Mark: Proves both vertical and horizontal roping. Mark: How interesting that both types of roping can occur in one puzzle. Also Mark: *proceeds to ignore vertical roping when disambiguating the pencil marks* Still a beautiful and very clever puzzle with a great solve.
I love how many different ways you can solve this puzzle My favorite that I thought of is after noticing the full roping and how each of the sets are all on different ropes, you can do mini sudoku in each box having 1 of each set in every row and column within the box
Phenomenal! Took me 42 minutes and I got there in a much more roundabout way than you did, but I did it! I actually assigned colours to each of the three triples and managed to colour the whole grid before using the white dots to solve which numbers to put in
Incredible. I did not think it was possible to solve by colouring. Thanks for sharing your approach. I focused on *box 8* and immediately understood that the two *diamonds* in its UL e LR corners could contain either *3-5* or *5-7.* The other combination (3-7) was incompatible with the white dots in the same box. Hence, the white dots contained either *1-2-3-4* or *6-7-8-9.* Surprisingly, however, *6-7-8-9* was not compatible with the white dots in *box 5,* due to striping.
Kind of an interesting thing. I accidentally mixed up the diamond and square constraints and the logic still resulted in a valid solve for those rules. Cool puzzle.
This was my fastest solve ever relative to Mark or Simon (5:31), and IMO there's an even nicer way to solve it than what Mark did. My approach was: 1. Realize that there is roping in every single row and column. Row roping is forced by the palindrome lines, and column roping is forced by the cages. 2. Notice that the blue diamonds must all be from {3,5,7}, since the 2s must all occur in green squares. Then the remaining three cells in columns 1,4,7 are all from {1,6,9}. 3. Color all {2,4,8} cells green, all {3,5,7} cells blue, and all {1,6,9} cells red. 4. Notice that by roping, every single group of 3 roping cells has exactly one cell of each color. If you ever had two cells of the same color sharing a row/column within a single 3x3 box, then the roping group for that row/column would be invalid. So from this, you can conclude that every row/column in each box has one cell of each color. So if you already have two cells of a color in one box, you can immediately place the third cell of that color. 5. Color all the cells in the first seven columns using that rule. This allowed me to very quickly color everything but the final two columns: i.imgur.com/0coNRwv.png And from there it's really easy to fill in all the digits using the white dots, remembering that every time you get a digit you can immediately fill it in four other places due to the row/column roping.
The colouring alone would take several minutes, no way screening, colouring and solving took 5 mins. Anyone who is that fast wouldn't be bothered with the colours 😆
I'm normally not that fast, but this one just flew by for me because I spotted the roping tricks immediately. I also always use the conflict checker, so filling in the grid after getting a few numbers is very quick and mechanical.
I did something similar, although you can fully colour the final two columns too, before using the white kropki dots. For example, at the end of row 2 you have a blue/red pair and at the end of row 4 you have a blue/green pair. (It's possible to see that the blue digit is the same blue digit from the blue triplet too.) That x-wing on blues will give repeated blues in the cage one way round, but not the other. That's enough to disambiguate the final two columns. You also don't need all the white kropki dots to complete the puzzle either. I just used the dots in boxes 5 and 8 (although other combos work just as well).
You came up with solving that there's roping both horizontally and vertically but then solved with X-wings instead of the easier roping eliminations for the vertical part! Very impressive still. Thanks for the video.
Mark, a cube (3 dimensions), is a number X to the third power (X^3), such as 2^3=8. Any "positive" power of 2 is any integer greater than zero, as an exponent. This limits the results to exponents of 1(2), 2(4), and 3(8). Zero is excluded as an exponent (which would always produce a value of 1), because zero is not a positive integer.
This was the most fun I've had in a while doing one of these puzzles. It was challenging to get into. But it was rewarding at the start, middle and end. There aren't many puzzles I go back and watch the solve, this was one of them.
It's also possible to fully populate the grid with A-I using the roping, then substitute for numbers based on the white dots. Which I find interesting because rarely is my solve path completely different, given that I learned variant sudoku from here.
That's the way I did it, too, but I messed up and get the rules mixed up (blue being a positive power of two and green being a prime number). It's still a valid puzzle that way (from what I can see), but obviously the solution checker doesn't work for it.
I finished in 26:17 minutes. This was such an interesting puzzle to solve. I had noticed that I was a getting a collection of three sets of triplets and was reminded of many coloring puzzles involving the modular and entropic rules. I started coloring it that way and it was a lot of fun to do. I like how the ending was to disambiguate everything using the cages. Then, I saw the message at the end. That is crazy that there are that many triplets possible. I really like this way of thinking and am glad to see it incorporated in a puzzle so elegantly. I think my favorite part in this puzzle was disambiguating which high set had to belong on the kropki dots in box 6. If it has a 9 on it, this forces both r5c1/r4c6 and r4c7 to both be 1. That was so cool to see. I hope more of this is explored as it was quite interesting. Great Puzzle!
49:26@#4422. Once I figured out that all the rows and columns were roped, it became mostly a coloring exercise. That was tricky, trying to keep track of everything. But once I got it sorted, the dots gave me the digits, and it filled right in.
