This is indeed the most beautiful equation in mathematics - it connects together the five most important constants in mathematics in one equation (!!!). And many thanks, Mark, for pronouncing Leonard Euler's name correctly. I remember as a freshman, wondering who this guy "Oiler" was, that all my profs were always talking about. BTW, multiplying by imaginary i (but no dot on top) has a nice geometric meaning. Consider the cartesian plane. When you multiply a point in the plane by (-1), the point get rotated 180 degrees. That's why (-1)(-1) = 1: two half-turns bring you back to where you started. Multiplying by imaginary i rotates the points 90 degrees counterclockwise. So i x i represents a rotation of 180 degrees. That is why i squared equals -1: two 90 degree rotations give you a rotations of 180 degrees, carried out by multiplying by -1. Also, Euler, the Prince of Mathematicians, is arguably one of the three greatest mathematicians of all time, along with Archimedes and Gauss. Up until the 20th century, he was the most prolific mathematician of all time. One of his many contributions was founding graph theory, the basis for computer networks. And he played around with Latin Squares, which can be seen as a precursor of the Sudoku grid 🙂
Very interesting, thanks @annikaQED! I also like that Eulers equation uses all of addition, subtraction, multiplication, exponentiation and equality exactly once.
If you think of a minus sign as an operator that spins a number 180 degrees on the number line (1 becomes -1 etc.), then i is an operator that spins a number only 90 degrees. Now the number has a vertical part to it, not just the horizontal part, but that is not too complicated (kinda like a sudoku grid).
Wonderful puzzle, went very smoothly and finished in 9:17 (plus 2-3 minutes for accidentally stopping the timer, yay! - conflict checker off), many thanks to Tobias for a beautiful puzzle that showcases a beautiful mathematical equation!
316 is included cuz Tobias clearly knew it was my favorite number. Also, now we just need a puzzle based around the Heisenberg Uncertainty Principle and Gödel's Incompleteness Theorem.
I have already published a puzzle on the "Uncertainty Principle" on Logic Masters that you can find if you search for that name (id=000HXV). In that case, the Sudoku connection is not visual, however, as in the current puzzle, but contains the mathematics of the Uncertainty Principle.
I finished in 17:22 minutes. It really is a beautiful equation. The simplicity sells it well. As for the puzzle, I think the same things can be said. I quite liked the interactions between the German Whispers on each other. I loved themed puzzles and I think this one nailed it. Great Puzzle!
This was another puzzle with straightforward rules and an interesting image in the grid, producing a very nice puzzle with a fun solve. Your video demonstrates all of this very nicely, Mark - I love watching your solves. (And I learned something about Euler tonight, though not how to pronounce his name, which I already knew and was amused by the insistence on the incorrect pronunciation during the stream you mentioned!)
If one is writing it by hand, I believe it is more usual to write it as a "script" i, without the dot. But if the font you're using has only dotted i's, nobody minds if you use a dotted i.
Yes, it is usually spelled as a lower-case "i" including the dot, though in electric engineering they often use "j" instead to avoid confusion with a current, denoted by "i".
It's funny, when I was in high school the teacher pronounced the name Euler (like Ruler) but when I got to college, I found out that you are indeed correct, it's Euler (like Oiler). You need to consider where a name is from in order to pronounce it correctly!
Thanks a lot for suggesting this! In my separate comment on the video, I refer to Wikipedia from which one can learn that apparently Euler himself used 6.28... for (an alternative) π in some of his work.
I like the suggestion (was it from Gauss first?) to call them ‘lateral’ numbers - because we can plot them on a vertical axis perpendicular to the real number line, forming the complex plane.
Complex numbers: pairs of real numbers. Real numbers: sequences of rational numbers. No wait; sequences of rational numbers with the Cauchy property. No wait! Sets of sequences of rational numbers with the Cauchy property equivalent under the relation that the termwise difference approaches 0, in the sense that for every epsilon > 0 there exists... And yet it's the complex numbers that people view with suspicion 🥲
The thing is, I can picture one banana, or three bananas. I have no idea what i bananas looks like. So in that sense, 1 and 3 are more real to me than i could ever be. (And if autocorrect could stop converting all my i's to I's, I would be most grateful.😂)
I think the 6 is on the 0 just because it's round. I wonder where you've seen the i without a dot. Like a Greek letter iota? I can name any number of textbooks that write it as a plain old i with a dot. Same for the Wikipedia article on complex numbers.
