The Doppler Effect: Another sudoku of crazy brilliance 

Hey everyone, Knickolas here. I’m very excited to watch this one! Thanks Mark and Simon as always for the feature and for the amazing content!!! An interesting thing I noticed while setting this puzzle is that there’s no valid way to resolve the galaxies if you have an even lengthed stretch of a galaxy along the perimeter (not including the corner cells). I was able to prove this using code but I wasn’t able to find any logical proof of why this breaks the puzzle. If anyone can find a reason for this I’d be very interested in learning why this is so :)

That Zippersnake poem is incredible!

I'm absolutely not going to be doing this every day (I can only imagine what it would be like trying to describe a wrogn puzzle in verse...) but I figured I'd do one more for the people who enjoyed yesterday's effort, this time in iambic pentameter: To fill the grid, regard the Doppler shift of distant spiral galaxies that drift and spin within the inky night-time sky, with rules to tell you where their arms must lie. Each galaxy remains, you'll surely learn, symmetrical when spun through half a turn. A further rule must always be obeyed: orthogonal connections must be made. A pair of regions must be rightly placed, adhering to these guidelines boldly faced: No galaxy may increase to accrue all four cells lying in a 2x2. The grid (in full) must occupy this map, yet only in one cell they overlap: the cell in row and column numbered fifth. The galaxies are different in their shift. One galaxy, its Doppler shift in blue, has values adding one less to each clue. One galaxy, its Doppler shift in red, has values counting for one more instead. Each cage's total indicates the sum of values which its contained cells become; no digit twice into a cage may fit, but shifted values may, if sums permit. The values which on arrowed lines you place must sum to match the value at its base; and finally, the rule that never shocks: in every column, every row and box there must appear the digits 1 to 9 with no repeats in any box or line. With such a grid, and regions red and blue, Doppler Effect's solution will be true.

For the concise and general proof that checkerboards always break, I think it goes something like this: the crossing rule means each galaxy has exactly two distinct arms, each connecting to the center cell from opposite sides. For two checkers of color A to connect to the same galaxy they must EITHER connect to the same arm or to opposite arms. If they're the connecting to same arm they will surround one or the other color B checker, leaving it no way to reach the center. If they connect to opposite arms those arms touch diagonally and so their galaxy forms a sort of figure-of-8 shape that fully encloses the color B galaxy, leaving one of the color B checkers fully blocked outside.

Now there's a challenge for setters..... rhyming rulesets!

Easy way of seeing the break-in with the perimeter: all coloured components of the perimeter separately have to connect to the centre. The centre only has four sides. This also makes it easy to see that there are no checkerboards - once those four components on the perimeter have used the four sides of the centre, there's no more sides available for any wierdness with checkerboards (if you prefer, we've divided into four smaller sub-boards, and we're playing normal ying-yang on each).

that crafty little 5 = 3+1 arrow all redshift nearly caught me out as I assumed it was "all red so all same" forgetting the two arrow digits were two shifts yet the circle only 1.

I feel like it's helpful to note that once you have all the colors of a killer cage in this ruleset which digits are in which color cells doesn't matter at all, the commutativity of addition means it's the same overall net modifier to the overall total regardless of how you move the digits around in it. Feel like Simon added difficulty for himself by continuing to keep track of which cells were getting +1 and which -1 while adding up rather than just mentally adjusting the overall target and then solving it like a normal killer cage...

One of the hardest puzzles I felt very able to tackle on my own, every breakthrough felt incredible to find and finishing it just leaves you with such an amazing feeling you can’t find anywhere else. Great puzzle.

I think a more elegant way to articulate the no-checkerboards rule might be as follows: In classic Yin-Yang, you can only have two changes of color on the perimeter and no checkerboarding because, at some point, the two groups will need to cross and you are not allowed any crossings. You would need at least one crossing for each pair of changes in the perimeter to have checkerboarding. In this puzzle, you allow exactly one crossing. Thus, you are permitted a max of four changes of color in the perimeter, but you would need a second crossing in order to allow checkerboarding.

