Pythagoras' Nightmare
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- čas přidán 14. 06. 2022
- ** TODAY'S PUZZLE **
A debut on the channel today for Matthias J. Raives and a simply wonderful puzzle involving pythagorean triples! At first it'll seem completely intractable but there is a way :)
Do give it a try at the link below:
app.crackingthecryptic.com/su...
Rules:
Normal sudoku rules apply. The sum of digits along each edge of a triangle must form a Pythagorean triple, with the largest number of the triple corresponding to the sum along the hypotenuse. There are 12 triangles in this puzzle. Some pythagorean triples include the following (and the multiples thereof): (3,4,5); (5,12,13); (8,15,17); and (7,24,25).
If you manage to solve this one and want more sudoku then give this puzzle from Sam Cavnar-Johnson a try:
app.crackingthecryptic.com/su...
It's a classic sudoku with a very cool trick. On Patreon we've released a solve video showing Mark's solve (not knowing the trick) and then Simon's explanation of how to go about it differently :)
For those of you interested in the puzzle Adventure by Jon Schneider (mentioned in the video) there is a video here:
• The Hardest Puzzle Eve...
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Melvyn Mainini
Thanks so much for featuring my puzzle! I came across your miracle sudoku video about a year ago, and have been hooked ever since.
My favorite part about this puzzle is how the little triangles interact with the big one. And its a property I discovered on accident! That's what I love about setting and solving new constraints, you get to work and play with them and by doing so discover how they work.
I'm definitely not done with this constraint, so keep an eye out on the Discord for more like it.
This was a fascinating puzzle and video - thank you for sharing it! And congratulations on your debut on the channel!
The two most interesting parts for me were figuring the min/max values for different size triangles and the overlapping 6 on 2 out of 3 number sets on the minis. Thank you. I really enjoyed this. Please play with this concept more.
Thanks for a lovely puzzle. Really enjoyable (and within my skill set)
Thank you for a fantastic puzzle 😃👍
A beautifully designed one. I thought how on earth to reduce the number of options and suddenly it clicked about the opposite of the hypotenuse being 123 on the little triangles and then the solution path flowed. I am glad this puzzle didnt involve having a superior numerical processor upstairs to crunch numbers.
Really like the troll of including the various pythagorean triples to make people (Simon) worry about them, when in the end everything was just a multiple of 3,4,5
Lol
I couldn't get my head around those legs on the big triangle being LONGER mathematically yet SHORTER geometrically..
Pythagoras is rolling over in his grave lol ].
I would have preferred not to have seen this spoiler to be honest. Not that I could have solved it anyway, but still.
@@rojavida What were you expecting? You looked at the comments section for a sudoku puzzle, we weren't going to be discussing the recent football results...
@@SellymeYT it was right at the top on the page. I didn't specifically click to read the comments. I'm not exactly apoplectic in my comment, it's only a puzzle.
In the spirit of xkcd 356: obviously the set of Pythagorean triples that can possibly fit Matthias' rule is bounded. The hypotenuse can't be larger than 81, for a start. But exactly which triples could he have used?
Simon: explains 'the secret" every episode in case someone doesn't know
Also Simon: everyone knows what a hypotenuse is that watches the channel, I'll skip over it.
First I thought it was a joke. Anyway... Since I don't know what a hypotenuse is, I won't even bother doing this puzzle.
@@SpeedyBozar The side of a triangle that's opposite the right angle (of a right-angled triangle). The "long" side
I've done one or two puzzles quicker than the video time, but this is the first puzzle that I've completed faster than Simon or Mark. I'm going to ignore the fact that Simon could go faster if he wasn't explaining things so well, but this has given me a feeling of achievement that I couldn't convey to anyone outside this channel! (28:17)
Thanks so much for the great content (both setters and Simon/Mark).
holy moly, 14:16, probably the biggest margin i've ever beaten simon by. i definitely appreciate these kinds of puzzles where the trick is more about math than about sudoku :)
first i figured out that for a triple (a, b, c) the corner cell must be ½(a + b − c), which made it quickly obvious that tiny triangles only 123/246/369; after that i pencilmarked all the tiny corners, eliminated some 1s, pencilmarked every hypotenuse that could only be 469, and the whole thing swiftly fell apart
me did very same!
Quite entertaining that Simon spends such a long time thinking about the bigger triangles before even looking at the small triangles that are what allow you to get started ... I think that helped a lot of us to beat his time today!
