal Sudoku rules applying. So, we're going to have to put the digits 1 to nine once each and every row, every column. Whoa.

[00:06:06]And every 3x3 box.

[00:06:09]Now, balanced x sums. Each arrow indicates that the sum of the odd digits must equal the sum of the even digits in the first X cells seen in the direction of the arrow where X is the digit in the cell next to the clue. So what does that mean? Let's put five in there and see if we can understand it. So five would be saying look at the five cells nearest to the clue.

[00:06:40]Okay. And the sum of the odd digits in those purples would equal the sum of the even digits in those purples.

[00:06:51]So that's quite peculiar, isn't it? Um I'm trying to think of a way of doing that so that it works. I think you could put seven in 2 4 6 something like that.

[00:07:03]2 4 6 is 12. 5 and 7 is 12.

[00:07:08]I think that would be a way of completing that clue. I'm going to claim and they No, I can't scroll. So, that is it. 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.

[00:07:27]Now, I say let's get cracking and I have absolutely no clue.

[00:07:36]Do we think it's going to be do we think there's going to be a constraint where arrows point at each other?

[00:07:49]Okay. All right. One one trivial thing we can say is that using the secret we can none of these arrow clues can have an actually this might be important.

[00:08:00]None of them can have a nine in their first position.

[00:08:06]And that's because um now this is a secret I share with only my very favorite people. I'm sure Rocky rower knew the secret already. Um and if you're watching this video, you're definitely one of my favorite people. So I'm going to share the secret with you.

[00:08:17]The secret of Sudoku is that all nine digits in any row, any column or indeed any 3x3 box. We know the sum of them because we know that any row, column or 3x3 box contains the digits 1 to nine once each. you sum those digits, you get 45. Now, one of the things that I can gro immediately from this rule set is that because the sum of the odds is going to equal the sum of the evens within any x sum clue, the total must be even for that clue. It must be because you're essentially adding x's worth of odd digits to x's worth of even digits. So, the sum of any x sum clue is 2x which is even. So 45 is odd. So 9 is impossible in any of these any of these arrow clues.

[00:09:09]I mean one is trivially impossible as well because obviously if we put one there we're saying that within that one cell the value of the odd digits which is one is equal to the value of the even digits which is zero. So that's definitely wrong.

[00:09:25]Two won't work either will it? Maybe we're going to have to actually work through this and see how it two won't work because this would have to be a two to but it just wouldn't work. Those two have to have well the odd digits in those two cells have to equal the even digits. It's just nonsense.

[00:09:46]Now what about three? Does three work?

[00:09:49]Three.

[00:09:54]Right. Okay, here is another principle that we can establish.

[00:10:03]The sum in in any let's just delete three for a moment. What whatever the length of the x sum is, there's going to be some odd digits in the x sum. Now, there's also going to be some even digits. What's the sum of the even digits going to be? Well, we might not know the quantity, but we know the nature of the sum. Because if you sum even digits, what do you get? you get an even number. So how do you create an even number out of the sum of odd digits? The way to do that is that you must have an even number of odd digits.

[00:10:37]If you had three odd digits and you added them up, you would get an odd number. But if you had two or four or six or eight, you would get an even number.

[00:10:48]So each x sum must have an even number of odd digits in it. So when we go back to thinking about three, we know that the the remaining we're going to need another odd digit in order to make the odd digit sum to an even number and one even digit. So we could do uh 1 four. That would work.

[00:11:15]We could do 58. That would work. 5 + 3 would equal 8. We can't do anything else cuz 7 + 3 is 10. And that's too much for the even digit. We can't put a 10 into any cell of the sudoku. So three, three, well, fundamentally three is possible. So it doesn't have the nature of 1, two, and nine. And then four.

[00:11:42]Four. Four is going to have to have two odd digits in it. Again, it's got to have an even number of odd digits.

[00:11:52]Okay, let's do a table. Um, so we're going to have the x number, the number of odd digits, the number of even digits, and see if we can find a pattern. So, we're going to three is the first one that's got two odds, one even.

[00:12:08]Four is going to have two and two. Five. Now five, you can't have four odd digits because four odd digits will add up to 1 3 57 which is 16. I can't write 16 into a cell. So that's going to have two odds and three evens.

[00:12:30]What's the minimum sum of three evens?

[00:12:35]2 4 6 which is 12.

[00:12:41]Now that is achievable.

[00:12:47]Okay. Don't know six. So six is either going to have two odds or four odds.

[00:12:55]So it can't have four odds because if it's got four odd.