I think it is quite funny to hear Simon refer to the two galaxies with all terminology conceivable *except* the terminology _actually_ used by the puzzle itself. He really only used the "shifted" terminology like twice in the entire solve.

I surmised that once you’ve got the two colours crossing in the middle, each half of the grid represents a separate grid, so there can’t be more than one change of colour on the border of each half and normal yin yang rules apply. I also made things easier when I realised that a cage sum with two colours could be worked out by ignoring the pair and adding 1 for each of the others

Simon spent 25 minutes convincing himself that there were only 4 border regions and there couldn't be checkerboarding, which was intuitively obvious to me since it made too many regions chase too few connections. However, his meticulous approach is doubtlessly superior for trying to get the right answer every time instead of trying to get the right answer quickly, but only most of the time.

In this case, the galaxy rules are just a more restricted set of yinyang-rules, adding the symmetry part. The central field that allows crossover just means that the perimeter may have 4 stretches of colour instead of only two, and would allow for one more checkerboard to be resolved. But galaxy symmetry rules dictate that there are always an even number of checkerboards. Therefore, no checkerboards are allowed.

I was VERY happy when you made the colours blue and orange, I had the same thought in the beginning of the solve, and was glad you found your way there ❤

Very fresh idea, beautifully composed and solved. I admire the poetry especially dealing with such a difficult subject. Altogether, wonderful Sunday logic show!

Another amazing solve, Simon - you way you tackled the ruleset, and juggled the constraints whilst finding that neat logical path. I aspire to be as clear-minded as that!

Another very interesting ruleset!

Not my first video, but sharing with you, your videos reached Brazilians!!

This was much, much easier, but if you don't know the trick, it's like trying a puzzle that needs SET, but you've never heard of SET. You're missing a powerful tool for puzzles like this. You need to think about them topologically. It avoids tying yourself in knots trying to think of ever more complicated ways of doing things, and worrying that you can't prove something because even though something seems impossible, you might have missed the one potential way of doing it. Thinking topologically means you can simplify the puzzle as much as possible. What do I mean by thinking topologically? The spiky head of a mace is topologically a sphere because there is only one surface with no holes. The spikes are irrelevant to the topology. A cube is also topologically the same as a sphere. A rubber band is topologically a torus no matter how many twists and knots you put in it. If you cut it, or punch a hole in it, it is no longer a torus. In application to this puzzle, it doesn't matter how circuitous you try to make the pattern, it is still topologically a simple cross. We know this, because there are at least four changes of colour on the perimeter, so we can be certain that each arm of the central cross must touch the sides. The puzzle can be represented as a horizontal green line filling R5 and a vertical yellow line filling C5. Now it's trivial to see that more colour changes are impossible. You only need to consider any single quadrant. Each quadrant contains a green arm and a yellow arm. All yellow cells in the quadrant need to connect to the yellow arm, and all green cells need to connect to the green arm, and they can't isolate any of the other colour in doing so. Topologically, all you are doing is widening both the green arm and the yellow arm until they meet. There is no way for any green cell to touch the other green arm in the other half of the grid. Ditto for yellow. The puzzle has now reduced to four yin yang puzzles, and yes, chequerboards are impossible. It doesn't matter how convoluted you make the boundary, it has to reduce to the same topology, so all you have to do is prove that it's impossible for the simple case to know that it's impossible for any shape of boundary. Armed with this, we know that between each marked boundary on the edge there must be a single colour. We know that chequerboards are not possible, the rules forbid 2x2s, and we can colour almost the entire grid. I was able to colour all but 8 cells - a domino in R2C6/7 which had to be one of each colour, and the vertical domino in the 19 cage (and their reflections). I coloured the regions green and purple until it became obvious which was red and which blue, which happened when the 13 and 14 cages had to be all one colour. They couldn't be blue, because they'd have to sum to 31, so they must be red. When it came to the numbers, you kept forgetting that the placement of the digits doesn't matter. For instance, in the 23 cage, you were saying "If the 1 goes there it counts as 2". It doesn't matter at all, because all the colours do is apply an adjustment to the target total. All you need to know is the effective total, which means the 23 cage has to contain digits which sum to 25. Even knowing the trick with the colouring, and realising that the placement of the digits in cages had no effect on the total, this was still a very interesting puzzle and brilliant setting. Depending on which elements they are on, the red shift and blue shift either had a subtle effect, or a profound effect (or none at all), which adds an interesting nuance to the puzzle. I'd love to see more like this.