I am overly critical sometimes about how Simon usually throws away smaller clues but it makes sense for most sudokus that bigger clues are usually more restrained - 3 cell arrows, long thermos, whisper/etc/lines, strings of kropkis etc. Even big sandwich clues and killer cages. In this case, the opposite is (we see now) true.
To tell the truth, when I do a puzzle, I don't worry about the time. I just work at solving the puzzle.
I am a teacher of Mathematics and find the logic in these puzzles truly wonderful. I try to encourage my students to access this channel and follow the solves. Particularly liked this Pythagorean puzzle and was questioning myself about the pronunciation of Pythagorean. I say it as follows; Pi - Thag - Or - Ian. Who cares though, great puzzle. Well done Simon.
As a US-born-and-bred math instructor, it is always interesting to hear his UK pronunciations for math(s) terms. Corollary is just one example.
I am so grateful for this channel. At first I never imagined I could even break into puzzles of this difficulty. I’m still very amateur but learning more every day thanks to you both. Now… let’s get cracking
stared at the “end” of the puzzle for 26 minutes before realizing I had disregarded the giant triangle. Like I said, amateur! 👍🏻
Simon wondered at one point about if the hypotenuse of a three-cell triangle spanning boxes to repeat - not only does that not work with the limited digits available, even if you could extend this ruleset to arbitrarily large cell values it would never work because it is impossible to create an integer-sided isosceles right triangle.
[This is related to the irrationality of the square root of 2; if you make a=b and substitute it in to the Pythagorean theorem you get 2a^2 = c^2
Which can be rearranged as such:
2=c^2/a^2
2=(c/a)^2
√2=c/a
If there were a solution where both a and c are integers, the square root of 2 would be rational!]
Yeah he mentioned a short bit later that it would be impossible because the two right angle sides would be the same and can't be.
When I thought a little about the puzzle theme, it occured to me that there is a nice geometric interpretation of the tiny triangles. If you draw the pythagorean triangle corresponding to that triple of cells, and if you then draw a circle around each corner with the radius of the corresponding cell, you will have a triple of kissing circles, (yes, that *is* a technical term for circles that mutually touch each other) and the ratio of any two circles is a rational number.
Clever!
An entertainng puzzle, not too difficult, based on a very novel idea. Thank you Matthias, congratulations on your debut.
Absolutely incredible! As a former math teacher and lifelong math nerd, I was enthralled. Thanks for the set and the solve!
Rules: 03:48
Let's Get Cracking: 05:40
Simon's time: 33m26s
Puzzle Solved: 39:06
What about this video's Top Tier Simarkisms?!
The Secret: 2x (21:05, 21:07)
And how about this video's Simarkisms?!
By Sudoku: 8x (26:11, 27:30, 28:58, 29:29, 29:56, 32:23, 33:04, 34:40)
Ah: 7x (15:20, 15:26, 19:23, 20:31, 22:37, 30:31, 35:21)
Clever: 5x (25:12, 26:11, 26:15, 39:09, 39:11)
Nonsense: 4x (17:23, 17:25, 17:33, 17:36)
Hang On: 4x (17:16, 29:35)
Beautiful: 3x (36:11, 36:15, 39:18)
Wow: 3x (09:38, 09:50, 39:07)
Good Grief: 2x (23:04, 29:08)
Sorry: 2x (13:19, 19:32)
Lovely: 2x (26:39, 31:47)
In Fact: 2x (27:52, 34:25)
Obviously: 2x (17:11, 17:23)
Goodness: 1x (24:18)
Bother: 1x (19:23)
Elegant: 1x (06:27)
Gorgeous: 1x (25:59)
Barbaric: 1x (02:47)
Take a Bow: 1x (36:15)
Shouting: 1x (01:58)
Progress: 1x (15:59)
Cake!: 1x (02:02)
Symmetry: 1x (23:50)
Most popular number(>9), digit and colour this video:
Fifteen (25 mentions)
Five (74 mentions)
Red (3 mentions)
Antithesis Battles:
Even (2) - Odd (0)
Higher (2) - Lower (0)
Row (6) - Column (4)
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!
19:36 Mowerick
That’s so ridiculously beautiful, that’s one of the best ones in this string of sudoku brilliancies for me. Great setting and solving, and I really want to see this concept/constraint used in future sudokus!