A codex isn’t necessarily a sacred work or a body of lore, it’s just the technical term for a book made from a stack of sheets bound on one side, as opposed to a scroll. Though nowadays it’s usually only used to refer to old books.
30:27 for me, including messing it up, checking the video to confirm I messed it up, and restarting. That was a fun puzzle, and I liked the note at the end explaining why the rules were what they were!
This puzzle solves beautifully through individual coloring; not only the double roaping becomes more apparent, but that more abstract logic for some reason is extremely appealing to me.
And 2, which Mark lists as valid, isn't a square of any integer. The positive squares are 1, 4, and 9, not 2, 4, and 8. The later are 2 raised to a positive power.
@@michaelnadler596 no this is correct, what you are listing are square numbers, not powers of two. Aka 2 is the base number and not the exponent, but that means 2^0 which equals 1 should still be a valid option as positive power
Amazing how there are different ways to get the same answer. I ended up having a jackson pollock of colours, then picked up on vertical roping. It wasn't until I had all the colours filled in that I was able to deduce what the digits were by the kropki dots. Very satisfying at the end just bashing in bulk numbers
When reading the rules I was flummoxed by "positive power of two". My first thought was that this included 1, since it is a) positive, and b) a power of 2 (namely 2^0). Apparently, this is not what the setter intended. Nevertheless, a quick view of the explanation of the rules by Mark set me right. Thanks!
Finished in 10:27. Fully coloured the first 7 columns in green/blue/pink and then starting with the white dots in box 8, everything just filled itself in
Perfectly designed experiment by *Glen Fletcher* with this new kind of asymmetric triplet: *{2, 4, 8}, {3, 5, 7},* and *{1, 6, 9}.* The stratospherically smart geometry of this construction creates gorgious interactions between clues with totally mind-blowing results. Thank you *Mark* for featuring this cosmic class masterpiece. 👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏
I used a different approach to solve this. I focused on *box 8* and immediately understood that the two *diamonds* in its UL e LR corners could contain either *3-5* or *5-7.* The other combination (3-7) was incompatible with the white dots in the same box. Hence, the white dots contained either *1-2-3-4* or *6-7-8-9.* Surprisingly, however, *6-7-8-9* was not compatible with the white dots in *box 5,* due to *roping.*
31:50, I assumed 2^0 wasn't a "positive power of two" and then kinda missed that 1 wasn't prime until the very end, surprised I wasn't super stuck before then.
13:15 As you said that one from 169 and one from 357 would end up in the bottom row of box 6, two of each in total would be off the dot pattern. In other words, only 1 of each could go into the dot sequence. 2345 takes 2 from 357, so only the 1234 option is left. Same applies to box 8.
14:49 finish. I used logic to speed me up in this one. Because there is roping in both directions, and only one blue diamond and one green square in each section of rope, then there can never be more than one green or blue cell in a row or column in a given box. This made it so much easier to color, and then I simply used letters for each color to differentiate. the numbers took less than 30 seconds at the end. Fun fun fun!
If you start with a solved puzzle and exchange any two rows within a band of three boxes (e.g. R7 and R9 are both in the bottom band), or two columns within a stack of three boxes (e.g. C4 and C5 are both within the centre stack), or even any two complete bands of three boxes (e.g. R4-6 and R7-9) or two stacks of three boxes (e.g. C1-3 and C7-9) within the whole grid, the puzzle remains in a solved state. (And you get some interesting distortions of the Phistomefel Ring and Aad van de Wetering's identity). This is because if you swap over two complete rows (or columns, of course) within a band (or stack) then every column in the puzzle still has the same digits in it afterwards as it had before, and every box within the band (or stack) has the same digits disposed differently. All you have done is swap over either complete Sudoku units, or pairs of digits within the same Sudoku unit. There is a finite (but large!) number of such transformations from any initial grid; so, even if there were enough clues to force the roping pattern both ways, you would definitely need some additional clues to reach a unique solution. And there's also probably a fiendish puzzle to be made by combining a SET identity, an invariant transform of the grid and a set of given digits 1-9 .....
I solved it with a mass colouring job. 3 starts to find a colour scheme that was interpretable. I did the numbers later and took one hell of a lot longer than Mark!
Rules: 02:03 Let's Get Cracking: 03:32 And how about this video's Simarkisms?! Sorry: 4x (09:27, 17:18, 19:56, 23:49) Clever: 4x (14:39, 17:12, 17:12, 19:24) Elegant: 3x (23:14, 23:17, 26:14) Obviously: 3x (05:08, 07:32, 13:06) In Fact: 2x (16:04, 24:36) Pencil Mark/mark: 2x (16:44, 19:54) Bother: 1x (11:37) Lovely: 1x (17:58) Beautiful: 1x (19:10) First Digit: 1x (15:01) By Sudoku: 1x (16:10) Hang On: 1x (19:50) Proof: 1x (19:01) Ah: 1x (22:02) Most popular number(>9), digit and colour this video: Fourteen, Twenty Three, Thirty Four, Eighty Four (2 mentions) Three (49 mentions) Green (10 mentions) Antithesis Battles: Even (3) - Odd (0) White (8) - Black (0) Column (12) - Row (11) FAQ: Q1: You missed something! A1: That could very well be the case! Human speech can be hard to understand for computers like me! Point out the ones that I missed and maybe I'll learn! Q2: Can you do this for another channel? A2: I've been thinking about that and wrote some code to make that possible. Let me know which channel you think would be a good fit!