This was, as usual, a very interesting solve. You are a master of the geographical puzzles, Simon, and I enjoy watching you solve them and I always try to learn from you. Thanks for this video!

1:30 Thank you so much 😊

Watching a long simon solve after getting out of bed always gets me ready for the day, so I certainly wouldn't mind if every puzzle was like this.

I am at the beginning and it's so funny to see Simon struggling with using "The galaxy that reduces the value of the digits" and "The galaxy that increases the value of the digits" in a sentence. Because the words are right there for him... redshifted and blueshifted lol

What an amazing puzzle 😻 I figured out the perimeter must have 4 sections (and remembered to use the galaxy rotation to determine where those sections were) and that there couldn't be a checkerboard pattern within about 10 minutes, managed to colour the whole grid correctly, and solved _most_ of the puzzle as well ... but some erroneous careless pencil marking led me astray with a couple of minor mistakes 😢

A really fun puzzle. It took me a while, but still a bit quicker than Simon so I'm really chuffed 😊

I prefer a balance of hard vs soft puzzles. I enjoy watching both - the difficult ones just to see how they unwind and the easier ones so that I can participate and telepathically send Simon an easy deduction I’ve made while he was conquering the main point of the puzzle.

the reason why checkerboards dont work is the centre only has 4 ways you can connect to it so you need 4 sperate reigions by having checkerboards you are creating 2 more reigions which wont be able to get into the centre

Brilliant solve Simon

I like how some creators are mixing some rules to make even better puzzles! 😀

Brilliant!

1:43:52 - It did take me nearly an hour to realise the checkerboard and perimeter implications but once I’d worked that out it flowed very nicely. Another gorgeous puzzle from @Knickolas

I haven't been able to solve a puzzle in a while, I don't know if I'm getting worse instead of improving with time. Or maybe the puzzles are getting harder on average, what do you guys think? I'm losing confidence

I thought he said sexual life at 52:15

Easy anti-checkerboard explanation: There's 4 branches from the middle square. (Easier, I think, to think of it as 4 distinct colours.) There's 4 sections on the outer boarder. Has to be a 1:1 correlation. A checkerboard either connects to the middle (which would require more branches from the middle). Or connects to 1 of the original 4 outer sections which isolates another section. Ergo: no checkerboards allowed.

44:01 for me. I was so happy when I realized early in it were two yin yang puzzles in regards to the edges. All the logic I knew from that ruleset made it quite a smooth solve for me.

Fantastic! Since the perimeter rule is usually a corollary of the checkerboard rule (or vice versa) in standard yin yang puzzles, i wonder if one checkerboard would have been allowed if there weren't 4 changes already in the perimeter. In other words, those forced "one of each" dominoes were required in the perimeter. Absolutely gorgeous!

The two length arrow I figured out... But then forgot and at the end I was staring at a deadly pattern... So that slowed my time. At the end it looked like a normal arrow... "Oh yes it can't be normal it's shifted.." that moment was a big relief.

One way of thinking about the perimeter and checkerboard at the same time: Think about 4 pyramids of color radiating from the central square to the perimeter. Ignore the 2x2 rule. There's no valid way to add or subtract from those pyramids to generate a checkerboard while maintaining 4 orthogonal connections from the center to the perimeter. No matter how skinny an orthogonal area gets, you can't checkerboard across it.

We even might have to think about sudoku... Bingo!! No Simon, that's yet another game

1:09:04 finish. I got stalled in the middle for a while, but once I found the break through to my block, things fell into place. Another unique and excellent puzzle!

A tip for the solving of such blueshift--redshift cages is, once the grid is colored up, whenever dealing with a cage, just calculate the total red+blue shift and apply it to the cage total, and treat the cage as if it was that instead, ignoring red and blue entirely. It will always add up no matter where you put the numbers, and, in general, there's no logic aside sudoku to distinguish which digit is hot and which one is cold.