Yes, this was a really cool 😎 fun puzzle.
Yes! And I would be interested to know if it is possible to have any of the other bases (not 3-4-5) used, because they seemed possible but are not used in this puzzle I believe
@@rosserobertolli I think 20-21-29 would be the only other possibility - made up of 714(8)25(6)797
The rest are too uneven
@@jonotick 8-15-17 would be possible with 52(1)6(8)45, 5-12-13 and 7-24-25 are not possible following Simons logic but yours would work again. So I think it should be possible?
To say this one was in my wheelhouse would be an understatement; at a time of 11:06, this may have been my fastest solve of a non-GAS, 9x9 sudoku at this site. Math *really* played a part in this one.
Fun puzzle!
“I am plagued by farmers and pilots” is my new favourite phrase of all time.
Wonders if Simon knows of Shawn the Sheep ... He must ... Baaaa .....
This was very interesting - and I felt, fairly approachable. I will definitely be making sure that my husband and son both see this one. A lovely and elegant solve, Simon. Thank you!
OMG! This is literally the first Sudoku I have ever solved faster than Simon! Yeah, maths! Simon still out-solves me 99 times out of 100 though. Great puzzle, Matthias!
I've written so much code for solving puzzles from this channel, especially for generating all killer/sandwich/X-sum/etc combinations. This was definitely a time when the code made things much easier!
I’ve come back to this after saving it months ago and getting overwhelmed. With your break in I was amazed I could solve it in about 70 minutes! Great puzzle
So simon always explains every basic thing again and again as of toddlers are watching these videos but he doesn't bother explaining the rules this puzzle when it was actually needed. Good job.
15:48 Found this one easy, probably because I realized right away the small triangles had to be (3,4,5) multiples regardless of box-spanning, then went for the fused pair + given digits and left the big triangle for the end.
31:56 for me. That was a fun puzzle! It went faster after I realized the only digits that could be at the right angle of a triangle with a length of 1 (which incidentally includes the large triangle thanks to overlapping sides) were 1,2, and 3. But, 1 could never be on certain triangles that see others!
Does it throw anyone else off that all of those triangles are 45-45-90 triangles and could therefore not possibly have pythagorean triple side lengths?
Not drawn to scale lol
LMAO 🤣
If Simon would use the pencil-Mark technique, he would be completing the puzzle in 10 minutes or less :)
Right!? I was surprised he started focusing on the large triangles when they are so under-constrained. Just pencil in the small ones and everything reduces quickly.
Got it faster than Simon for the first time in my life. Bucket list item crossed off. Only took 100s of attempts >.
Without Simon's preliminary work (calculating the possible combinations of digits for the small triangles) I would certainly not have known where to start. But with this help I was able to solve it in less than an hour. A great puzzle.
@@herrpausr7008 👍🙂
“Plagued by farmers and pilots” Lolol that cracked me up
12:50 - Ah Simon, always proving things don't work in the most difficult way. The nine in the corner is much easier.
new-to-sudoku undergraduate maths student here; loved this puzzle.
Decades away from math(s) classes, I did have to look up hypotenuse 😞 (also looked up Pythagorean triples) - please don't judge. 🙃 ok- just watching I could have figured it out... excuse - it's really hot here today in SE USA - over 100F. I'm on my 3rd glass of ice tea.
Scorching hot by me today long way, over 100 also. I am sure you are not alone in having to look up hypotenuse..:-) be well and safe my friend. Keep cool
@@davidrattner9 Thank you, David! Same to you!
Great concept for the puzzle! and after finding the start you rolled through it Simon, nice one!
What are you talking about the hardest puzzle of all time, all you have to do is program your character to program your character to program you to program your character... I don't know why anyone would find that complicated?
I loaded the puzzle, paused the timer to get a drink, and forgot to turn it back on XD Loved the puzzle and actually got through it quite quickly, the small triangles needing a 345 or multiple thereof was an amazing way to open this.
“…are made up of little triangles’ hypotenuse…es!”
I think the word you want here is “hippopotami.”
😄
I thought the plural of hypotenuse was hypoteneese.
22:13 for me, lovely puzzle, found it very intuitive to tackle the little triangles first.
Loved every minute of this Puzzle
I have started to try the puzzles included in the sudoku program you use and they are hard!!! I’m solving them at around 30-60 minutes when I do solve them. Anyone wanting to try advanced sudokus should definitely get that app/program and use it as a stepping stone
With that title, I expected beans.