[00:13:02]Ah that's interesting. It also can't have two odds.

[00:13:08]Well that's not strict. Ah, we might have to be careful with that.

[00:13:14]Ooh, right. We're going to have to be careful with this one. Six. Can it be?

[00:13:23]Oh, hang on. Hang on. I I am going to have to be careful because I've just noticed there are diagonal clues. And I've just been assessing this from the perspective of clues that can't repeat digits.

[00:13:39]Ah. Oh, I've got bits of tree in my hair. That's um Yeah. Okay. Well, let sorry. I'm gonna I'm gonna now slightly change this. I'm going to say this table is for horizontal and vertical x sums only. I'm not sure about this one where you can repeat digits. There's a few of those actually, isn't there? There's three of them at least.

[00:14:05]So, so let let's think about six. If it say it was in this row. Now what we can say is it can't be two odd digits because if it was two odd digit if it was two odd digits and four even digits those four even digits would all be different. And I know that the four even digits in sodoku sum to 20. 2 + 4 + 6 + 8 is 20. So that's not going to work. So I can't make two odd digits sum to more than 16. 9 + 7. So that doesn't work.

[00:14:35]which means it would have to be four and two the other way round. But here we get a different problem. The minimum sum of four odd digits if they have to be different is 1 357 which is 16. And I can't make two even digits sum up to more than 14. 8 and six.

[00:14:53]So six doesn't work which is very strange. So seven.

[00:15:00]Now, seven again, if we're just thinking about a row, it can't be two odd digits because there'd be five even digits in the row and there are only four even digits in Sudoku. So, it's going to have to be four odd digits and three evens. Now, does this work?

[00:15:18]Four odd digits are going to sum to a minimum well, all the odd digits sum to 25.

[00:15:29]So a minimum of 16.

[00:15:32]Oh no, you can do that. Um because three the three even digits the three even digits will sum it can certainly sum to 16. In fact, they're going to have to be five different. That's weird.

[00:15:53]That is a weird thought I've just had.

[00:15:56]Yeah. Okay. Here is a thought.

[00:16:00]So in Sudoku there are five odd digits and there are four even digits. Now the nature of a seven x sum clue is saying I'm going to remove from the odd digits one of them because we've only got four out of the total of five. And the even digits I'm going to remove one of them.

[00:16:19]So I'm only going to have three instead of the total number we could have which is four. And yet somehow the sum of those has to be the same.

[00:16:27]But the natural sum of the odd digits is 25. And the natural sum of the even digits is 20. So the digits that I'm sort of going to be playing around with, I'm going to have to make sure that the odd digit I take out, I don't include, is five more than the even digit I don't include.

[00:16:55]So it's either 9 and four or seven and two and it's not seven. Oh, this is so be that's beautiful actually. That's my favorite one so far. That's absolutely beautiful. So seven only has one way of being concocted because you have to have five difference between the let the outies if you like the one the digits you're leaving out.

[00:17:19]You're sort of trying to correct a sum of 25 and a sum of 20 by removing one digit from each of those such that the totals are the same.

[00:17:30]You're going to have to you could remove a nine from the odds that gets you to 16 and a four from the evens. Or you could remove a seven from the odds and two from the evens. that you can't rem if if we had a seven here and then we said, "Okay, we're going to leave seven and two out." You'd you'd have a seven and you'd have a seven repeated in the column. You can't do that. So seven only works one way and that is you leave out a four and a nine.

[00:18:03]So this one is very this one is the most restricted one so far.

[00:18:12]I think I certainly didn't notice any restriction when we thought about fours and fives of that nature. Let's think about eight. So eight, we know you can't have a nine.

[00:18:27]Oh, eight's easy. Yeah, eight's going to have to be four and four.

[00:18:34]um because well we need an even number of odd digits. So we we can't only have two cuz we we can't have more than four evens if you see what I mean. But that means we're using all of the even digits which sum to 20. So the odds are going to have to sum to 20. And by the secret therefore the digit left out of an eight clue in this puzzle will be a five. So if you put an eight there that would be a five.

[00:19:05]Um so so 7 and 8 are the most restricted I think and 1 2 6 and 9 if we're not talking about diagonals simply can't exist.

[00:19:27]So what we want to do is find a it's nearly well no I can I can actually do that in this column because what this cell can't be 126 or 9 I mean the arrow is going vertical but it's still the same principle that can't be 1269 that can't be that can't be that can't be and that can't be. So those digits have to be 1 2 6 9.