10:45 finish. It took me a while to figure out where in my equations to put the pi and e for the solve, but eventually I guessed that the puzzle was imaginary, so I added the two together and ate my dessert. Sorry, but I'm not a fan of chocolate cake. 😝😝😝
Rules: 05:57 Let's Get Cracking: 06:56 Mark's time: 14m51s Puzzle Solved: 21:47 What about this video's Top Tier Simarkisms?! Three In the Corner: 1x (18:48) And how about this video's Simarkisms?! Beautiful: 4x (01:58, 05:20, 21:46, 21:57) Pencil Mark/mark: 3x (12:00, 12:58, 18:08) Sorry: 2x (04:45, 11:29) Lovely: 2x (22:06, 22:08) By Sudoku: 2x (15:48, 16:17) Ah: 2x (10:00, 17:53) Stuck: 1x (17:53) Brilliant: 1x (21:48) Incredible: 1x (04:03) Extraordinary: 1x (04:14) Astonishing: 1x (02:27) Hang On: 1x (07:39) Unbelievable: 1x (01:05) In Fact: 1x (11:47) Intriguing: 1x (03:34) Most popular number(>9), digit and colour this video: Forty Nine (4 mentions) Three (43 mentions) Green (8 mentions) Antithesis Battles: High (9) - Low (7) Even (5) - Odd (3) Column (10) - Row (8) 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!
14:02, didn't really need a lot of math to solve it, just some subtraction to see if the digits had a difference bigger than 5. And I guess I divided some digits by 2 to see if there was a remainder once or twice.
Ah, what a brilliant way to quickly decide whether it's even or odd. I usually convert it to binary and look at rightmost bit, but that turns out to be way more cumbersome
Why can't I seem to ever get through these puzzles without blowing it halfway through? I got 30 minutes into it, only to hit an unrecoverable conflict. I must've taken a wrong turn at Albuquerque or something. After restarting, though, I was able to blast through it in another 19:30@#3454. It flew by much more smoothly since I had already worked out most of the logic.
I'm an American PhD mathematician and professor, having taught university math for over 40 years. I imagine there are more mathematicians in the US than in UK. So I think a strong case can be made that those who have devoted their lives to mathematics can call it whatever they please. I'm fine with Brits calling it maths. But all I claim is that "math" can be seen as an equally correct abbreviation; 🙂
In the US, the correct term is 'math.' "Why do you persist with these Britishisms" is an equally valid (or invalid, I would argue) question? The international, language- and culture-independent nature of sudoku guarantees that we will get many ways of expressing oneself in the comments on these videos, and I find it quite interesting and fascinating, and I think it broadens my appreciation for the different kinds of people who love sudoku and love this channel.
This is indeed the most beautiful equation in mathematics - it connects together the five most important constants in mathematics in one equation (!!!). And many thanks, Mark, for pronouncing Leonard Euler's name correctly. I remember as a freshman, wondering who this guy "Oiler" was, that all my profs were always talking about. BTW, multiplying by imaginary i (but no dot on top) has a nice geometric meaning. Consider the cartesian plane. When you multiply a point in the plane by (-1), the point get rotated 180 degrees. That's why (-1)(-1) = 1: two half-turns bring you back to where you started. Multiplying by imaginary i rotates the points 90 degrees counterclockwise. So i x i represents a rotation of 180 degrees. That is why i squared equals -1: two 90 degree rotations give you a rotations of 180 degrees, carried out by multiplying by -1. Also, Euler, the Prince of Mathematicians, is arguably one of the three greatest mathematicians of all time, along with Archimedes and Gauss. Up until the 20th century, he was the most prolific mathematician of all time. One of his many contributions was founding graph theory, the basis for computer networks. And he played around with Latin Squares, which can be seen as a precursor of the Sudoku grid 🙂
The comments section of Cracking the Cryptic is truly a youtube unicorn. Love it! 😅
Thank you @annikaQED 👌
Thank you for these additional explanations and also mentioning the Latin Squares!
Very interesting, thanks @annikaQED! I also like that Eulers equation uses all of addition, subtraction, multiplication, exponentiation and equality exactly once.
As native German speaker I approve of everything Mark said about Eulers name and pronounciation.
This one is right up my maths-loving alley!
And thank you for the proper pronunciation of the Swiss-German name Euler.