More like Pythagorass, amirite?
13:42 for me. I tend to do well on the more mathematical ones.
My Greatest Hits of CTC arrived today. I was one of the few who was unable to order during the Kickstarter,; I was one of the ones who ordered it the first day it was available from your website.
I have a degree in Mathematics and absolutely love your longer videos.
This has been my favorite puzzle to date
This is a great example of Simon's abhorrence for pencil marks making the solve so much harder. Once he decided that the small ones were 1-2-3, 2-4-6, or 3-6-9, he could have immediately pencil marked all the little ones (with 1-2-3 on the right angle and 2-3-4-6-9 on the 45s) and started eliminating possibilities. The puzzle breaks down as soon as you reach that step, but he didn't want to write a bunch of pencil marks in and spent 20 minutes opining about the larger triangular sums. Turned a 15 minute solve into a 35 minute one.
His approach allows him to be more skilled at tackling harder puzzles I suppose.
One of few of Simon's puzzle I actualy could finish without wtaching the solve! Took me just over 55 minutes.
For once, I'm not at all surprised to beat Simon's time; this one is rather in my wheelhouse. Of course, I'm still no speed demon, at 25:50.
That big triangle really wasn't so scary, was it?
Lol. My daily challenge has become to beat Simon's time, even though I kind of cheat because I generally have to watch his explanation of the rule set before I start. (More often than not my dopey brain can't decipher new variations until I 'see' the rules explained on a grid.. as was the case today. And Simon's explanations are wonderful.)
He's still undisputed GOAT when it comes to difficult break ins, but Simon has officially become "Postman Plod." when it comes to basic sudoku skills.
I beat him too for first time ever. The small triangles were a great help.
17:03 for me. It took me quite a while to understand the idea of the puzzle, but once I did it led to a fantastic solvepath. Really great logic in here, it leaves me wanting more of it.
Such a lovely idea, great puzzle!
Love the Puzzles where Maths logic is involved, reminds me of the Harshad sudoku from the CTC book which also left me with a smile from ear to ear. :)
A very satisfying 16:05 for me as I realised how constrained the small triangles were quickly.
And, if anyone is interested, every Pythagorean triple is generated by [k(b^2 - a^2), 2kab, k(b^2 + a^2)] for some combination of three positive integers a, b, k with a < b. For example, a = 1, b = 2, k = 1 gives the 3-4-5 triple.
Easier to recognize them perhaps is that for the simplest ones the two larger ones differ by 1 and add up to the square of the smaller one. e.g. 24+25=7*7. The 8,15,17 is just a 4,7.5,8.5 in disguise ;)
@@andrelglinnenbank2856 or notice that every "simple one" is an odd numbered side squared then half it... round the answer down for the shorter remaining side and round up for the hypotenuse.. (and if you want, double all sides or treble all sides for some other values). other variants are available which give 20, 21, 29 where you square, divide by 8 then add and subtract 4, rather than half :) the formula above isn't the only one out there
The first time I really flew through a puzzle! Having the triples handy was super helpful and finding the property of the small triangles just jumped at me, but with that I finished it in 17:34 :)
For any two positive integers m and n with m larger than n, a Pythagorean triple is formed from (m^2 - n^2, 2mn, m^2 + n^2). So for example with m = 7 and n = 4, the resulting triple is (33, 56, 65).
Though, not all Pythagorean triples are formed this way, for example (9,12,15) is not of that form. But if you include multiples of those, then you indeed get all Pythagorean triples. Also, small nitpick, but you probably want to write it as (m^2-n^2, 2mn, m^2 + n^2) since m^2+n^2 is always the length of hypotenuse.
@@Ennar You’re right, the formula isn’t exhaustive as it stands, but it can generate all base cases and scalar multiplication takes care of the rest. I’ll rearrange the terms as you suggested just because I’m a stickler for order. 😆
@@woodchuk1 yes, it does generate all primitive Pythagorean triples. In fact, the triple it generates will be primitive if and only if you pick m and n to be coprime and not of the same parity, which is very neat :)
@@Ennar That’s interesting…not only in that making m and n coprime also makes the triples coprime, but that the generating integers can’t both be odd for this to be the case! Never would have expected that!
