A Mathematical Miracle

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** TODAY'S PUZZLE **
Leonhard Kohl-Lörting returns to the channel today with an absolutely GREAT sudoku. We're reluctant to say too much about it for fear of spoiling the experience but just give yourself a treat and give this one a try!
Play the puzzle at the link below:
sudokupad.app/P3DDj2thmF
Rules:
Normal sudoku rules apply. Clues outside the grid are X-Sum clues and indicate the sum of the first X cells, where X is the value of the first cell in that column. The total sum of all X-Sum clues is 165. Digits in cages sum to their clue.
** NEW SUDOKU HUNT ON PATREON **
We're delighted to share with you a brand new sudoku hunt themed around snake egg puzzles by Glum Hippo. If you've never tried snake egg logic puzzles before, prepare for serious fun - these sudokus are outstanding! Finish by May 20 to enter the competition!
Other treats on Patreon include:
- the Sumgeons & Diagrams sudoku by sunnyjum;
- Simon's latest forays into the world of Islands Of Insight;
- Mark's video looking at the new OneUp puzzle from Rodolfo Kurchan;
- his solve of Region Geometry by Emre Kolotoğlu (3hr 36min long...!);
- and Mark's latest solve of The Times Club Monthly cryptic crossword
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▶ Contents Of This Video ◀
0:00 Theme music & puzzle intro
1:32 Closing Date For Competition
2:33 600k Subscribers Sudoku Pack
3:15 Happy Birthday
3:54 Rules
4:55 Start of Solve: Let's Get Cracking
▶ Contact Us ◀
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14 июн 2024

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Комментарии : 243

@Rubrickety 25 дней назад

Some peak Simon there near the end, when he _finally_ decided he could delete his test examples and random colors, and unconcernedly left behind a single blue square to drive us all mad. 😂

@stangerrits6712 26 дней назад

Once the 1-9 cycles where found, you can use the modularity principle that Simon often explains in the 1-5-9 indexing puzzles: in row 1 you need a low, middle and high digit in each box, to make sure you won’t get the same digit twice within a box.

@Sidnv 25 дней назад

Not quite low-middle-high, it's more powerful than that. 5 and 7 can't be in the same box in row 1 for instance, anything in the same box has to be separated by at least 3. So you end up forced with 147, 258, 369 in the same box as the only possible arrangement of digits, as otherwise some number would be within 3 of something else in the same box.

@HunterJE 25 дней назад

Indeed, by the action of the rules and clues this *is* a 159 puzzle, though rotated 90º from the usual orientation

@Zerotan 25 дней назад

@@Sidnv mod 3 go brrr

@AREmrys 25 дней назад

@@HunterJE It's even more than that. Every line is an indexing line.

@user-gj5uc4yx3i 25 дней назад

Every cell is an indexing cell for its column.

@emilywilliams3237 26 дней назад

I never, ever, ever get any puzzle faster than either you or Mark, Simon. But this one just tripped in my mind and I had it done in 3 minutes. I actually started with thinking about where the 1 would have to be, and then the 2, and recognized something that made me think about the sum of 165, and then a tiny moment gave me modular sets, and then it was simply finished. Very, very fun. I feel as if I can explain it logically, but I could not have done so while solving it because my brain was simply seeing what had to happen. It is so interesting how different people's minds work. Thanks for the wide variety of puzzles featured here on CtC.

@UnaturalShadows 25 дней назад

well he does have to explain himself as he does it

@pouletbelette 25 дней назад

Congratulations Emily! That's quite impressive.

@psymar 25 дней назад

Basically: once you figure out what the 9 ?s have to be (without consideration for their order), The top row has 1..9. The 2 must have a 1 directly below it, the 3 must then have a 2, the 4 must then have a 3, and so on up to the 9 having an 8, leaving only the possibility of 9 under the 1. Similar logic gives you how all 9 columns go and now it's just a matter of what order they go in.

@emilywilliams3237 25 дней назад

@@pouletbelette It was an epiphany. And quite fun. But I still enjoyed watching Simon solve it and explain in words what my brain, today, in this one isolated instance, just saw had to be true.

@emilywilliams3237 25 дней назад

@@UnaturalShadows Right! Which is why he has a sudoku-solving channel and I don't!

@HunterJE 25 дней назад

Weird thing to notice: This setup results in a secret 1-5-9 puzzle, though rotated 90º from how those are usually presented - the "slot machine" arrangement results in every digit in row 1 indexing the position of the 1 in its column, every digit in row 5 indexing the position of the 5, and every digit in row 9 indexing the 9

@attker89 25 дней назад

I think that pattern extends to every row and every digit. Row 2 indexes all the twos, row 3 indexes all the threes, etc. Very interesting properties in this puzzle.

@jack_2612 25 дней назад

nice spot, I didn't notice

@Darkstar2342 25 дней назад

16:34 he basically did the correct reasoning befire to show that the 13 cage cannot be 8-5... so it is 9-4. so if the 4 was in the left cell of the cage, it would have a 5 above it which would clash with the 5 in the center green strip. So the 4 is in the right cell

@emanuelfer456 25 дней назад

I was shouting that for 3 minutes while Simon just looked at 9 instead of looking at 4

@spatulamahn 25 дней назад

Came here for this comment! LOL - Simon saying that we need to look at the 11-cage when the 13-cage was giving him the answer already! :)

@In_42_Space 25 дней назад

Surely Simon this has to be one of your favourite puzzles as you were not really forced into doing any actual sudoku 😂😂.