Loved it! Nice with a new set of triplets and I enjoy roping Eta: I didn't even realize the horizontal roping. I solved through triplets and vertical roping, using colored columns and letters. Looking forwards to trying it again with this new knowledge
I solved it by acident by swapping the blue and green. Took me a while before I realized I made that mistake. It was quite confusing seeing a seemingly fully functional Sudoku not tick complete when I solved it.
I'm not sure how much it would have made the solve quicker but you can place some digits when at 5:52 situation based on the roping. For example the 357 digit at R3C7 has to go in R1C2 because it's certainly not 248 or 169 in purple. It also has to go in R2C5 for the same reason.
Yes, thats how I did it. Placed a few more cells in the same way in the top 3 boxes and then worked on the dots in box 3 - and BTW, 2345 is not a possible fill for the dots in box 3 or it would break the 248 cells (10:20)
@@huxflaxI didn't need them either. I fully coloured the grid using the full roping properties before even considering the white dots. Then the white dot sequences in boxes 5 and 8 were sufficient to assign digits to all the colours.
This was a quick solve, but could have been even faster. As soon as Mark discovered the horizontal roping, he could have placed the square digits from Column 7 onto their respective palindromes and likely concluded each had to be different within their set of three rows.
(I haven't watched the video yet) I've worked out that the whole grid has both vertical and horizontal roping, and my colour schemes are going crazy! I thought with roping going both ways that it would lead me to the location of every number, but I just can't work out what to do. I feel so near, yet so far. Time to watch Mark solve it.
This was very fun to watch, Mark, and a fun puzzle, I am sure. Another one for my list to solve sometime before long! (But it may not be in the month of December!). Thanks for this video, I loved it.
A truly elegant puzzle. Very fun! My only complaint: The green square rule is ambiguous, because sometimes "power" is used to mean the exponent to which a number is raised, and sometimes it means the result of raising a number to its exponent. So, depending on what the player was taught, some will interpret the rule to mean 2 to a positive power (excludes zero exponent) gives {2, 4, 8}, and some will interpret it to mean power of 2 that is positive (includes zero exponent) giving {1, 2, 4, 8}?. It becomes clear soon enough the intention is the former interpretation.
@@RichSmith77 Redundancy is not entirely unknown in rule sets, so while what you say is technically true, it doesn't resolve the ambiguity here because 1 is still a positive power of 2 by that interpretation. Though I see now the rules in the description of been updated to clarify the set is {2, 4, 8}.
I didn't know whether the green square rule referred to a positive "power of two" or a "positive power" of two. The difference is that the former includes 1 while the latter doesn't.
Very interesting puzzle. I went full ham with 9 colours from the start! I never used the white dots in box 6, and it appears they are not required for a unique solution. Probably means I over complicated the solve
I'm pretty sure I've seen more than one puzzle on the channel where the solution has been fully roped in both directions. Usually neither yourself or Simon notice
Zero is not considered positive or negative; it is neutral. Therefore 1 doesn't work. I had considered it myself, until I remembered hearing about zero.
The "positive" here applies to the exponent; "a positive power of 2" really means "2 to a positive power". Since 0 is not positive, 1 = 2⁰ doesn't qualify.
@@markjreed I agree based on the solve but not on the language. The language is ambiguous. There is nothing in the that forces it to reference the exponent. It could reference the result of the calculation.
Heh - I solved the puzzle and was told I got the solution wrong. Spent a few minutes trying to figure out my error. It didn't click until I watched the video - I swapped the rules for green squares and blue diamonds.
Lovely puzzle! ...Although this strengthens my conviction that we need to use hyphens more in English, because I initially thought the rules meant a positive power-of-two, not a positive-power of two. xD
Interesting puzzle. The rule was ambiguous; it was unclear whether to read it as "a positive [power of two]", i.e., 1, 2, 4 or 8, or "[a positive power] of two", i.e. 2^1, 2^2 or 2^3. I guess in case the first reading had been correct, the qualification that the power be positive wouldn't have been necessary.
I started by filling the middle grid with the letters A-i then working outwards from there. The thing I'm wondering about is, seeing that this fully vertically and horizontally roped puzzle uses the same sets of roping digits in each column and row of boxes, is this always the case? I suspect it is.
Every power of 2 is positive though. Saying (positive) (power of 2) is no different than simply saying (power of 2). So it only makes sense to include the qualifier "positive" if it's read as (positive power) (of 2). Thereby ruling out 2⁰ (=1) since 0 is not a positive power.
@@stevewithaq There's nothing in the rules that says that "positive" refers to the exponent. The rules could have read "power of two with a positive exponent" or something like that. Or just "2, 4, 8" which is shorter than the current ambiguous rule.