[00:19:58]Uh, and we can do the same in column one. I just didn't see it. You've got a vertical clue, but in both of these positions. So neither of those can be 1 2 6 or 9. Those can't be 1 2 6 or 9. So these are 1 269.

[00:20:16]Now in this row these can't be 1269.

[00:20:25]No, that one can be.

[00:20:31]So there's five for the four digits there. And the top row is top row is useless, isn't it?

[00:20:40]So this clue can't be eight. because we knew eight had to have a five across from it.

[00:20:49]Yeah. Okay. Then if you've been watching the channel for a while, you're going to laugh at me for this, but I'm actually going to fill this in. 3 4 5 7 8. Now, what we learned before, what we learned when we were doing the examples is that eight and seven were the difficult ones to place because they had to eight had to have a five opposite it and seven had to have sort of a 49 pair.

[00:21:27]Okay. Yeah, that's good. So, seven can't be there or there.

[00:21:33]That's wrong. That's wrong. Sorry, didn't see it. It's absolutely could be there because that could be a 49 pair bobbins. That can't be because if this was a seven, we know the other side of the column would have to be the four and the nine and they can't be 49. So, that can't be seven.

[00:21:59]Uh that can that could be seven.

[00:22:03]Oh dear. This hang on. This is I'm not doing making a good fist of this at all.

[00:22:10]Eight. If you put eight in it had to have five at the other end. But that could be five. Those could be a 58 pair.

[00:22:18]I think.

[00:22:25]How did three work? Three was three was either with one four. Ah, okay. No song for this this cell because if if this was three, those two cells would either be a 1 four pair or a 58 pair. 3 + 1 is four. 3 + 5 is 8. And that's neither of those is possible. So that's not three.

[00:22:50]And what about this one? That can't be.

[00:22:52]That's the same. That can't This This cannot be 14 or 58. What about No, that one could be. That's That's only pointing that way. This one.

[00:23:04]Uh that one could be. That could be three with a 58 beneath it.

[00:23:09]Bobbins. Um Bobbins.

[00:23:17]Uh, okay.

[00:23:24]It's quite tricky actually, isn't it?

[00:23:26]Even though we've done some thinking. I mean, I've got a horrendous sequence of pencil marks here.

[00:23:33]I'm desperately looking around saying where where is restricted? Oh, that can't be eight cuz this can't be five.

[00:23:41]That can't be eight cuz that can't be five. That can be eight cuz that can be five. That can't be eight.

[00:23:46]because this can't be five.

[00:23:54]Um, if that was five, we'd know the sums because we'd be summing those. We know that's a 1269.

[00:24:06]No, that No, that doesn't work at all.

[00:24:08]Look, look at that. I'm saying if Oh, that's a horrible color actually choice.

[00:24:12]If this was a five, then the two even digits.

[00:24:19]Well, no, there hang on. There should be three evens.

[00:24:23]It does it doesn't even work from a parity perspective if this is five because you you'd actually be adding three odd numbers. You'd be adding 1, 9, and five together. So, you're going to get an odd total. That's just absolutely absolute nonsense even before you get to the arithmetic.

[00:24:41]So this digit's actually very restricted.

[00:24:45]Can that be five?

[00:24:50]If that's five, I've got to have a second odd digit and three even digit. Oh, the three even.

[00:25:04]Oh, that's lovely. Actually, no, no, no, that doesn't work. Right, the way to see this is par. Again, if you make this an eight, we know eight goes opposite five because we're summing those eight digits up. And we know that we need a 20 total for the evens and a 20 total to the odds. So, you've got to have a five here to make the column add up. Now, now you're summing these digits and you're trying to equate the odds and the evens, but we know that we need an odd number, sorry, an even number of odd digits in these green cells, which means we need three even digits, not including eight.

[00:25:41]So, we know that those digits need to include 2, 4, and six. Well, they can't.

[00:25:46]Where's the six going to go? This one could be two or six, but it can't be both. And none of those can be anything other than four if you want them to be even. It just doesn't work.

[00:26:00]Um, so this is good. This is going to get us our first digit, which is going to be this one. That is a four.

[00:26:10]So these are not four.

[00:26:13]And now we're summing those four up to be equal to something useful.

[00:26:20]So, we need two odd digit. Oh, we need Yeah, there you go. You need two odd digits in that sum.

[00:26:26]Well, what are those odd digits going to be? They're going to have to look at the options. They're going to have to be one and nine summing to 10. So, you're going to need a six with those. So, this is a two. This is lovely. All right. That's not a two by Sudoku. That's not a four by Sudoku.