@@woodchuk1 if they are both odd, difference and sum of their squares are both even. Combined with the fact that 2mn is even, it means all three numbers in the triple end up being even, so the triple is not primitive.
Sudoko meet Pythagoras; Pythagoras meet Suduko. I believe you have much to talk about! Hope you both enjoy the gathering. ...
This was a good fun puzzle and I had a pretty good solve time, of course I didn't have to take the time to explain it. My break in point was the large triangle / small triangle interaction in boxes 4 and 7.
29:50 for me. Looks like a lot of us are finally getting our first "I beat Simon!" badge today, hahahaha.
This puzzle was by far the easiest to start of any of the puzzles I've completed on this channel. It was immediately obvious to me that the 2x2 triangles could only be 3-4-5, which limited their sudoku numbers to {1,2,3}, {2,4,6}, and {3,6,9}. Eliminating 1 from the right angle and 6 from either end of the hypotenuse cut down the possibilities significantly, so I was able to flood the grid with plenty of solved numbers right off the bat. I got a little too carried away looking for Pythagorean triples, when they all turned out to not only be in the list provided, but also just 3-4-5.
First solve without needing help from the video. Finally!!!
I have worked out, though it's probably useless, that a+b+c=ab/c where c is the right angle cell and a and b are the other two on a small triangle
I don't know what the implications of this are, besides the fact that I can do simple algebra.
4:05 "Everybody who watches Cracking the criptic will know what's the hypotenuse..."
Did Simon forget there are 5 y-o watching too? (I don't really know if you get thought the pythagorean theorem at that age, but I'm strongly inclined to believe that's not the case...)
I once taught a three-year-old to sing the Pythagorean Theorem. Though, to be fair, I don’t think he understood what it meant…
What is almost annoying is the almost red herring with the other pythagorean triples never being used just 3 4 5. If I knew ahead of time it was always 3 4 5, I could have solved this so quickly.
One of those gems that youtube filters back.
Was hoping this one could god the distance in a letter/colour solve but that medium tri was impossible to break...also hoping u could solve it without placing a 578 but that didnt happen either
Did it in 24:43 - I know Simon was explaining to us as he was going but this is first time I have ever beaten him (I know he'd have been quicker if not explaining). The logic around the small triangles was great and loved ow it locked numbers in quickly even if you wern't 100% on which sequence each little triangle was.
Adding another faster than usual time here: 23:21 After working out that the small triangles only had 3 options, half the grid filled in easily. It was nearly just all sudoku left after that
I learned "pi-THAG-o-REE-n" as the correct way to say that word.
I wondered if Simon's "pi-tha-GO-re-un" was a Britishism, because I know there are other long words that are stressed differently in American and British English (for ex. "veterinary," which is "VET-rin-er-y" in the US, but my late British colleague said "vet-RIN-er-y" ). But the OED only gives forms with primary stress on "RE," secondary stress on either the first or the second syllable (so either pi-THAG-o-REE-n or PI-thag-o-REE-n).
It's fortunate Simon isn't a true Pythagorean, or he would surely be exectued for revealing The Secret.
Very clever puzzle. I like how we were given a bunch of different examples of pythagorean triples, but they all turned out the be 345 multiples. Tricky tricky....
26:41 for me. Once the possible digits on the small triangles are deduced, it's not that difficult to solve.
I concentrated on the tiny triangles which interact with the 4, 6 and 9 and made quick progress. I only looked at the large triangle as an afterthought once it was largely filled in.
Fun Puzzle. First time I have ever beaten Simon to a break in. Couldn't have done it without his tutelage.
Fascinating and funny that it's verging on approachable.
at 15:00 and after you have to stop thinking about the pythagorean triple an instead think about the three cells on the tiny triangle: 123 on the corner and 23469 on the hypoteneuse. 2/3 on the hypoteneuse go together, and 469 go together (as a 4 and 6 or 6 and 9) and they don't mix. If you can't have a 2 you can't have a 3 and so on. If one can only be 9 or 4 the other must be 6 and the converse.
If you just pencil all these in, eliminate the ones made impossible by the given cells, and do Sudoku you quickly get to having everything filled in except for 5s, 7s, and 8s and a couple of 3s. Then you attack the big triangle, then finally the 2 by 2 where you tried to start.
I'd like t share with you my joy! this is the first time I solve a puzzle in a time comparable to either Simon or Mark: 31 minutes. Normally, it takes me like 4 or 5 or more times the time it takes them!