Is 1 prime or not? Sometimes in these puzzles it is, and sometimes it isn't. I happened to learn that 1 is special and not prime. But I've met a lot of younger folks that seem to have been taught that 1 is prime. Then there's the powers of 2. 2^0 is 1, which is a positive number that is a power of 2. That is a positive power of 2. Does the word "positive" apply to the result - positive (power of 2) - or just to the the power - (positive power) of 2? Lots of ambiguity here, making the puzzle unsolvable without additional assumptions.
1 is never prime, and the people who think it is are just wrong. In this case (positive power) was meant, not positive number that's a power of 2, but maybe it's slightly confusing.
@@Tahgtahv one has been on and off considered a prime over history and it's completely arbitrary. The only reason one isn't considered a prime is because it makes things awkward to word when you have to exclude 1 from rules involving prime, but the definition of primes end up excluding 1 arbitrarily which is just as awkward. If you don't include that exclusion then 1 is prime so anyone who is just taught the rule in a simplified way will learn that 1 is prime cause that's the only conclusion you can have if you just look at the general pattern.
@@Laezar1 Right. The general rule for prime numbers is that they are divisible by only 1 and itself. 1 fits that definition, so 1 should be prime. Some teach that it is. Some teach that it's special. Kind of like the people that think their way is the only right way.
1 is not a prime number as it doesn't meet the definition of a number that has exactly two integer factors. Anyone who claims it is or can be a prime is either mistaken or deliberately making trouble for the sake of it. As normal sudoku rules apply, there is no possibility of having digits that are _not_ positive numbers, so in the case of the green squares, positive must be referring to the exponent or else it would be completely redundant.
@@emilywilliams3237 You still need the cage to disambiguate the x-wings, although the extra cell is not necessary, since it sees all other cells in the cage. Of course what he did was valid, although he needed to use different logic later in the puzzle.
The rules refer to a positive power of 2, and 0 isn't positive so 2⁰=1 isn't a valid every on a 🟩. This limits them to {2,4,8} and so 2 isn't available as an option for the 🔷s.
@stevieinselby I don't disagree with that. I was mainly commenting on the necessary conditions needed to use these as valid triplets based on the end comments.
I think the rules ought to be a bit clearer, when i tried this, i took one as a power of 2 as it is 2 to the poeer of zero. Wasted a lot of time because of that. :[
I've been meaning to ask for a while - could you refrain from putting the rules on that bright green background please? Something about it and the black text really messes with my eyes 😢
Mark: Proves both vertical and horizontal roping.
Mark: How interesting that both types of roping can occur in one puzzle.
Also Mark: *proceeds to ignore vertical roping when disambiguating the pencil marks*
Still a beautiful and very clever puzzle with a great solve.
Horizontal and Vertical roping should be called a basketweave
Yes, obviously.
I love how many different ways you can solve this puzzle
My favorite that I thought of is after noticing the full roping and how each of the sets are all on different ropes, you can do mini sudoku in each box having 1 of each set in every row and column within the box
Phenomenal! Took me 42 minutes and I got there in a much more roundabout way than you did, but I did it!
I actually assigned colours to each of the three triples and managed to colour the whole grid before using the white dots to solve which numbers to put in
I did that too
Incredible. I did not think it was possible to solve by colouring. Thanks for sharing your approach.
I focused on *box 8* and immediately understood that the two *diamonds* in its UL e LR corners could contain either *3-5* or *5-7.* The other combination (3-7) was incompatible with the white dots in the same box. Hence, the white dots contained either *1-2-3-4* or *6-7-8-9.*
Surprisingly, however, *6-7-8-9* was not compatible with the white dots in *box 5,* due to striping.
I did the same thing, the cages really helped in figuring out which digits were the same colours and eventually the white dots gave us the digits.
Kind of an interesting thing. I accidentally mixed up the diamond and square constraints and the logic still resulted in a valid solve for those rules. Cool puzzle.
This was my fastest solve ever relative to Mark or Simon (5:31), and IMO there's an even nicer way to solve it than what Mark did. My approach was:
1. Realize that there is roping in every single row and column. Row roping is forced by the palindrome lines, and column roping is forced by the cages.
2. Notice that the blue diamonds must all be from {3,5,7}, since the 2s must all occur in green squares. Then the remaining three cells in columns 1,4,7 are all from {1,6,9}.
3. Color all {2,4,8} cells green, all {3,5,7} cells blue, and all {1,6,9} cells red.
4. Notice that by roping, every single group of 3 roping cells has exactly one cell of each color. If you ever had two cells of the same color sharing a row/column within a single 3x3 box, then the roping group for that row/column would be invalid. So from this, you can conclude that every row/column in each box has one cell of each color. So if you already have two cells of a color in one box, you can immediately place the third cell of that color.
5. Color all the cells in the first seven columns using that rule. This allowed me to very quickly color everything but the final two columns: i.imgur.com/0coNRwv.png
And from there it's really easy to fill in all the digits using the white dots, remembering that every time you get a digit you can immediately fill it in four other places due to the row/column roping.
Exactly the way I meant for you to solve the puzzle.