[00:26:47]Now, so there there's a diagonal one there as well. That's that's a diagonal clue. But I I think the diagonal clues are going to be horrible because you can repeat digits on them. You know, there's nothing to say this can't be a four. I mean, there might there might be some sodoku reason I'm not noticing, but I I think that could be a four.

[00:27:15]Oh no. So, what do we do now?

[00:27:19]Um, I don't know.

[00:27:29]Right here is a tiny point.

[00:27:33]This four has removed the ability of these cells to be four. Therefore, I'm going to allege I think correctly that none of these can be seven because we know that seven on the other side of the row, you're going to get a 49 pair.

[00:27:51]So, I don't think that's going to work anymore cuz none of these can be four or nine.

[00:28:00]So, is there a reason? Now, that can be seven bobbins. I was going to say maybe that can't be seven. Then I would have known this was seven. I'd have known this was nine and this was four.

[00:28:14]But that can be seven because I think that can be 49.

[00:28:24]Um.

[00:28:27]Oh dear.

[00:28:31]Can that be five? We're going to know the sum.

[00:28:38]Yeah, let's think about that. If that's five, no, there aren't enough. No, sorry, that's not five. There aren't enough even digits in that sequence as two and eight. They need to be three. Oh, and now whatever I've been cutting down outside is giving me hay fever. Ah, so that's three, seven, or eight.

[00:29:03]Oh, it's not eight. I could I can I can see that because if that's eight that would be five. So this is right down this is down to three or seven. Now if it was three that would have to be five eight and this would be seven. Oh it nearly doesn't work. That would be seven. This would be four. This would be nine. But if that was seven that would be nine.

[00:29:28]Oh.

[00:29:30]Oh no. I was thinking if that was there is a diagonal clue there as well. So if that was three, this would be have to be 58 and that would have to be 14.

[00:29:41]But that could could be true.

[00:29:50]Um, oh, Robbins, I don't know. Then how do we do this?

[00:30:00]Eight has to be in this column in one of those two cells.

[00:30:04]So we know eight because eight is in one of those two. One of those two is five because eight must be opposite five. So that's not five in that corner or five here or five here. So that one's come right down. That's seven or eight.

[00:30:23]I'm just going to quickly I'm just going to do that just to say there's an eight. There's an eight in one of these and there's a five in one of those. I don't want to do that because I'll I'll forget what it means.

[00:30:37]No. Oh, no. I was going to say where's seven? But that doesn't work.

[00:30:41]Wait, that can't be eight cuz that can't be five.

[00:30:46]Hang on.

[00:30:50]That's okay. That's really cool. So actually this eight here, it's bizarre. It doesn't because it puts five in one of those two. It doesn't just get rid of five from that cell. It also gets rid of eight from it because we know eight would be opposite five on the other side of the That's lovely.

[00:31:09]That's got to be seven. This is huge.

[00:31:12]This is absolutely huge because now Oh well well no it is huge because those have to be 49 and we know the order.

[00:31:24]So these are not four this is three by sudoku now. So these are now five and eight and that four has to be the sum of those. Look, let's use that color.

[00:31:42]This three has to be the sum of those.

[00:31:45]Oh, lovely. That three can't be 58 there because that would kill off this square.

[00:31:50]So, that's got to be with a 1 four pair, which means this is not a one or or or a nine. That's just a naked old six sitting there with a one two pair here.

[00:32:02]This is not a one.

[00:32:07]One of these is eight. So, one of these is five. So, that's not a five. So, I've got a three seven pair now in column one. This is It's absolutely beautiful.

[00:32:17]Oh. Oh, no. That's okay. I was going to say I've got a deadly pattern of fives and eights there. But it's not really deadly, is it? Because contrast that to this situation. Now, that would be deadly if that existed.

[00:32:32]ignoring x sum clues because you couldn't determine whether the fives would be there or there.

[00:32:38]Either way round, you'd still end up with two fives in every row, column, and box of the grid and 28s in those columns as well. But this, if that was a five, for example, it would disambiguate all of that sequence. So, this isn't deadly.

[00:32:55]Um, but okay, it's fine. It's fine. This can't be a three in the corner now because what would we put into those two cells?

[00:33:10]We know these. If this is three, there should be 1 four or 58. And they can't be those because that one can't be five or eight. So that's got to be a seven in the corner with a three here.

[00:33:22]Now this this three clue can't be 58. So that's got to be one four the same as that one.

