Unusual Sudoku Logic

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Cracking The Cryptic

Cracking The Cryptic

4 жыл бұрын

You can try today's Sudoku here:
cracking-the-cryptic.web.app/...
You can try the two Aquarium puzzles by Atanas Georgiev here:
cracking-the-cryptic.web.app/...
cracking-the-cryptic.web.app/...
Atanas' website, which has a number of playable logic puzzles:
www.puzzle-loop.com
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▶SOFTWARE◀
We sometimes use Duncan's Sudoku Solver, which is available at www.littlegogs.com/index.shtml
Or our own software, which you can access by clicking on the links under the video.
▶ABOUT US◀
Hi! We're Simon Anthony and Mark Goodliffe, two of the UK's most enthusiastic puzzle solvers. We have both represented the UK at the World Sudoku Championships and the World Puzzle Championships. We're also "cryptic crossword" aficionados. Mark is the eleven-time winner of The Times championship and Simon is the former record holder for most consecutive correct solutions to The Listener crossword. We hope we can help your puzzle solving while also introducing you to some of the world's best puzzles.
Thank you for watching!
Simon and Mark

Пікірлер: 114
@matthew456turner128
@matthew456turner128 4 жыл бұрын
Simon, you are a total genius. I watched alot of your videos and you make solving extremely difficult puzzles look easy. I love how you take the time to explain your logic. You're like a teacher and I feel like a student. I'm so glad I found this channel.
@siccodewilt1
@siccodewilt1 4 жыл бұрын
I love it when he points out a pattern and asks if you can see the effect :D I always pause the video and try to find it.
@tadperry1817
@tadperry1817 4 жыл бұрын
"Schroedinger's cell". Haha. I had no idea that solving Sudoku puzzles could get that involved. I've been watching for about three weeks. Started as a random KZfaq suggestion. I found watching Simon solve Sudoku to be surprisingly entertaining. Wasn't trying to solve them myself at first, but then wanted to be able to do what Simon does. I've always loved the cryptics page in the paper, especially the crossword puzzles. I love Merl Reagle's puzzles and find them fun and entertaining. The Sudoku doesn't always stump me, but sometimes. I find it quite satisfying when I can finish it along with all the other puzzles.So I started attempting some of these. Maybe not the best way to start, like jumping into the deep end of a pool to learn how to swim, because the puzzles Simon is tackling never straightforward. Right now, I try it on my own and when I take it as far as I can, I turn on the video. This time, every time I needed to watch it, Simon was also slowing down and wondering "what next?" But of course, he quickly found it every time and I couldn't on my own. I have another problem that I make logical errors in my notation that can lead to contradictions, or miss a fact or two, and have to update the notation to match Simon and then continue. After he found the AIC and set Column 3 Row 8 to an 8, I went forward on my own, making some kind of mistake on the way and ended out with a contradiction 3 different times in a row. I went back to this step each time and tried again. Ended out leaving it overnight and trying again the next day and finally did it right.
@ricksick81
@ricksick81 4 жыл бұрын
I love how he still is excited about solving these puzzles or finding something new, really passionate about it
@LeoninMinecraft
@LeoninMinecraft 4 жыл бұрын
its crazy the difference between what a pro can see and what a casual sees. That method of finding the 6 through the symmeterical 9s just absolutely baffled me. The things you do using logic!
@Anthony-vu8bl
@Anthony-vu8bl 4 жыл бұрын
That 2 thingy is sooooo clever *.*
@pmnt_
@pmnt_ 18 күн бұрын
I really like these old videos (it was 4 years old when I watched it). I still understand what's going on, even if Simon finds something I would have missed. It's amazing how Sudoku and Simon have envolved over time.
@Wecoc1
@Wecoc1 4 жыл бұрын
The trick involving the 28 is called an AIC (Alternating Inference Chain), or more specifically a Discontinuous Alternating Nice Loop.
@Reb0rnLP
@Reb0rnLP 4 жыл бұрын
Yes.
@JuanMataCFC
@JuanMataCFC 4 жыл бұрын
u might be right, but let's be real "Schrödinger's cell" is a much *cooler* name for it 😂
@buzzly108
@buzzly108 4 жыл бұрын
Let it be known henceforth
@janisgustavian
@janisgustavian 4 жыл бұрын
no, the trick is called the cell that wants to be two digits. better than "alternating inference chain".