The colouring alone would take several minutes, no way screening, colouring and solving took 5 mins.
Anyone who is that fast wouldn't be bothered with the colours 😆
I'm normally not that fast, but this one just flew by for me because I spotted the roping tricks immediately. I also always use the conflict checker, so filling in the grid after getting a few numbers is very quick and mechanical.
Hahah ok, good job then
I did something similar, although you can fully colour the final two columns too, before using the white kropki dots.
For example, at the end of row 2 you have a blue/red pair and at the end of row 4 you have a blue/green pair. (It's possible to see that the blue digit is the same blue digit from the blue triplet too.) That x-wing on blues will give repeated blues in the cage one way round, but not the other. That's enough to disambiguate the final two columns.
You also don't need all the white kropki dots to complete the puzzle either. I just used the dots in boxes 5 and 8 (although other combos work just as well).
You came up with solving that there's roping both horizontally and vertically but then solved with X-wings instead of the easier roping eliminations for the vertical part! Very impressive still. Thanks for the video.
Mark, a cube (3 dimensions), is a number X to the third power (X^3), such as 2^3=8. Any "positive" power of 2 is any integer greater than zero, as an exponent. This limits the results to exponents of 1(2), 2(4), and 3(8). Zero is excluded as an exponent (which would always produce a value of 1), because zero is not a positive integer.
This was the most fun I've had in a while doing one of these puzzles. It was challenging to get into. But it was rewarding at the start, middle and end. There aren't many puzzles I go back and watch the solve, this was one of them.
I've become fascinated by sudoku grid theory ever since you and Simon started explaining the phistomefel ring and various other SET type things
It's also possible to fully populate the grid with A-I using the roping, then substitute for numbers based on the white dots. Which I find interesting because rarely is my solve path completely different, given that I learned variant sudoku from here.
That's the way I did it (only with colours rather than letters).
That's the way I did it, too, but I messed up and get the rules mixed up (blue being a positive power of two and green being a prime number). It's still a valid puzzle that way (from what I can see), but obviously the solution checker doesn't work for it.
Coloring the roping for whichever area I was working on really helped my solve!
I finished in 26:17 minutes. This was such an interesting puzzle to solve. I had noticed that I was a getting a collection of three sets of triplets and was reminded of many coloring puzzles involving the modular and entropic rules. I started coloring it that way and it was a lot of fun to do. I like how the ending was to disambiguate everything using the cages. Then, I saw the message at the end. That is crazy that there are that many triplets possible. I really like this way of thinking and am glad to see it incorporated in a puzzle so elegantly. I think my favorite part in this puzzle was disambiguating which high set had to belong on the kropki dots in box 6. If it has a 9 on it, this forces both r5c1/r4c6 and r4c7 to both be 1. That was so cool to see. I hope more of this is explored as it was quite interesting. Great Puzzle!
49:26@#4422. Once I figured out that all the rows and columns were roped, it became mostly a coloring exercise. That was tricky, trying to keep track of everything. But once I got it sorted, the dots gave me the digits, and it filled right in.
22:52 you can use the roping. 5-6-8 can only go in the second column of box 3.
A codex isn’t necessarily a sacred work or a body of lore, it’s just the technical term for a book made from a stack of sheets bound on one side, as opposed to a scroll. Though nowadays it’s usually only used to refer to old books.
30:27 for me, including messing it up, checking the video to confirm I messed it up, and restarting.
That was a fun puzzle, and I liked the note at the end explaining why the rules were what they were!
This puzzle solves beautifully through individual coloring; not only the double roaping becomes more apparent, but that more abstract logic for some reason is extremely appealing to me.
Finished in 27:55 ...... I was very surprised about my quick solution. A very enjoyable puzzle.
"positive power of 2" is a bit ambiguous isn't it? perhaps you mean "2 to a positive power" else you could argue 1 is a positive power of 2
Maybe should be even powers of 2 - one is a positive power of 2.
Yeah, how does Mark know the green squares can't be 2 to the zeroth power? How are we supposed to know???
And 2, which Mark lists as valid, isn't a square of any integer. The positive squares are 1, 4, and 9, not 2, 4, and 8. The later are 2 raised to a positive power.
@@michaelnadler596 no this is correct, what you are listing are square numbers, not powers of two.
Aka 2 is the base number and not the exponent, but that means 2^0 which equals 1 should still be a valid option as positive power
Well, given that standard sudoku rules apply, it follows that we're talking about whole numbers from 1 to 9...
Amazing how there are different ways to get the same answer. I ended up having a jackson pollock of colours, then picked up on vertical roping. It wasn't until I had all the colours filled in that I was able to deduce what the digits were by the kropki dots. Very satisfying at the end just bashing in bulk numbers
I love the primes + 2 powers + etc triplets. After many low/med/high and modular(3) puzzles, a new triplet is very fun.
When reading the rules I was flummoxed by "positive power of two". My first thought was that this included 1, since it is a) positive, and b) a power of 2 (namely 2^0). Apparently, this is not what the setter intended. Nevertheless, a quick view of the explanation of the rules by Mark set me right. Thanks!