55:07 for me, what a creative way to obfuscate otherwise familiar ruleset and its secrets in a shroud of uncertainty, with a fun value constrain to top it all off

in a galaxy far far away... there were two forces... One used blue lightsabers and the other, red... Would the world of sudoku break? Or will they find the perfect balance?..

55:52 for me. That was a mental workout! I didn't absolutely rigorously prove the logic, but I saw that the four runs of colour on the perimeter were the maximum, then convinced myself that adding checkerboards would be equivalent to adding more runs of colour to the perimeter (in terms of ability to connect everything). When dealing with the cages, it's much easier to combine the shifts in the cage before considering individual digits. If there's a four-cell cage with three blue cells and one red cell, its digits will be increased once and decreased thrice, for a total of -2. Mentally add 2 to the clue in the corner, and you can now treat it as a bog-standard killer cage.

an interesting way you could've thought about the checkerboards is the same way you thought about the wall you made through the center with the 1 cell arrows. the rotational symmetry created by the 1 cell arrows sort of acted as checkerboards with space between them.

Being “chocolate tea-potted” is my favorite new Simonism.

To me, a better title would be 'topological Doppler effect' - because the main argument for excluding checkerboards is topological by nature.

Well done! Total gobbledegook to me!

I thought of each galaxy as made up of two colors: the original and the reflected version (red-A and red-B, blue-A and blue-B), which all have to touch the center. Then you can fall back on the rules of a "four-color Yin-Yang" (if that's not a contradiction) puzzle, where each color can only appear on the boundary in one stretch. I thought that meant that you *could* have a "checkerboard" so long as it involved at least three of these colors (blue-A, blue-A, red-A, and red-B for instance), but some experimentation suggests that you need all four colors to work, and it *won't* work given the added rotational symmetry of the puzzle, because on one side blue needs to enclose red, and on the other red needs to enclose blue. No proof though.

The checkerboard is in principal the same as dividing the perimeter in more segments. If you only consider the segments seen from the midle.

Fantastic puzzle. I almost messed it up because somehow I had convinced myself that in the 16-cage the 9 had to decrease and the 7 had to increase. Luckily I spotted the mistake.

The checkerboard doesn't work in this for the same reason it doesn't work in other similar colouring puzzles. Even though both colours can go through the center, in order for one colour to connect orthogonally back to itself, it must wrap around the center. Cutting off one side of the other colour's central crossing in the process.

Reading the rules, I imediately noticed that this was effectively a mirrored yin-yang puzzle (fill the grid with 2 regions, avoiding 2x2s). Therefore, it was fairly easy to conclude that there couldn't be more than 4 colours on the perimeter (the normal 2 mirrored/doubled) and there couldn't be checkerboards.

There are generic rules for connectivity between regions. It's called graph theory. Now the specifics of the galaxy sudoku aren't clear to me, but let's say without the possibility of a "bridge" you usually can have only 1 color swap along the border of the grid (meaning 2 continuous stretch of the different colors). With 1 bridge, you are allowed 2 additional stretches. In this puzzle when Simon was wondering if he could have a checkerboard there already were 4 regions along the border, meaning he could treat it as regular coloring yin-yang and couldn't create checkerboard patterns.

Lovely video, funny thing you increased with the red and decreased with blue. Murphy's a bless. If it's at random it will be the always opposite.

Wish I had seen the VIP pack, in the kickstarter sooner.

Creating a checker board creates a new perimeter with 8 different colour changes if you like, so it can never work.

Is it fair to say that the galaxy rule turns a 4-region ying-yang into a "half grid" regular ying-yang? If so, that's pretty sweet

puzzle hack: skip half an hour of pontification by simply intuiting that yin yang secrets (no more than two segments of color around the perimeter (plus two for each time they cross eachother)) and no checkerboards (except for the big one on the perimeter; that one's using the overlap and we don't have any more) and then not actually proving it

So my years of studying graph theory comes to practice. It is a known thing! It's a problem called planar graphs and it is well studied by Euler.

29:04 for me. Awesome puzzle, loved it!!