The little book is "1089 and All That: A Journey into Mathematics" by David Acheson
Thanks
00:30:01 - beautiful! Thank you very much!😊
If you think of a minus sign as an operator that spins a number 180 degrees on the number line (1 becomes -1 etc.), then i is an operator that spins a number only 90 degrees. Now the number has a vertical part to it, not just the horizontal part, but that is not too complicated (kinda like a sudoku grid).
Very nice explanation with the rotational Imagery for the Imaginary number i
Thanks to you , I can now see the connection between i and Pi. Now the e part ?
Yes, thank you!
I think that the book title that Mark was looking for is "1089 and All That: A Journey into Mathematics" by David Acheson
Wonderful puzzle, went very smoothly and finished in 9:17 (plus 2-3 minutes for accidentally stopping the timer, yay! - conflict checker off), many thanks to Tobias for a beautiful puzzle that showcases a beautiful mathematical equation!
So glad you liked it!
316 is included cuz Tobias clearly knew it was my favorite number. Also, now we just need a puzzle based around the Heisenberg Uncertainty Principle and Gödel's Incompleteness Theorem.
I have already published a puzzle on the "Uncertainty Principle" on Logic Masters that you can find if you search for that name (id=000HXV). In that case, the Sudoku connection is not visual, however, as in the current puzzle, but contains the mathematics of the Uncertainty Principle.
Yes, and we want a Sudoku that's both solvable and unsolvable at the same time 🙂
Would the Heisenberg puzzle have Schrödinger cells that have different numbers depending on whether one observes the row, column, or block?
@@JohnRandomness105 😂
@@JohnRandomness105 Obviously.
I've never heard native English speakers say Yuler (or maybe I just never noticed).
I finished in 17:22 minutes. It really is a beautiful equation. The simplicity sells it well. As for the puzzle, I think the same things can be said. I quite liked the interactions between the German Whispers on each other. I loved themed puzzles and I think this one nailed it. Great Puzzle!
Thanks a lot!
Impeccable pronunciation of Euler! (I am bilingual).
Finished in 11:12. Great puzzle!
Thanks a lot!
8:07 ... Euler would be proud
Nice puzzle!
This was another puzzle with straightforward rules and an interesting image in the grid, producing a very nice puzzle with a fun solve. Your video demonstrates all of this very nicely, Mark - I love watching your solves. (And I learned something about Euler tonight, though not how to pronounce his name, which I already knew and was amused by the insistence on the incorrect pronunciation during the stream you mentioned!)
Glad you enjoyed the puzzle!
I managed the puzzle, although I got stuck more than I should have. The equal sign was superfluous pseudokowise.
13:32. Beautiful!
14:46 for me. Another fun one. I could've done it a couple minutes faster if I didn't make a pencilmarking mistake.
13:34 for me. What an exquisite puzzle!
Happy to hear you enjoyed it!
17:45 to solve. Excellent puzzle!
Nice to read you liked it!
@ superb construction!
The imaginary unit is represented, usually, by a lower-case i, so the dot is appropriate.
If one is writing it by hand, I believe it is more usual to write it as a "script" i, without the dot. But if the font you're using has only dotted i's, nobody minds if you use a dotted i.
@annikaQED I have never seen anyone write a mathematical expression with a script i.
Yes, it is usually spelled as a lower-case "i" including the dot, though in electric engineering they often use "j" instead to avoid confusion with a current, denoted by "i".
Yes, although I can see why that might have seemed a bit, ahem, odd to him. 😉
It's funny, when I was in high school the teacher pronounced the name Euler (like Ruler) but when I got to college, I found out that you are indeed correct, it's Euler (like Oiler). You need to consider where a name is from in order to pronounce it correctly!
5:41 6 could be the most significant digit in 2π=6.28.. which makes sense to be at 0, where the full circle returns to the beginning
Thanks a lot for suggesting this! In my separate comment on the video, I refer to Wikipedia from which one can learn that apparently Euler himself used 6.28... for (an alternative) π in some of his work.
Liked just for the correct pronunciation of Euler.
Tasty 14:54 snack, smooth start and a bit stuck momentarily in the middle
What I love about i is that you can use it in formulas so it must be a number, just our number line method of writing maths doesn't like it.
Despite the title of imaginary, the number i does exist. As much as the number 3, 1, 0, π, and all the other real numbers ;)
I like the suggestion (was it from Gauss first?) to call them ‘lateral’ numbers - because we can plot them on a vertical axis perpendicular to the real number line, forming the complex plane.