Finished in 10:27. Fully coloured the first 7 columns in green/blue/pink and then starting with the white dots in box 8, everything just filled itself in
Perfectly designed experiment by *Glen Fletcher* with this new kind of asymmetric triplet:
*{2, 4, 8}, {3, 5, 7},* and *{1, 6, 9}.*
The stratospherically smart geometry of this construction creates gorgious interactions between clues with totally mind-blowing results.
Thank you *Mark* for featuring this cosmic class masterpiece.
👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏
I used a different approach to solve this.
I focused on *box 8* and immediately understood that the two *diamonds* in its UL e LR corners could contain either *3-5* or *5-7.* The other combination (3-7) was incompatible with the white dots in the same box. Hence, the white dots contained either *1-2-3-4* or *6-7-8-9.*
Surprisingly, however, *6-7-8-9* was not compatible with the white dots in *box 5,* due to *roping.*
31:50, I assumed 2^0 wasn't a "positive power of two" and then kinda missed that 1 wasn't prime until the very end, surprised I wasn't super stuck before then.
13:15 As you said that one from 169 and one from 357 would end up in the bottom row of box 6, two of each in total would be off the dot pattern. In other words, only 1 of each could go into the dot sequence. 2345 takes 2 from 357, so only the 1234 option is left.
Same applies to box 8.
16:38 ... solutions like this I always find fascinating
Nice puzzle!
The interaction between the dots and cages was nice.
14:49 finish. I used logic to speed me up in this one. Because there is roping in both directions, and only one blue diamond and one green square in each section of rope, then there can never be more than one green or blue cell in a row or column in a given box. This made it so much easier to color, and then I simply used letters for each color to differentiate. the numbers took less than 30 seconds at the end. Fun fun fun!
If you start with a solved puzzle and exchange any two rows within a band of three boxes (e.g. R7 and R9 are both in the bottom band), or two columns within a stack of three boxes (e.g. C4 and C5 are both within the centre stack), or even any two complete bands of three boxes (e.g. R4-6 and R7-9) or two stacks of three boxes (e.g. C1-3 and C7-9) within the whole grid, the puzzle remains in a solved state. (And you get some interesting distortions of the Phistomefel Ring and Aad van de Wetering's identity). This is because if you swap over two complete rows (or columns, of course) within a band (or stack) then every column in the puzzle still has the same digits in it afterwards as it had before, and every box within the band (or stack) has the same digits disposed differently. All you have done is swap over either complete Sudoku units, or pairs of digits within the same Sudoku unit.
There is a finite (but large!) number of such transformations from any initial grid; so, even if there were enough clues to force the roping pattern both ways, you would definitely need some additional clues to reach a unique solution.
And there's also probably a fiendish puzzle to be made by combining a SET identity, an invariant transform of the grid and a set of given digits 1-9 .....
I solved it with a mass colouring job. 3 starts to find a colour scheme that was interpretable. I did the numbers later and took one hell of a lot longer than Mark!
Rules: 02:03
Let's Get Cracking: 03:32
And how about this video's Simarkisms?!
Sorry: 4x (09:27, 17:18, 19:56, 23:49)
Clever: 4x (14:39, 17:12, 17:12, 19:24)
Elegant: 3x (23:14, 23:17, 26:14)
Obviously: 3x (05:08, 07:32, 13:06)
In Fact: 2x (16:04, 24:36)
Pencil Mark/mark: 2x (16:44, 19:54)
Bother: 1x (11:37)
Lovely: 1x (17:58)
Beautiful: 1x (19:10)
First Digit: 1x (15:01)
By Sudoku: 1x (16:10)
Hang On: 1x (19:50)
Proof: 1x (19:01)
Ah: 1x (22:02)
Most popular number(>9), digit and colour this video:
Fourteen, Twenty Three, Thirty Four, Eighty Four (2 mentions)
Three (49 mentions)
Green (10 mentions)
Antithesis Battles:
Even (3) - Odd (0)
White (8) - Black (0)
Column (12) - Row (11)
FAQ:
Q1: You missed something!
A1: That could very well be the case! Human speech can be hard to understand for computers like me! Point out the ones that I missed and maybe I'll learn!
Q2: Can you do this for another channel?
A2: I've been thinking about that and wrote some code to make that possible. Let me know which channel you think would be a good fit!
34:06 for me - took an embarrassingly long time at the end trying to solve "ambiguities" that were created by me forgetting that 1 isn't prime.
Finished in 28:24. Wonderful puzzle aside from the ambiguously worded rules.
Loved it! Nice with a new set of triplets and I enjoy roping
Eta: I didn't even realize the horizontal roping. I solved through triplets and vertical roping, using colored columns and letters.
Looking forwards to trying it again with this new knowledge
Instead of using the x-wings at the end, I used the vertical roping.
Fantastic puzzle! Really enjoy these colouring puzzles.
Really enjoyed thus. Spotted the pattern quickly, 14:38 for me
I solved it by acident by swapping the blue and green. Took me a while before I realized I made that mistake. It was quite confusing seeing a seemingly fully functional Sudoku not tick complete when I solved it.