The galaxies are red shifted and blue shifted, so I did color the middle cell blue/red. Now staring!!

i hate myself, i got stuck at the 5>31 arrow, i just thought "they are in the same color, no need to think of the shifting" so i was stuck with either 41 or 32 and it not resolving, it's the only thing i needed the video for, so close...

You can think of this grid as a yin-yang puzzle with a mirror through the center. Easier to visualize if you have a mirror that follows one of the arms of a galaxy. You get two times the number of color changes on the edge of the grid. The rest of the rules (example, no checker board) apply.

77:36 for me. Brilliant puzzle!

Mind bending ... but brilliant.

Here is a simpler way to show that *2x2 checkerboard patterns* are not allowed. *IN SHORT* Consider only one "side" of the grid, and treat it as it were a *Yin-Yang* puzzle❗ (all yellow cells are orthogonally connected; all green cells are orthogonally connected; no 2x2 region can be completely filled with a single colour). *MORE DETAILS* As Simon shows @29:50, there must be a *green wall.* Just remove whatever you find on the left (or right) side of that wall. Everything else (including the green wall) is just a Yin-Yang puzzle, where checkerboards are not allowed because there is no way to connect two corners without segregating the other two corners from each other. For instance, if you connect the *yellow* corners to each other, it becomes impossible to connect the *green* corners.

Checkerboarding would be possible without the rotational symmetry. The crossing point increases the degrees of freedom to allow checkerboarding, but then the rotational requirement splits the grid into two reduced grids which cannot overlap and without the overlap, checkerboarding is impossible.

Rules: 06:18 Let's Get Cracking: 10:14 Simon's time: 1h16m51s Puzzle Solved: 1:27:05 What about this video's Top Tier Simarkisms?! The Secret: 7x (1:07:13, 1:07:18, 1:07:25, 1:07:30, 1:07:44, 1:07:49, 1:08:02) Bobbins: 4x (05:50, 06:06, 10:30, 10:33) Three In the Corner: 2x (1:24:21, 1:26:17) Chocolate Teapot: 2x (1:21:37, 1:23:49) The Raven: 1x (01:40) ​Scooby-Doo: 1x (22:04) And how about this video's Simarkisms?! Checkerboard: 29x (34:44, 34:57, 34:59, 35:10, 35:12, 35:25, 41:15, 41:39, 41:47, 41:49, 42:38, 42:51, 43:29, 44:13, 44:33, 45:28, 45:44, 46:08, 46:16, 47:24, 47:45, 48:47, 48:49, 49:25, 49:33, 50:29, 51:13, 51:41, 51:44) Ah: 15x (12:44, 16:53, 26:04, 26:04, 26:14, 30:26, 38:51, 49:16, 1:00:50, 1:03:57, 1:03:57, 1:04:28, 1:04:33, 1:15:20, 1:19:41, 1:24:35) Hang On: 12x (13:19, 19:08, 27:28, 30:37, 37:28, 37:41, 50:30, 50:30, 50:30, 1:11:41, 1:22:55, 1:24:18) By Sudoku: 11x (11:07, 57:53, 1:06:40, 1:08:46, 1:09:48, 1:12:28, 1:13:57, 1:14:19, 1:24:18, 1:25:24, 1:25:41) Symmetry: 11x (00:47, 06:29, 16:28, 26:04, 26:25, 37:25, 37:31, 37:37, 46:50, 48:53, 1:27:52) Weird: 8x (21:16, 30:43, 30:46, 31:17, 57:15, 1:04:06, 1:08:27, 1:08:39) Sorry: 7x (04:38, 04:38, 04:55, 19:12, 57:15, 1:11:29, 1:20:16) In Fact: 7x (21:08, 58:14, 58:26, 1:04:28, 1:15:40, 1:25:05, 1:27:41) Obviously: 5x (23:54, 26:11, 32:59, 46:39, 59:05) What Does This Mean?: 5x (29:47, 39:36, 59:34, 1:10:58, 1:16:35) Beautiful: 4x (1:07:18, 1:21:56, 1:23:22, 1:27:41) Snake: 4x (02:07, 02:15, 02:25, 03:03) Bother: 3x (1:16:35, 1:17:18, 1:20:08) Brilliant: 3x (52:28, 1:06:52, 1:06:54) Good Grief: 2x (47:59, 1:03:23) I Have no Clue: 2x (21:44, 1:06:05) Incredible: 2x (1:27:17, 1:27:41) Unbelievable: 2x (51:11, 51:13) Box Thingy: 2x (1:14:42, 1:14:42) Have a Think: 2x (26:07, 27:31) What on Earth: 1x (1:19:41) Goodness: 1x (52:22) The Answer is: 1x (40:03) Clever: 1x (12:53) Naughty: 1x (1:18:24) Bingo: 1x (56:11) Lovely: 1x (57:45) Insane: 1x (37:17) Fascinating: 1x (48:29) Ridiculous: 1x (1:25:21) Gorgeous: 1x (1:24:51) Take a Bow: 1x (1:28:41) Shouting: 1x (01:42) Nature: 1x (35:31) Pencil Mark/mark: 1x (1:12:31) Cake!: 1x (06:04) Most popular number(>9), digit and colour this video: Twenty Seven (9 mentions) One (120 mentions) Green, Yellow (85 mentions) Antithesis Battles: Low (6) - High (1) Even (16) - Odd (4) Black (3) - White (1) Row (11) - Column (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!