Complex numbers: pairs of real numbers.
Real numbers: sequences of rational numbers. No wait; sequences of rational numbers with the Cauchy property. No wait! Sets of sequences of rational numbers with the Cauchy property equivalent under the relation that the termwise difference approaches 0, in the sense that for every epsilon > 0 there exists...
And yet it's the complex numbers that people view with suspicion 🥲
The thing is, I can picture one banana, or three bananas. I have no idea what i bananas looks like. So in that sense, 1 and 3 are more real to me than i could ever be.
(And if autocorrect could stop converting all my i's to I's, I would be most grateful.😂)
16:00 for me as #1898 solver
Euler's formula in the sudoku 😂
The i can be written with or without the dot - purely a stylistic choice.
That's so interesting to read. Maybe this is some cultural variation? I have never seen it without a dot...
I think the 6 is on the 0 just because it's round.
I wonder where you've seen the i without a dot. Like a Greek letter iota? I can name any number of textbooks that write it as a plain old i with a dot. Same for the Wikipedia article on complex numbers.
10:45 finish. It took me a while to figure out where in my equations to put the pi and e for the solve, but eventually I guessed that the puzzle was imaginary, so I added the two together and ate my dessert. Sorry, but I'm not a fan of chocolate cake. 😝😝😝
Rules: 05:57
Let's Get Cracking: 06:56
Mark's time: 14m51s
Puzzle Solved: 21:47
What about this video's Top Tier Simarkisms?!
Three In the Corner: 1x (18:48)
And how about this video's Simarkisms?!
Beautiful: 4x (01:58, 05:20, 21:46, 21:57)
Pencil Mark/mark: 3x (12:00, 12:58, 18:08)
Sorry: 2x (04:45, 11:29)
Lovely: 2x (22:06, 22:08)
By Sudoku: 2x (15:48, 16:17)
Ah: 2x (10:00, 17:53)
Stuck: 1x (17:53)
Brilliant: 1x (21:48)
Incredible: 1x (04:03)
Extraordinary: 1x (04:14)
Astonishing: 1x (02:27)
Hang On: 1x (07:39)
Unbelievable: 1x (01:05)
In Fact: 1x (11:47)
Intriguing: 1x (03:34)
Most popular number(>9), digit and colour this video:
Forty Nine (4 mentions)
Three (43 mentions)
Green (8 mentions)
Antithesis Battles:
High (9) - Low (7)
Even (5) - Odd (3)
Column (10) - Row (8)
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!
00:23:30
14:02, didn't really need a lot of math to solve it, just some subtraction to see if the digits had a difference bigger than 5. And I guess I divided some digits by 2 to see if there was a remainder once or twice.
Ah, what a brilliant way to quickly decide whether it's even or odd. I usually convert it to binary and look at rightmost bit, but that turns out to be way more cumbersome
Why can't I seem to ever get through these puzzles without blowing it halfway through? I got 30 minutes into it, only to hit an unrecoverable conflict. I must've taken a wrong turn at Albuquerque or something. After restarting, though, I was able to blast through it in another 19:30@#3454. It flew by much more smoothly since I had already worked out most of the logic.
Happens to me all the time as well
10:41 for me
nice puzzle
8:24
Euler is definitely pronounced like Oiler! Whoever insisted it was like ruler was speaking rubbish! 😂
11:38 for me.
14:34 for me
34:30 for me and solver #2848.
Sorry but the correct term is Maths. Why do you persist with these Americanisations?
I'm an American PhD mathematician and professor, having taught university math for over 40 years. I imagine there are more mathematicians in the US than in UK. So I think a strong case can be made that those who have devoted their lives to mathematics can call it whatever they please. I'm fine with Brits calling it maths. But all I claim is that "math" can be seen as an equally correct abbreviation; 🙂
In the US, the correct term is 'math.' "Why do you persist with these Britishisms" is an equally valid (or invalid, I would argue) question? The international, language- and culture-independent nature of sudoku guarantees that we will get many ways of expressing oneself in the comments on these videos, and I find it quite interesting and fascinating, and I think it broadens my appreciation for the different kinds of people who love sudoku and love this channel.
Throughout the entire video, I did not hear Mark ever use the word "math" instead of "maths".
@@QuLogic Yeah, I opened the transcript and did a search. Mark never used "math" as a standalone word.
Most Brits that I know are fine with calling it math. Australians on the other hand....
00:22:24