@beansock
@beansock 4 жыл бұрын
Haha as always I get distracted by seeing the easy clue and frustrated about it, even though filling it in wouldn't help solve the puzzle!
@Kingcob7
@Kingcob7 4 жыл бұрын
Ask him for the last puzzle in the book next :D
@tobortine
@tobortine 4 жыл бұрын
_"...let me know what I missed..."_ I wish.
@lakarto19
@lakarto19 4 жыл бұрын
Took me 36:28... one of the most difficult classic Sudokus I've ever encountered on this channel (unlike the variants, most of the classic Sudokus on this channel I found relatively easy, to be honest) I did manage to spot the trick with the XY-chain to put the 6 into r4c8, but quickly got stuck again and required much more complicated chaining to eventually solve the puzzle... Simon's 'Schrödinger' technique was much more elegant; once again hugely impressed with how quickly he managed to spot the pattern.
@JuanDi_SDK
@JuanDi_SDK 4 жыл бұрын
I really love how this channel is growing and how the content is still feeling fresh and fun. However, I feel like you guys are at the point in which you should definitely get a better microphone. Having good quality audio goes a long way, even more so than a better webcam, into making your videos look way more professional and prepared. There are some cheap usb mics like the behringer C-1U or the Yeti or [the one shaped like a ball whose name I always forget] that are produced precisely for stuff like podcasts/streaming/youtube vids that sound good and require very little setup or technical knowledge.
@fishandthesticks
@fishandthesticks 4 жыл бұрын
What a treat! Great puzzle. Also, I don't mind when a puzzle stumps you guys for a bit, gives me a chance to catch up!
@CrimFerret
@CrimFerret 4 жыл бұрын
I was able to get to the point you got at around 10:00 after that it went beyond my current ability. Considering it seemed to even be a tough one for you, I don't feel too badly about not being able to solve this one on my own. I did learn a few more tricks at least
@SaintNine
@SaintNine 4 жыл бұрын
I'm new to your videos, having watched it as a random KZfaq recommendation, and I haves to say that I'm impressed. I'm not a Sudoku addict and not entirely familiar with some of the terms you used (I haven't come across ‘skyscraper’ before, but I'll look it up). I was also very impressed with the software you use, particularly the pencilling options. Did you develop it yourself and is it possible to get a copy? I used to be a software engineer myself and specialised in the ergonomic aspects (one of my early projects was a version of Minefield I wrote for an Amiga - which shows how long ago that was - in which I improved the interface considerably. Some of them are now more common, but I still haven't found a version that has all of them), so I'd be very interested to see the code, if it's open source. I didn't think that watching someone solve a Sudoku would be as entertaining as you made it. It's very interesting to see your thought process (with your clear and excellent explanation) and to see how it differs from my own, ie what draws your attention and where you focus. Well done and thank you.
@filam7371
@filam7371 4 жыл бұрын
I thought I was getting better at this, but I see I'm still a child when it comes to Sudoku...
@ala4sox02
@ala4sox02 4 жыл бұрын
At 9:02, my puzzle looked almost identical to yours, pencil marks and all. Feel pretty proud of that.
@calebtadlock5956
@calebtadlock5956 4 жыл бұрын
Am genuinely loving the thumbnails as of late!
@dondebortoli2713
@dondebortoli2713 4 жыл бұрын
I agree that was brilliant solving loved watching it.
@Tolle44
@Tolle44 4 жыл бұрын
This sudoku lost me 2 hours of sleep. Interestingly, I found the 2 in R7C9 very quickly somehow, but after backtracking I have no idea how I did it. Got stuck on the part requiring the "schrödinger cell" or AIC as someone pointed out. Ended up watching you find that before I could proceed :/
@milkom2021
@milkom2021 Жыл бұрын
👀 I can't believe my eyes! Simon using two chaining techniques (XY-Chain and AIC) one after the other!
@dav3ston3
@dav3ston3 4 жыл бұрын
I tend to go thru puzzles numerically so the first thing I did was fill in all the 1s without needing any other numbers. Not long after that I missed that the 2 in the top left cube was obvious so I floundered there’s for a while and I completely missed the weak box in the center. Nice catch there.