Had to use coloring (among the 2/3s, then the 4/9s), but still finished in 20:49. If you pencil in 123 on the right corner of each tiny triangle, you can very quickly start to narrow down and rule things out. Great puzzle, fun ruleset, it’d be interested to see if you could set one where not all triangles were a multiple of the smallest 3-4-5 Pythagorean triple.
I caught on to the fact the little triangles had to be (123/246/369) almost immediately, and finished in 24 minutes. A rare time I beat Simon, and I think by my one of my best margins.
can't believe Mark didn't take his chance to color the grid.
.
all these pairs all over the place and night a bright hue to be seen.
This was the first time that I could solve a puzzle faster than you :-)
This is the first time I've ever beaten Simon. Even if he was going slow explaining all his choices, I'm taking this W
This is a wonderfully unique and interesting puzzle! This reminds me, I need to call my grade school math teacher to thank him and let him know I finally used the Pythagorean Theorem outside of class 😂
As an addendum... for those of you who are interested in what Pythagorean triples are all about, see the 3Blue1Brown video from May 2017. As ever for you matheads out there, if you don't know his videos, then go and watch a bunch of them! Always fascinating and beautifully created!
Yes, a bit of a fan, me!
Yep. Pythagorean triples are integer solutions, indeed. And there are an infinite amount of them.
Note that angles in a semicircle are 90º, so if you move a point from the end of one diameter around the semicircle to the opposite side, the locus of that point traces out an infinite amount right angled triangles. So any rational points on that line, can be suitably scaled to give you an integer pythagorean triple.
This looked intriguing so I did it myself and put off watching the video until I'd finished it. Like everyone else here, I started with the highly contrained little triangles. I didn't *quite* beat Simon's time because I made a dreadful arithmetic error at one point (decided 32-20 = 8, which got me into some trouble) and had to backtrack a bit. Very cool puzzle, though.
17:25 for me. the constraints can work better if you study them first in a noteblock
Simon's Drinking Game
If Simon says "Let me ask you a facetious question" .............................................Take a drink
If Simon brings out his ridiculous scrabble bag analogy ........................................Take a drink
If Simon says "Schrodinger cell"................................................................................Take a drink
If Simon breaks into The Raven.................................................................................Take a drink
If Simon says "And that means..." when he has no idea what it means .................Take a small sip. I don't want you to die from alcohol poisoning.
Remember don't tell Simon about the game or he will plan to do a 2 hour video and after each deduction he will quote the raven nevermore.
Just finished it; exactly 26 minutes for me, yay! Doesn't happen often that I can beat Simon's time but for this one, I could see relatively soon that the small triangles can only take multiples of (3, 4, 5) and from there, it was actually pretty easy :) btw I love your channel, only discovered it a few weeks ago and I'm already super addicted ^^
To be fair he does have to talk through his thoughts so that way the viewers understand what he’s doing, but still 26 min is pretty fast(depending on how many difficulty stars it has on Logic Masters Germany)
@@tangrybird04 Your absolutely right and I by no means would compare myself in that way to Simon :D I'm just happy that everything I have learned from this channel already translates in a much faster time for myself because the first puzzles I've attempted took me almost always twice as long as his time so this is just a neat way for me to see how much progress I've made so far :) (I'm not really familiar with Logic Masters Germany but thank you :))
I'm sure the constructor was trying to incorporate a 5, 12,13 combo. Had to be content with multiples of 3,4,5.
23:32 finish, in honor of all of the 2-3 pairs with which we started off the grid. Another "fun with math" sudoku, excellent!
Small triangles must have 1,2,3 on the right angle and 2,3,4,6,9 on the hypotenuse. Goodliffe all the small right triangles and it all falls into place quickly.
Fun Fact: In China they call it the Gougu theorem instead of Pythagorean theorem.
a^2 + b^2 = c^2
I see you watch Mind Your Decisions too. 😉
@@rontyson6118 and others
So many triangles ^_^" it might lead away but I think you enjoy these challenges!
Did anyone try the classic sudoku linked in the description of the video? I'm stuck!
OK, some algebra I found helpful...
For the small triangles, if we're looking for a,b,c that corresponds to the Pythagorean triple {X,Y,Z}, such that (a+b)=X, (a+c)=Y, (b+c)=Z, then add all three equations together to get 2(a+b+c)=X+Y+Z.