I'm not sure how much it would have made the solve quicker but you can place some digits when at 5:52 situation based on the roping. For example the 357 digit at R3C7 has to go in R1C2 because it's certainly not 248 or 169 in purple. It also has to go in R2C5 for the same reason.
Yes, thats how I did it. Placed a few more cells in the same way in the top 3 boxes and then worked on the dots in box 3 - and BTW, 2345 is not a possible fill for the dots in box 3 or it would break the 248 cells (10:20)
Nice puzzle. I never used the white dots in boxes 3 and 6.
So where did your disambiguation start? If not in box 5 and then 6 as the first steps
@@huxflaxI didn't need them either. I fully coloured the grid using the full roping properties before even considering the white dots. Then the white dot sequences in boxes 5 and 8 were sufficient to assign digits to all the colours.
I solved it in a different way, and I'm kinda proud of it :D Interesting puzzle!
SQUARES are to the power of two. CUBES are to the power of three.
This was a quick solve, but could have been even faster. As soon as Mark discovered the horizontal roping, he could have placed the square digits from Column 7 onto their respective palindromes and likely concluded each had to be different within their set of three rows.
(I haven't watched the video yet) I've worked out that the whole grid has both vertical and horizontal roping, and my colour schemes are going crazy! I thought with roping going both ways that it would lead me to the location of every number, but I just can't work out what to do. I feel so near, yet so far. Time to watch Mark solve it.
Oh, I did it. Brilliant! A bit slower than you Mark, but sound logic utilised nonetheless.
This was very fun to watch, Mark, and a fun puzzle, I am sure. Another one for my list to solve sometime before long! (But it may not be in the month of December!). Thanks for this video, I loved it.
Very nice puzzle. 27:00.
A truly elegant puzzle. Very fun!
My only complaint: The green square rule is ambiguous, because sometimes "power" is used to mean the exponent to which a number is raised, and sometimes it means the result of raising a number to its exponent. So, depending on what the player was taught, some will interpret the rule to mean 2 to a positive power (excludes zero exponent) gives {2, 4, 8}, and some will interpret it to mean power of 2 that is positive (includes zero exponent) giving {1, 2, 4, 8}?. It becomes clear soon enough the intention is the former interpretation.
There is no power of 2 that is not positive though, so saying the result of the power of 2 needs to be positive would be pretty redundant.
@@RichSmith77 Redundancy is not entirely unknown in rule sets, so while what you say is technically true, it doesn't resolve the ambiguity here because 1 is still a positive power of 2 by that interpretation. Though I see now the rules in the description of been updated to clarify the set is {2, 4, 8}.
16:53! The patterns in this one were very fun.
8:21 for me. Don't know why it took me that long, we've seen this idea many times before.
That was fun. Took me a while, mainly due to obsessive colouring...
51:19, all by myself)
27:26 - Happy with that.
I didn't know whether the green square rule referred to a positive "power of two" or a "positive power" of two. The difference is that the former includes 1 while the latter doesn't.
Very interesting puzzle. I went full ham with 9 colours from the start!
I never used the white dots in box 6, and it appears they are not required for a unique solution. Probably means I over complicated the solve
I'm pretty sure I've seen more than one puzzle on the channel where the solution has been fully roped in both directions. Usually neither yourself or Simon notice
1 is a positive power of 2
Zero is not considered positive or negative; it is neutral. Therefore 1 doesn't work. I had considered it myself, until I remembered hearing about zero.
Zero is classified as "non-negative," not positive, so by normally-accepted definitions 1 is not a positive power of 2.
The "positive" here applies to the exponent; "a positive power of 2" really means "2 to a positive power". Since 0 is not positive, 1 = 2⁰ doesn't qualify.
Thx for the answers
@@markjreed I agree based on the solve but not on the language. The language is ambiguous. There is nothing in the that forces it to reference the exponent. It could reference the result of the calculation.
Heh - I solved the puzzle and was told I got the solution wrong. Spent a few minutes trying to figure out my error. It didn't click until I watched the video - I swapped the rules for green squares and blue diamonds.
Lovely puzzle!
...Although this strengthens my conviction that we need to use hyphens more in English, because I initially thought the rules meant a positive power-of-two, not a positive-power of two. xD
26:35 for me. Quite neat, though not a fan of the power of two rules wording.
Interesting puzzle. The rule was ambiguous; it was unclear whether to read it as "a positive [power of two]", i.e., 1, 2, 4 or 8, or "[a positive power] of two", i.e. 2^1, 2^2 or 2^3. I guess in case the first reading had been correct, the qualification that the power be positive wouldn't have been necessary.
Titi's Time: 32:48
20:50 from this point couldn't you just use the vertical roping?
Did Mark never notice that the solution to Mitchell Lee's Miracle Sudoku (4.3M views) had both horizontal and vertical roping?
I started by filling the middle grid with the letters A-i then working outwards from there.
The thing I'm wondering about is, seeing that this fully vertically and horizontally roped puzzle uses the same sets of roping digits in each column and row of boxes, is this always the case? I suspect it is.