@imenja
@imenja 4 жыл бұрын
I'm new to sudoku, so I try methods as I see them on your channel. I did the washing machine with ones fairly early into the puzzle, and that gave me all the ones plus some other digits. Then I was stuck again, but when you showed us that 6, I was able to solve the puzzle right away because of the washing machine method that I'd done earlier.
@StephanSpelde
@StephanSpelde 4 жыл бұрын
That was AWESOME!
@SvviftDeath
@SvviftDeath 4 жыл бұрын
Puzzle took me 32min. If this is the first puzzle of the book, I don't want to know how hard the ones in the back are.
@michelvandersmissen4512
@michelvandersmissen4512 3 жыл бұрын
18:44 "Therefore it can never be a two in the first place. And that..... is NICE"
@Sevicify
@Sevicify 4 жыл бұрын
I done it in 22 minutes. Going back through my logic though I made a big mistake at the start of the puzzle placing an 8 into r6c1 thinking it was the only spot for it without realising it could have gone into r6c3, thankfully through a stroke of luck it worked out to be the correct spot and having it so early made the puzzle a lot easier.
@alexortiz9777
@alexortiz9777 4 жыл бұрын
Had to restart the first aquarium puzzle 3 times because I kept mixing up horizontal and vertical logic on that rightmost column.
@timdolan4591
@timdolan4591 4 жыл бұрын
nice trick at 14:45 for the 6. At 15:45 I would be sooo "a-scared" being stuck - "We'll be right back after a short commercial break :) :)". Of course there were hundreds for followers YELLING cuz you missed some "obvious" spot.
@bigfatkittymeow5237
@bigfatkittymeow5237 4 жыл бұрын
More aquarium puzzles!
@rabidsamfan
@rabidsamfan 4 жыл бұрын
48:30. Whew. I lost twenty minutes to missing an obvious deduction, but I got there. Now to see how Simon does it.
@Tahgtahv
@Tahgtahv 4 жыл бұрын
You erased a penciled 7 in box 7 without filling in the other possible cell. Looking back now though, I'm not sure it would've helped much.
@cindyshirey8561
@cindyshirey8561 10 күн бұрын
Love it !
@RoderickEtheria
@RoderickEtheria 4 жыл бұрын
Two techniques I didn't have awareness of, so I don't think I would have finished it successfully before now. Now I'll have to be more careful to look for those logic tricks.
@suka_sukaGaming
@suka_sukaGaming 4 жыл бұрын
My brain screams after watching this video. What a jenius way to solve a puzzle.
@chaotix37
@chaotix37 4 жыл бұрын
I was stuck in the 1st 10 minutes but now I realized that theres triples,. In addition what technique was used on 13:37-14:43? Schrodinger's cell... @_@ that was pretty new to me. Finally solved it in 47 minutes
@Stephen-Fox
@Stephen-Fox 4 жыл бұрын
The new spreadsheet let me find the two bonus Aquariums from before I found the channel.
@longcat
@longcat 4 жыл бұрын
very nice, thanks...
@XoIoRouge
@XoIoRouge 4 жыл бұрын
Wish I had answers to the Aqua puzzles, cuz I'm not confident in mine. I mean, though, if there's only one answer, the numbers do add up... I just hope I'm not missing something.
@TzeroX
@TzeroX 4 жыл бұрын
Getting better, took me only 21 minutes this time :) i did "bruteforce" at one point tho
@reddcube
@reddcube 4 жыл бұрын
Have you done a Futoshiki puzzle on the channel yet?
@_KingRaz
@_KingRaz 4 жыл бұрын
Also, I absolutely love the aquarium puzzles. Makes me feel really smart! Haha did them both in about 4.5 minutes each.
@KDOTott
@KDOTott 4 жыл бұрын
How do you do it? I've been on the first one for an hour, I have a handle on some of the basic logic required and I still can't do it, I even went back and watched the first video he did on Aquariums.
@_KingRaz
@_KingRaz 4 жыл бұрын
@@KDOTott do you understand the rules? Lets get that out of the way first
@KDOTott
@KDOTott 4 жыл бұрын
@@_KingRaz Yep! I actually solved the second one with relative ease, clocking in at about eight minutes. the first one just had me stuck really badly.
@sonalita_
@sonalita_ 4 жыл бұрын
@@_KingRaz I am also stuck on 1st puzzle :(
@_KingRaz
@_KingRaz 4 жыл бұрын
@@sonalita_ Without totally giving it away, the trick that broke it for me was to look at the way the 5s on the bottom, and the 5s in the middle rows interact. Specifically with the top left cell, and the z looking cell in the mid left.