[00:33:36]And that's oh the seven clue in the column. We know sevens just as we did here go opposite fours and nines. So this is lovely. That's nine. That's six.

[00:33:45]That's one. That's one. That's two.

[00:33:53]This is a beautiful puzzle. It's It's I know exactly what those people who've recommended it mean. It's sort of it's just got a a gorgeous sort of underpinning logic to it. Now, that's seven. We've got diagonal sevens pointing at each other here, which is a bit weird. Let's put that in.

[00:34:20]I mean, that's that's horrific, isn't it? That is horrific cuz there could be so many repeated digits on those diagonals.

[00:34:33]So, what have I used all the clues that we've been given here?

[00:34:43]Or is it sedoku one has to be in one of those two cells because it you can't it can never be a clue cell can it? These two clue cells have to be I'm going to say from 358. We know they can't be 1269 and they can't be 47. So these are from 358.

[00:35:06]Now, so if this was three, this would have to be a 58 pair.

[00:35:20]If it's five, you got to have a lot of You got to have two evens.

[00:35:32]No, you got to have No, you got to have three evens. I keep thinking five the wrong way round.

[00:35:41]You got to have three evens.

[00:35:45]So, so they'd have to in that in that situation it would have to be because you can't put twos in any of those cells. It would have to be this is five. If it would have to be 468, which is it doesn't work. Doesn't work at all.

[00:36:08]The five Oh, I tell you something I hadn't realized when we looked at five earlier. Five, a 5x sum clue has to have has to have two odd digits and three even digits. Now, if those three even digits didn't include a two, it wouldn't work.

[00:36:32]4 + 6 + 8 is 18. And you can't make two odd digits sum to 18. So, there is always a two in any five clue. So, this is not five because those can't be two and that can't be two. So, can that be five?

[00:36:50]You could have a two. Yeah, that's easier. That's a lot easier. Although, can that be two?

[00:36:58]Because that would make No, if that's a two, this four clue would break because these would have these two digits would have to be two odd digits summing to six. They can't be double three or one or five. So, that is not a two. So, so if this is five, that is a two.

[00:37:26]Um, I don't think that's a problem. Is five restricted in some more meaningful way that I'm not understanding that. Can I just think about five a bit harder?

[00:37:37]So, five is it's two odd digits. I keep getting this wrong. And three evens including a two.

[00:37:47]So it could be 2 4 6 which is 12. Can I make two odd digits sum to 12? I can in two different ways. 57 or 39. Oh, but I can't use No, I can. No, but I'm using five in the clue.

[00:38:03]So So it would have to be 572 46, right?

[00:38:06]Let me just write that down. So one option for five is 57 2 4 6 wouldn't work in that row. Now, what about what about if we went higher than that?

[00:38:22]Then 59 would be the only other way of doing it because we know the three evens must add to at least 12 and we know one of the odd digits is a five. So 5 and 9 is 14.

[00:38:37]So 2 48 is the other one. So 2 4 8 5 9 is the other one. So there are two ways of doing fives and they both have four in and they both have two in.

[00:39:08]Okay.

[00:39:12]Um, bobbins. I don't know. How am I meant to do this then? Uh, is there is there any way to see how this works?

[00:39:40]I don't know.

[00:39:43]Okay, they both have they both have two and four in and five.

[00:39:49]Four in that row is in one of two three places.

[00:39:59]So whichever one of these is five is going to need a four.

[00:40:06]So one of those is a five and it's going to need a two.

[00:40:13]So if that was a five, two would be in one of two couldn't be in one. Ah, there you go. There you go.

[00:40:20]There's some that's it. That's actually it. Okay. So look at look what we have to do is a bit of sodoku, but not much.

[00:40:28]There's a two here.

[00:40:30]Now, where's two in box six? It's in one of those two cells. Now, can this be five then? No, because in those five cells, we'd need to have a two to make the maths work, and we can't have one because of this. So, that's got to be the eight. So, 8 55 58.

[00:40:50]Now, I think the eights are going to be particularly useless from that. We're not going to get any more information, but the fives we do have some more information to g to gain from. For a start, this five needs a four in it. So, the four must be in one of these two.

[00:41:07]This five needs a four in it. So, that's got to be in one of those two.

[00:41:17]And this one, this five can't have eight in it. So it must be 2 4 6 and five and seven. So I know what these digits are.

[00:41:26]They are 2 4 6 7 and this is not got four in it or six. So this is a 46 pair.

[00:41:38]It's very very beautiful isn't it? Now they are 8 n therefore by sudoku and these digits are 139 by sudoku.