I think you can prove the checkerboard logic with graph theory similar to the 7 bridges of konnisburg but that might be a little unnecessarily abstract for you

52:15 *surprised pikachu😮*

This must be the first time that there has ever been a horizontal or vertical one cell arrow in a sudoku puzzle.

I was moved by The Raven, presented by Simon. Which other stories by Edgar Alan Poe should I look for in my library?

And the next variant will be ying yang sudoku with 1 crossing... Which would allow one checker board or 4 changes of colour on the perimeter. Then we will see two crossings...

I think, if you look at this not as 2 intersecting snakes, but 4 different snakes that start at one point, counting of color switches and is checkerboards allowed would take less time. Because you already knew that.

Around 1:28... I can' get all this nonsense about the green going all the way to R4C5. On its way it already touches R6C5, which does the job without needing to continue, or R7C5, which isolates the yellow cells in the lower left.

Was the Agamemnon puzzle a trojan?

One must be careful when drawing examples of galaxies lol

64:14 for me

Will Simon conclude this is a yin yang with a single cross?

I've been watching the channel for 4 years and I've always wondered, what the heck is a chocolate tea pot? Is it a tea pot made of chocolate? Is it a tea pot that's being used for chocolate? I'm very confused. I get that it's useless, which is ironic because the typical digits marked this way only need a single digit to resolve the group. So in sudoku it's not useless.

Oh the world of symmetrie Sudoku's me oh the misterie La la la Can this blue be red La di da Doppler rules the set

I understand why r9c5 and r9c6 must be in different galaxies (same for r2c9 and r3c9). At 26:50, I don't see why it's assumed that r9c5 & r3c9 are in the same galaxy as each other. Why wasn't the other relationship also considered (r9c5 and r2c9 being in the same galaxy as each other)?

I understand your concern for colourblind viewers, and I hope there is an answer available that works for everybody, but could you please stop doing the blue and orange colouring? 🙏 Please please please please please? 🙏🙏🙏 I'm not sure if it's too much contrast or too little, but it feels like the colours are shouting at me in my brain, competing for attention, can't separate them properly, and I can't see the rest of the puzzle because of it. As a neurodivergent (probably on the spectrum), the combination is *physically* painful. Sharp pains in my eyes, and a headache. Whereas the very obvious colours for this puzzle, the red and blue, would look lovely, maybe even soothing (the yellow and green are also fine for me -- the orange and purple are not). Again, my sympathies to my colour-blind brethren, and I hope there's some option that would work for everybody, but from an accessibility/anti-ableist perspective, surely 'not causing physical pain to viewers' is at least equal to 'make sure viewers can functionally see the puzzle', right? (Because viewers who have pain in their eyes are not going to be able to functionally see the puzzle either. I couldn't watch, but only listen to, the second half of the video.)

Why don't you post Classic Sudoku anymore?