@Urroner
@Urroner 4 жыл бұрын
A little late see and commenting on this video. This was great. The problem I had with this puzzle wasn't really the puzzle, it's my marking. I'm from the old marking school, but I like this notation style a lot better. It's cleaner and faster. I'll fill a square with the number and there another square in the block which has two marks in it, the number I just entered and another number, so, in the old way, the contents of that cell must be the other number. No, not necessarily. And as I get towards the end, I end up with two numbers in the same row, block, or column and who knows how long ago I made the error. Again, a great learning video. And Simmon said he needed to slow down in the middle because he didn't want to miss anything simple, I too need to slow down and think when I see two pencil marks in a cell. They might not be binary. They might just be possibilities.
@Mephistahpheles
@Mephistahpheles 4 жыл бұрын
@14:30 Is there a name for that pattern? I think I've seen Simon use it before. Double y-wing? I dunno.
@watergod321
@watergod321 4 жыл бұрын
You lost me at the 7 8 leap, the test made sense
@jimdetry9420
@jimdetry9420 4 жыл бұрын
Of all the KZfaq channels I've seen, this one has the most informative, polite comments of all (except for the occasional dim bulb who types "first.")
@lawrencekallal6640
@lawrencekallal6640 4 жыл бұрын
Quite a challenging puzzle. Nice forcing tests with 29 and 218 to crack it open.
@brianrichter8255
@brianrichter8255 4 жыл бұрын
Dang, that was hard. Had to resort to bifurcating, and even then I couldn’t see the logic that would have solved it for me until I watched the video. Some years ago I gave up doing Sudoku because of puzzles like this, because there was no web app to make bifurcating so easy (bifurcating with pencil and eraser is really messy) and there was no KZfaq channel to show you how you could have solved it!
@JuanMataCFC
@JuanMataCFC 4 жыл бұрын
the Sudoku was too hard for me, but i managed to solve both the Aquarium puzzles. the second Aquarium was a lot easier than the first! :)
@eddaumpeddaump4509
@eddaumpeddaump4509 4 жыл бұрын
How do I submit a puzzle to you? I'm stuck and can't logic it out.
@_KingRaz
@_KingRaz 4 жыл бұрын
First attempt I misclicked and it screwed me over way at the end of the puzzle because I didn't notice it. 2nd attempt took me 11 minutes. To find the mistake on the first attempt though, took me like 30 minutes. Very frustrating haha
@johnhumberstone9674
@johnhumberstone9674 3 жыл бұрын
I am stuck on the first Aq puzzle. Do rows on the bottom and left but can't see any logic to move me forward. Is there a solution anywhere?
@missioncardiac7599
@missioncardiac7599 4 жыл бұрын
Ok, now this is very funny. I found an alternate solution. actually two alternate solutions if you count exchanging 2 numbers in 4 corresponding squares.
@ThatGuy-dj3qr
@ThatGuy-dj3qr 4 жыл бұрын
Does anyone know if there is a name for the strategy used at 13:50? It seems to require four bivalue cells, but interestingly no remote pairs are required.
@lakarto19
@lakarto19 4 жыл бұрын
I think they're called XY-chains.
@milkom2021
@milkom2021 Жыл бұрын
@@lakarto19 First time I've seen Simon resorting to a chaining strategy (bifurcation)
@jaguh9069
@jaguh9069 4 жыл бұрын
How you get 6 for r4c8 blown my mind!
@r0bw00d
@r0bw00d 4 жыл бұрын
I tried to solve this with the limited knowledge that I have. After two hours and nearly having the board complete, I realized I made a mistake somewhere and was unable to complete it.
@gposchman
@gposchman 4 жыл бұрын
I actually need information. At the beginning of the puzzle, I noticed that there were 3 1's in boxes that fit something you had shown earlier, I don't remember what it's called but a number in a box where the box in its own column and row. I walked it through and found two solutions that were different and there were three cells that a 1 could go in but none of those three cells were part of either solution. I pencil marked the 1's leaving those cells out. As I worked the puzzle I got stopped so I watched you and I had the triplet in the upper left box, but I didn't see it. The point is that as soon as I found one of the 1's the rest followed. Was my discovery with the 1's valid, or a fluke? Gene
@orNICOrynque
@orNICOrynque 4 жыл бұрын
What you are refering to is called the « slot machine », sometimes it can help you to break the puzzle before even starting it. Here, it isn’t really usefull because you can’t really use it to find something with the ones. There is also a slot machine on the 8 but again, it doesn’t do much...