[00:41:51]Um, oh, and it's the same logic in this row because the eight's on the other side, we can't use that option that was the 248 for the even digit. So, we're So, here we're also using 246 for the evens and the extra odd digit is a seven. So, this this is not 47. So, this is this is 47.

[00:42:20]This is 26. And let me guess that's 89.

[00:42:23]And it is right now.

[00:42:32]Yeah. And now this four clue comes into its own because what's the other even digit we're going to put into this four clue? It can't be eight anymore. Now if it was six, we'd have two even digits summing to 10. So the odd digits would have to either be 1 n. Nope. or 37.

[00:42:49]Nope, that doesn't work. So, they have to be um 1, five, two.

[00:42:56]Four and two is equal to 1 + 5. And these have to be six and three.

[00:43:06]Um, now there is a three in one of those. What? What are those digits? Are they one? They're 139 again, aren't they? It's exactly the same as the row beneath it.

[00:43:18]Okay. And they can't be nine by Sudoku.

[00:43:21]So this is a nine. This is a 1. Three.

[00:43:25]So this is 25. And that is a one. This is gorgeous. It's just gorgeous.

[00:43:31]Right. What are those digits then? 67.

[00:43:35]No joking. That's that's a meme I'm sure that Rocky Row has come across once or twice uh in the classroom. Uh 28 five I think.

[00:43:50]Does that look right in row two? I think so. By Sudoku.

[00:43:54]Okay. So, which let let's once we've got a clue, we should uncolor it to make sure that we don't rethink about it.

[00:44:05]So, we haven't done this clue. We haven't done these two clues.

[00:44:11]That's a very dodgy sound outside. Um, what does this mean? So, if this was eight, that would be five. This can't be eight anymore. Can't have a five above it.

[00:44:28]So, this is three or five.

[00:44:34]So, if it's five, what do we work out?

[00:44:36]It had to contain two and four.

[00:44:41]So, it would have to be No, it doesn't work. It doesn't work.

[00:44:51]Um, how to explain that? If this was five, we'd be summing these up.

[00:44:58]And what we said earlier is that whenever you have a five clue, you definitely involve the digits 2 and four. And then you either have a 67 pair to go with that or you have an 89 pair to go with that. So have a look at this column now and let's think how that works.

[00:45:17]If this were was the 248 uh 59 variety, that cell can't exist.

[00:45:28]And if it was the 24657 variety, that cell can't exist. So this has to be three I think. And we can shorten our our length of green. Now those two digits don't involve one. So these aren't one four. So they're five.

[00:45:45]This is gorgeous, isn't it? So that's now a two up here.

[00:45:51]Oh, okay.

[00:45:56]Okay. Maybe that. Oh, this is an eight now. That's going to do something up there. That we know eights go opposite fives. So that gives us a little bit more sedoku to benefit from. Look, that's seven. That's six.

[00:46:11]That's eight. That's nine. Using the power of this eight.

[00:46:17]And we've done those clues now. So, so they don't earn colors. So, we have now almost finished all of the It must be Sedoku or it's either Sedoku or these these remaining um colored clues, but they seem very underpop populated to me.

[00:46:41]So, I'm s I'm slightly too scared of them to try and work them out. Um, can we do any more basic Sudoku?

[00:46:50]Let's think about that probably.

[00:46:56]Although I don't exactly see where this is 35. Oh, that's a three. So there's a six at the top of column four.

[00:47:06]So 6 3 that isn't three.

[00:47:15]There's no X sums clues at the top of columns four, five, or six.

[00:47:25]So these digits are from 1 N or seven.

[00:47:31]That can't be nine. So this is a one.

[00:47:34]This is from a snooker maximum. That can't be one or four. Actually, that is seven or nine. Maverick's just taken off. Must be a hot day to be Maverick today.

[00:47:44]Um, one I can do one in the bottom row. It's got to be there.

[00:47:56]So, one of these two cells is a one the Okay, let's look at the bottom row then.

[00:48:01]2569. Can we Let's pencil mark it. 2569.

[00:48:06]Can we do We can Where's nine in the bottom row? There's enough nines that we can take care of business there. So, this becomes a nine.

[00:48:19]This can't be five because of this 358 triple. So, that's two or six. Five is definitely in this domino.

[00:48:33]It's probably this clue, isn't it?

[00:48:35]Because at least we've restricted this row 6, column 4 a little bit.

[00:48:44]Well, okay. What's the even? Yeah, this is fine. What's the even digit that's going to accompany the four? It's definitely not a two because that's already going to swamp it cuz we'd be trying to add the two odd digits to six.