Fun note, if you flip the rules of green squares and blue diamonds the puzzle also solves. I found this out by accident.
13:01 very easy and fun :)
Surely 1 is a positive power of 2? It's 2^0 and it's positive. If the setter meant 2^n where n>0 then maybe they'd have said so?
Every power of 2 is positive though.
Saying (positive) (power of 2) is no different than simply saying (power of 2). So it only makes sense to include the qualifier "positive" if it's read as (positive power) (of 2). Thereby ruling out 2⁰ (=1) since 0 is not a positive power.
Was hoping the design could be letterable/colourable,but even if u can do dots in B1 and B8. Still no way to differentiate 789
Fun design tho
1 is a positive power of 2. It's a positive number and it's two to the power of zero.
Zero is not a positive number.
Positive refers to the power. 0 is not positive.
@@stevewithaq There's nothing in the rules that says that "positive" refers to the exponent. The rules could have read "power of two with a positive exponent" or something like that. Or just "2, 4, 8" which is shorter than the current ambiguous rule.
the power is what needs to be positive, not the result of taking 2 to said power
I read the rule different way, that the exponent needed to be positive. Which would exclude 2^0. But it was a bit ambiguous.
I also made the 2^0=1 mistake, after all, the answer 1 is positive as required by the ruleset.
You didn't make a mistake. The rules contain the mistake :)
Is 1 prime or not? Sometimes in these puzzles it is, and sometimes it isn't. I happened to learn that 1 is special and not prime. But I've met a lot of younger folks that seem to have been taught that 1 is prime. Then there's the powers of 2. 2^0 is 1, which is a positive number that is a power of 2. That is a positive power of 2. Does the word "positive" apply to the result - positive (power of 2) - or just to the the power - (positive power) of 2? Lots of ambiguity here, making the puzzle unsolvable without additional assumptions.
1 is never prime, and the people who think it is are just wrong. In this case (positive power) was meant, not positive number that's a power of 2, but maybe it's slightly confusing.
@@Tahgtahv one has been on and off considered a prime over history and it's completely arbitrary. The only reason one isn't considered a prime is because it makes things awkward to word when you have to exclude 1 from rules involving prime, but the definition of primes end up excluding 1 arbitrarily which is just as awkward. If you don't include that exclusion then 1 is prime so anyone who is just taught the rule in a simplified way will learn that 1 is prime cause that's the only conclusion you can have if you just look at the general pattern.
@@Laezar1 Right. The general rule for prime numbers is that they are divisible by only 1 and itself. 1 fits that definition, so 1 should be prime. Some teach that it is. Some teach that it's special. Kind of like the people that think their way is the only right way.
@@Tahgtahv How do you know that (positive power) of 2 was meant rather than positive (power of 2)? Other than the puzzle solves using that assumption?
1 is not a prime number as it doesn't meet the definition of a number that has exactly two integer factors. Anyone who claims it is or can be a prime is either mistaken or deliberately making trouble for the sake of it.
As normal sudoku rules apply, there is no possibility of having digits that are _not_ positive numbers, so in the case of the green squares, positive must be referring to the exponent or else it would be completely redundant.
00:31:02
The extra cell in the cage doesn't actually change the logic of the puzzle, since it sees all the other cells by sudoku.
Mark used that "extra" cell to prove which way the x-wings had to be solved. That would not be clear from just the sudoku, would it?
@@emilywilliams3237 You still need the cage to disambiguate the x-wings, although the extra cell is not necessary, since it sees all other cells in the cage. Of course what he did was valid, although he needed to use different logic later in the puzzle.
Is 1 not considered a prime number?
The powers of 2 are 1,4 and 9
Those are square numbers. They are not powers of 2, they're digits "to the power of 2”.
Primes only count as a triplet when you eliminate the even prime. Powers of 2 only count when you eliminate 2^0.
The rules refer to a positive power of 2, and 0 isn't positive so 2⁰=1 isn't a valid every on a 🟩. This limits them to {2,4,8} and so 2 isn't available as an option for the 🔷s.
@stevieinselby I don't disagree with that. I was mainly commenting on the necessary conditions needed to use these as valid triplets based on the end comments.
Forget that. Try to have some fun with it, that’s the whole point.
Was thinking the same. People are so tense 😂
Why 1 is not a positive power of two? 2º=1
Zero is not positive.
24:48 for me
Oops - SOLVES with blue diamonds powers of 2 and green squares prime - too!
22:32 for me. i overcomplicated the solve.
21:30 for me and solver #1739.
12m 35s
Someone tried to solve considering 1 as a power of 2? Please lemme know :)
1 is prime gang here
I think the rules ought to be a bit clearer, when i tried this, i took one as a power of 2 as it is 2 to the poeer of zero. Wasted a lot of time because of that. :[
Not a fan of having to know number theory to understand deep details about primes, just say what the numbers are in the cells
WAIT, 1 is definitely a power of 2, it's 2^0. I banged my head on this sudoku going nowhere because of this. Really bad set of rules in my opinion
I've been meaning to ask for a while - could you refrain from putting the rules on that bright green background please? Something about it and the black text really messes with my eyes 😢