@gposchman
@gposchman 4 жыл бұрын
​@@orNICOrynque, that's it. Slot machine. I Penciled in all the 1"s. Then I followed the paths of all three double 1's using a unique color for each path. Six. The paths that completed successfully, left the same three 1 cells open. I also noted that paths that didn't complete also left those cells open. I deduced that those cells would not contain a 1. Later when I resolved a 1, the rest of the 1's resolved. It worked. Was it a fluke or was the deduction valid?
@SofaKingBeast11
@SofaKingBeast11 4 жыл бұрын
dang
@WhatsInTheRain
@WhatsInTheRain 4 жыл бұрын
Does anyone know what book this puzzle came from? I've been looking to get a sudoku book and would love to do more like this!
@matthew456turner128
@matthew456turner128 4 жыл бұрын
www.amazon.com/gp/aw/d/1540653021?psc=1&ref=ppx_pop_mob_b_asin_title
@WhatsInTheRain
@WhatsInTheRain 4 жыл бұрын
@@matthew456turner128 Thank you!
@timdolan4591
@timdolan4591 4 жыл бұрын
@@matthew456turner128 You will certainly get your moneys worth. ??This was puzzle number ONE?? Let us know how you did on Page 71.
@daveflamenco4085
@daveflamenco4085 3 жыл бұрын
Excuse me for the rookie comment or my lack of experience solving sudokus, but what’s the logic process involved to find the very first two ? I couldn’t understand it
@JohnSmith-rf1tx
@JohnSmith-rf1tx 3 жыл бұрын
He's just considering what numbers are possible to put in that specific cell. Sudoku only uses 1 to 9 and you can't have duplicates in a row, column, or 3x3 box. He sees that there are 5 digits in the column and when he compares those 5 to the 3 digits in the row, he notices that there isn't any overlap. I.e. the cell which gets a 2 is already "seeing" 8 different digits. Ergo, there's only one possibility left. In this case it was a 2.
@emiolty1875
@emiolty1875 4 жыл бұрын
How did you know at about 8 minutes that the 1 - 8 had to be a 1 (middle box on top) ? There was no certainty to exclude it could be a 7 ?
@angusl1694
@angusl1694 4 жыл бұрын
8:08 the 1 in the top right and the pencil-mark 1s in the top left sees the top 2 rows of the puzzle. Therefore 1 in the box is forced into the 3rd row, and the only vacant square in the box is where he put the 1. Hope this answers your question.
@terminatron2
@terminatron2 3 жыл бұрын
I was also stuck on that shordingers cell part but it would have taken me much longer to figure that out ( i cheated and watched him at that point)
@Robin19358
@Robin19358 4 жыл бұрын
I took 45 min to fill in the middle row, and got stuck after
@user-A168
@user-A168 4 жыл бұрын
Good
@pedrodias7839
@pedrodias7839 4 жыл бұрын
the aquarium ones were easy but fun, where can I find more of them?
@dusaprukiyathan1613
@dusaprukiyathan1613 4 жыл бұрын
He said. www.puzzle-loop.com
@pedrodias7839
@pedrodias7839 4 жыл бұрын
@@dusaprukiyathan1613 thank you, missed that
@Deathranger999
@Deathranger999 4 жыл бұрын
DuSAprukiyathan No, he didn’t. He said that’s the website run by the person who sent him the aquariums, not that that site has more aquariums.
@dusaprukiyathan1613
@dusaprukiyathan1613 4 жыл бұрын
@@Deathranger999 Sorry, you're right; however, he did say that the site had quite a lot of puzzles and also said that the guy made a program to generate aquarium puzzles. There are, in fact, aquarium puzzles there, and nearly infinite, so uh... 50% ain't bad? -Except it is.-
@Deathranger999
@Deathranger999 4 жыл бұрын
DuSAprukiyathan Fair enough. I hadn’t checked the site for a few days, so I didn’t notice that they had added a beta for aquariums.