[00:49:00]Now, actually, maybe we could do it. Okay. So if if the even digit here is a six, then we know this would be a seven with a three. So this would be a 3 six pair.

[00:49:16]Now is there a reason that doesn't work?

[00:49:22]Not sure. I don't think so. And if the even digit on the other hand is eight in these two cells, then we then the odd digits are adding to 12.

[00:49:33]So, it' either be seven with a five in one of these or nine with a three in one of those.

[00:49:43]Oh, bobbins. That just that seems completely open-ended then.

[00:49:52]Oh, I'm sorry. I don't think that is how to do it then.

[00:49:59]Okay.

[00:50:01]All right. I'm going to All right, I'm going to change tuck entirely. I'm going to do three by Sudoku in the middle box there. I've said it. It goes there.

[00:50:12]Um, so we need 1 1457 1457.

[00:50:27]That's a 4 six. So that is not a four there.

[00:50:35]So this is ah that's weird. Oh I see where no simple question. Where's four in column four? That's been available forever. That's four. That's seven.

[00:50:49]So in box five, what what digits have we not placed? 25. Is it 258 again?

[00:50:57]That's quite handy. Oh, that's really handy. That is really handy. 258. That digit sees two and eight. That's got to be five.

[00:51:07]That does Oh, that does the five and the eight. The eight and the two.

[00:51:13]So, Sedoku has suddenly come to our rescue for once. This is right. Now, what do we need in this row then? 1 147 again. 147 with the four definitely being over here.

[00:51:28]And in this row, 579 with the five definitely being over here.

[00:51:38]79.

[00:51:41]Okay, let's let's check out column five then. What do we need there? 469.

[00:51:47]Yeah, where's nine? We can do that.

[00:51:49]That's got to go there.

[00:51:53]And in this column, it's slightly less populated, isn't it? 2 4 6 7 2 4 6 7 And that can't be two.

[00:52:10]Now, what if anything does this tell us?

[00:52:17]I don't know. I'm not sure. Oh dear.

[00:52:26]I don't want to look at these. These look really complicated. Um, I think we're going to have to though, aren't we?

[00:52:35]So, how's this going to work? It must be subject to the same principles. Just we've just got to be careful with the fact that we've got repeating digits.

[00:52:43]So, for example, a seven clue, it is going to have to have an even number of odd digits.

[00:52:50]So, it's going to have to have I'm going to claim.

[00:52:55]Well, actually, look look if we look at the look at this top left clue.

[00:53:01]That's already got That's definitely odd. That's definitely odd. And that's definitely odd. That's three odd digits.

[00:53:07]That's not enough. There must be one more odd digit.

[00:53:12]Um because we definitely can't make six odd digits work because they would have to Well, there's there's two even digits anyway. It just wouldn't work. So, okay.

[00:53:24]So, so the the indigo seven, this one has got four odd digits on it, which are that one, that one, that one. I wonder if that can be nine. That would be two.

[00:53:37]21 for those three. So, at least 22.

[00:53:47]I think it could work. No, actually, no, it can't. Uh, hang on. I'm not sure. I'm not sure, actually.

[00:53:55]No, it can't. It actually can't work.

[00:53:57]That's weird.

[00:53:59]That's a bit tricky. Okay, so what we're going to do is we're on the indigo clue, we're going to I'm going to change the color of this. Um, and change the color of that to green. Let's label the odds.

[00:54:12]They are Those are definitely odd.

[00:54:15]This one could be odd.

[00:54:18]This is definitely even. This is definitely even.

[00:54:23]That one could be even. This one could be even. So, let's try and max out the value of the evens that we could possibly have. We could have a six here, an eight here, that's 14. But this can't be bigger than four. It sees six and eight and that can't be bigger than four. So the absolute maximum value of the evens on the indigo diagonal is 6 8 and another four which is 18. So if this was 9, it you'd never get big enough cuz that's already 21 for those three. So this is one, this is seven, this is nine.

[00:55:02]Now hopefully that will do something sedoku wise for us. It might affect the green clue.

[00:55:10]Um yeah, we can't have six here anymore because it would need 1 nine to to match it off which won't work. So we're going to be looking at we're going to be looking at 48 adding up to the same as 39. So this is a 38. Ah it doesn't doesn't quite do it. This is a 38 pair.

[00:55:35]That's incredible. That hasn't actually done it at all.