Rearrange to get,
a = (X+Y+Z)/2 - (b+c)
= (X+Y+Z)/2 - Z
If we let S=(X+Y+Z)/2, then we can say,
a = S - Z, b = S - Y, c = S - X
E.g. For the {3,4,5} Pythagorean triple, S=(3+4+5)/2=6; a=6-5=1, b=6-4=2, c=6-3=3. So (a,b,c) is (1,2,3)
Since a,b,c are all < 10, this only gives valid solutions for the {3,4,5} Pythagorean triple and it's first two multiples ({6,8,10} & {9,12,15}).
That got me 1,2,3 on all the right angles of the small triangles, with either 2,3; 4,6 or 6,9 on the corresponding corners.
Another equivalent way to look at this is that (wlog in any permutation):
X-Y = a-c, thus
X-Y+Z = 2a
So it's very easy to check the other triples and see that they can't work.
Nothing like a good ole math puzzle. 👍
The name of this puzzle makes me think of the fact that sqrt(2) is irrational. This discovery was something of a nightmare for Pythagoras since a core tenet of his philosophy was that all things are in simple ratios with one another.
I recently learned a very cool proof of this fact that channel viewers should enjoy. Consider two positive numbers b and c such that b^2 = 2c^2. Take a square of side length b and inscribe two squares of side length c, one in the bottom left and the other in the top right. The overlap of these two squares is a square of side length 2c - b. Not covered by these squares are two squares of side length b - c in the top left and bottom right. Since two squares of side length c have the same area as one square of side length b we can use SET to show that the area of the overlap square equals the total area of the two uncovered squares, i.e. (2c - b)^2 = 2*(b - c)^2. This is the same relation we started with but with smaller numbers. Now, if b and c are whole numbers then 2c - b and b - c are also whole numbers, so we can repeat the construction again and again to get smaller and smaller whole numbers, violating the principle of infinite descent.
Very cool. All the other Pythagoras triples were red herrings it turns out. Only used the 3 4 5 set
It would have been interesting if the creator had found a way to use the almost-square combo of (20, 21, 29). But he took the easy way out and used -- okay, I won't give it away.
3:30 I did Sam Cavnar-Johnson's puzzle in lieu of yesterday's(?) puzzle, when I couldn't figure out how to start that one. Cavnar-Johnson's puzzle was doable. I don't know whether the pattern (a sum-rule with 9s being an exception) was the trick. (Does anyone know whether that was the trick?)
5:30 It's a fun exercise in algebra figuring out various patterns -- for example m² + n² = (n + 1)². These lead to (1,0,1), (3,4,5), (5,12,13), (7,24,25), (9,40,41), (11,60,61),... m has to be odd, and n = (m² - 1)/2. Then there's m² + n² = (n + 2)² -- half of these are double the (n + 1) numbers, and the other half are new, such as (8,15,17). Then there's (m, n, n+p) where p is an odd prime. All of these are p*(the n+1 numbers).
5:55 "I can tell you this is not a 345 triangle because..." Does anything prevent it from being a multiple of 345? (I've already done the puzzle, and the answer is ...)
10:15 You have a large number of small triangles, and they have only three possibilities.
14:50 It's easier to think of the cells rather than the sums -- (1,2,3) and multiples thereof.
25:00 A little earlier, "Just place a digit, Simon." Now, you've placed not one but two.
Some Pythagorean-triple trivia: In every triple, the "legs" include multiples of 3 & 4, and one of the three "sides" is a multiple of 5. (There's only one multiple each of 3, 4, & 5 if the three sides are relatively prime, i.e. their greatest common factor is 1. One leg may be the multiple of two of these numbers (as in 5,12,13, or 8,15,17), or even all three (as in 11,60,61).)
I spent about 20 minutes to find possibilities of various of pythagorean triples. Then it is not that hard to finish the grid.
nice puzzle!
Wonderful
36:30 for me but I spent the first 15 minutes bumbling down the wrong path as I made an incorrect assumption. Definitely my kind of puzzle!
25 minutes for me, a rare occasion when I beat Simon.
Fun creative puzzle, I consider this very approachable.
But did you explain the “secret” in your time? lol
@@bobh6728 the answer to life the universe and everything + the number in the corner = the secret. There add 10 seconds.
@@zegres good one!
39 minutes for me. A fun idea with a surprisingly simple solution.