@lukeczerwin1390
@lukeczerwin1390 4 жыл бұрын
What’s the intro song name
@markmcknight9601
@markmcknight9601 4 жыл бұрын
Simon, I watch you a lot, probably too much, but you drive me crazy by not making one pass at first for the low hanging fruit. Drives me crazy. Obviously not easy to do this with folks watching, but....
@mnbabariya43
@mnbabariya43 4 жыл бұрын
The table is too short and figures aren't seen correctly. CAN SOMETHING BE DONE ???
@r0bw00d
@r0bw00d 4 жыл бұрын
This one really pisses me off. I've done it twice and failed both times due to an error that I can't find, probably because it's too far back in the progression for me to be able to.
@ericveneto1593
@ericveneto1593 4 жыл бұрын
Who was screaming "8 in bottom of 3rd cell!"?
@Jason1.1
@Jason1.1 4 жыл бұрын
Please make a dark mode in your software and use it!
@JohnWalshLegend
@JohnWalshLegend 4 жыл бұрын
Took me over an hour for this one, but I'm always faster on Pencil and paper rather than Screen and mouse for some reason....Anybody else get this?
@ashemaclean5545
@ashemaclean5545 4 жыл бұрын
Ahahahah, yes, I have that issue too. I think it's cause I have to click, check that I'm typing it correctly with the annotations I want, and then analyze what I just added, where as on paper, I'm doing it all at once, with fewer input errors.
@csvidyasagar
@csvidyasagar 2 жыл бұрын
Sir, At video time 18:43 you mentioned that if cell R8-C3 (2,8) were to be 2, then cell R2-C3(1,2) becomes 1. Till now I agree with you. But you mentioned that cell R9-C1 becomes 1. I have not understood this logic. How the cell R2-C3 got any interaction with cell R9-C1? When you make cell R8-C3 as 2, you have to see in box 7 there is no 8 as you mentioned cell R9-C1 contains 1. Therefore whether you make cell R8-C3 as 2, still cell R9-C8 will contain only candidate 8. I have not understood your logic making cell R9-C1 as 1 if cell R8-C3 were to be 2.
@willh1655
@willh1655 2 жыл бұрын
if the (2,8) is a 2. then the (1,2) above would be a 1 correct? now if that (1,2) is a 1 the normal sudoku says the (1,8) must be a 1, its the only place a 1 can go in column. that means the 8 cannot be place anywhere, breaking the puzzle. so it cannot be a 2..
@selueen3028
@selueen3028 4 жыл бұрын
7 minutes in, I have 7 numbers on the grid, not great, but decent start for persumable hard puzzle. 25 minutes in, I have 8 numbers on the grid. That's not going well at all.. 49 minutes in... no progress at all, I gave up. I'm curious to see what I've missed.
@ArthurSavage
@ArthurSavage 4 жыл бұрын
This is a pretty hard puzzle. Not uncommon but Andoku after finding a few 3-4 cell chains couldn't finish.
@martinmpop19
@martinmpop19 4 жыл бұрын
I was completely stuck until you showed me the 6 in the middle right.
@boydegg
@boydegg 2 жыл бұрын
Tough!
@dusaprukiyathan1613
@dusaprukiyathan1613 4 жыл бұрын
Z-Wing?
@BritishBeachcomber
@BritishBeachcomber 4 жыл бұрын
Like you, I spotted the centre column, then the rest just fell into place. Not very difficult.
@joek511
@joek511 4 жыл бұрын
My program rates it at 2400 / extreme. I did it in about 2 min. center column is the key.
@call2020
@call2020 4 жыл бұрын
He talks to himself
@XeonSX
@XeonSX 4 жыл бұрын
Does it not feel like cheating using corner / centre and even colours? Surely those Sudokus are hard because it’s hard to keep everything in your head. Using these aids is taking all that complexity away.
@jeffking
@jeffking 4 жыл бұрын
I gave this video 3 likes; one for each puzzle!
@pepshaven6520
@pepshaven6520 4 жыл бұрын
'What am I missing?" he says. How loudly was I screaming at the sceen to tell him what he's missing??? And which program are you using?
@Fern527
@Fern527 3 жыл бұрын
No
@jamesharrer9971
@jamesharrer9971 4 жыл бұрын
I had it mostly solved in my head while you were still hemming and hawing about placing annoying little numbers in!! Don't need all those little numbers---its plain simple logic!!
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