[00:55:43]All right, let's go back to this again, then. Um, so at the moment, our odd digits are adding to 13, and we've got one more to come.

[00:55:54]I don't know where that is. It could be here or it could be there.

[00:56:00]If that is even, it has to be four.

[00:56:10]And that would Okay, so if that was true, this would be a one. We'd have 7 51 and another one 14 8 four and that would be two and that would work. bother.

[00:56:28]I mean, there might be a technical reason doesn't work, but it's certainly not obvious to me what that is.

[00:56:37]Okay, let's have a think about that. I haven't thought about this clue at all.

[00:56:42]Now, that clue that's definitely got the same OB. It's got the one and the five in box five.

[00:57:00]It's got that one's definitely even.

[00:57:02]That one's definitely even. So, there must be one even. This must be one even and one odd.

[00:57:17]Okay.

[00:57:18]I'm going to have to pencil mark them, aren't I? I don't want to do it, but I think I'm going to have to. Um, right.

[00:57:25]Let's check this one out first. 1 3 4 6.

[00:57:29]Okay. So, 3 4 or six. It can't be one.

[00:57:33]So, this is also 3 four or six.

[00:57:36]Therefore, just from the column logic in this column 1 3 4 5. So, these are both 3 4 5.

[00:57:51]Now, what on earth does that mean?

[00:58:00]That's the question.

[00:58:04]Uh, how do we how on earth do you do this?

[00:58:14]What you can do if that's odd.

[00:58:26]Yeah, that's quite interesting. If that Let's just think about if this is odd for a moment because there is an interesting thing that happens then with the two seven clues together. If this is odd. So just let's just label it odd for the sake of exa example. You can then see that the odd digits that are counted in this clue are going to be the seven, this one, this five, and that digit. But the it's the same here. It would be the seven, this digit, that digit, and that digit.

[00:58:58]So the odd digits for both of these clues would sum to the same number. So the even digits ought to sum to the same number. And from this one's perspective, the even digits are then going to be this one, this one, and that digit.

[00:59:13]And from this one's perspective, they're going to be that one, that one, and that digit. Now because those are common that means that digit and that digit would be the same and they would both be four.

[00:59:32]Um now okay that does do something.

[00:59:45]might not it might not be much but it does something. So what we've just demonstrated is that if this is odd both of these digits have to be four for both seven clues to work. Now the other alternative is that this is not odd. And if this is not odd it is four. So in this so in this box four is either there or there. So four has to be in one of those and those are not four.

[01:00:17]Right. And now I've got a 358 triple in that row which means this is not three. So that is even now uh that is great. That is great. That is actually that sequence is great because this is now so beautiful. this being even.

[01:00:42]Let's look at this clue. Now we now know that there are four odd digits on this for this clue and they have to be that one, that one and that one. So we are in a situation where this is odd. So this is odd which means these two digits are the same. Let's go go through that again just to check that we agree that these two cells have to contain the same number. So the even digits on this one are that one and that one and obviously this one. But the odd digits are that one, that one, that one, and that one. And the odd digits from this one are that one, that one, that one, and that one. So they're completely common because the sevens are the same.

[01:01:24]And the even digits from this one's perspective are that one and that one plus one more, which is this one. But those two were in this blue. So these have to be the same digit and they have to both be four.

[01:01:38]And that's a one. And that's a one. And that's a four. And that being that's going to do some stuff over here. Look, it's going to do lots of stuff. Six.

[01:01:50]Six here as well.

[01:01:52]6 4 two.

[01:01:59]So I've got a four. I've got a 47 pair which is resolved actually.

[01:02:05]And this 358 thing, what's that doing?

[01:02:07]Where's two? Two's got to go there. So that's two and that's six. That's six and that's five. That's five and that's seven.

[01:02:19]These two digits now are what's that? That's a seven by Sudoku.

[01:02:25]Uh that doesn't So it's maths, is it?

[01:02:27]Now do we have to I have to make the diagonal work. Okay. So, how do we do that? Let's use this clue. This clue tells Well, let's do the evens. We've got 18. So, 12 and 1 is 13. So, that should be a five. So, 53 58 83 39 98. That is such a lovely, lovely finish. Wow. Wow. Absolutely beautiful. It almost got harder as we went through actually. it. I love the fact that there are some weird discoveries to be found. Like six doesn't work as a clue. It's not obvious at all.

[01:03:09]And you can then play around and get some stuff done. But actually that bit, the seven's bit was gorgeous.

[01:03:19]This is a brilliant puzzle. Rocky Row, it's lovely to see you again. 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.