@BryanSarlo 25 дней назад

Reminds me of a sudoku featured on here about a year ago on which the rules were along the lines of 9 invisible vertical thermos whose bulbs are in row 9. Similar solves in this one and that one from last year, but both are great and this comment is not meant to take anything away from this puzzle!

@MattYDdraig 26 дней назад

Sum of the first n triangular numbers is the nth tetrahedral number: n(n+1)(n+2)/6

@markfinnicum8340 25 дней назад

Fun Fact for Parties: nth tetrahedral number is a 3rd degree polynomial because the summation of a polynomial function results in another polynomial function with one higher degree. The nth triangular number is a 2nd degree polynomial: n(n+1)/2. And obviously, the nth natural number is a 1st degree polynomial: n

@andy-kg5fb 25 дней назад

Another fun fact for parties, summation of tetrahedron numbers is n(n+1)(n+2)(n+3)/24 and the pattern repeats!!

@Anne_Mahoney 25 дней назад

@@andy-kg5fb I want to get invited to your kind of parties! 😺

@HunterJE 25 дней назад

Fun coincidence, friends of the channel Numberphile released a video about tetrahedral numbers today!

@MattYDdraig 25 дней назад

Thanks for that!! Fun bonus coincidence: The current year 2024 is a tetrahedral number. The most recent previous year this was true in was 1771, and the next will be 2300, so revel in our current tetrahedral glory! (For typical western calendars, of course)

@vandelay33 26 дней назад

4:21 for me. Definitely the fastest ive ever done a CtC puzzle but very fun nonetheless

@davidenas 25 дней назад

After you conclude how to fill in an entire column from just a single digit and doing it for column 5, you can actually conclude that columns 4 and 6 must start with 1 and 7 in some order to not break box 2. And only one of those orders avoids breaking a cage, so you fill those in. Then the cages solve the rest of the puzzle, that part is the same.

@Turalize12 24 дня назад

Exactly what I did!

@ilyrm89 26 дней назад

So happy Simon finally removed the blue highlight at 19:19, thank you!

@damadclown 26 дней назад

3:40. First time I managed to get a decent time compared to other viewers. It helped that this puzzle reminded me of another one with a similar "structure" and everything fell in place from there

@jurgnobs1308 25 дней назад

yea there was a similar logic one a few days ago, right? where it would also end up with the numbers being in order

@ironalice9452 26 дней назад

Beautiful logic! As always a joy watching you ponder and solve as well as listening to your explanations! I don't really solve Sudokus nor am I good at math. But seeing you solve these manages to relax me every time and tingle just the right parts or the brain. 😊

@vanguard2960 24 дня назад

Never get a puzzle done faster than you guys, but this one just worked for me and I noticed the pattern + that the 3 numbers in the same row in a given box had to be 3 apart. Very fun puzzle!

@megusta9268 25 дней назад

This is probably in the family of the first sudokus ever to be written, I always make this one when I'm thinking about sudoku in my head. Its cool that these rules denote this exact sudoku though.

@Kirbyfan87827 26 дней назад

Finished in 21:00 with extensive help from the video.

@spin-rg9ib 25 дней назад

i had to revert to the puzzle i figured out that the sums were a minimum but couldnt think of how to go from there. i knew there was going to be a 147, 258, 269 combo in the top row and that they would go in descending order but got stuck on where to go from there. then i checked the video and just needed the detail that the bottom of the grid will also follow suite with the order after the x sums.

@HunterJE 26 дней назад

Simon started looking at the disambiguating cages on the right then changed course to look at the left first instead, but it's also possible to solve the right first on its own-stepping off from the deduction that the 13 cage must be a 49 pair you can then see that it breaks if you put the 9 in r8c7; it would have an 8 below it, which would need a 1 in r9c8 to finish the 9 cage, BUT the 9 r8c7 puts another 1 in r7c7 which clashes in the box. Therefore the 9 has to be in r8c6 and things flow from there.

@willemm9356 26 дней назад

Even simpler, if it's 49, the 4 would have a 5 above it which clashes in the box.

@IguitarVreakI 25 дней назад

Haha, I looked at the 4 instead and realized that it would have a 5 above it and you'd have two 5's in Box 8

@user-gj5uc4yx3i 25 дней назад

I worked out that the 6 in box 8 was flanked by 3 and 9. Try putting a 3 into a two cell 13 cage.

@derekpangelinan4229 6 дней назад

The blue square!! The most uncomfortable moment in any CTC video ever!!

@chipsounder4633 26 дней назад

Vertical and horizontal drive entropic roping ❤ yet another masterpiece... Oops , i always mix up entropy with modularity.. but the roping still holds true 🎉🎉

@Ravenishish 25 дней назад

The formula of the sum of triangular numbers is (n(n+1)(n+2))/6 and you're right, you can take multiple copies of them and make a cube.

@tezza7 25 дней назад

I get excited when I see a shorter video because it means i'll have a chance of solving the puzzle..and this time I did, yay!

@samwortham2385 26 дней назад

This one filled itself in very nicely! It took me a few minutes to figure out the logic, but it was quite satisfying when I did! Thank you Leonhard!

@goldenera7090 18 дней назад

wow congratulations to Simon and Mark almost reaching 600k subscribers... I remember how this channel started so well done on your journey

@stevieinselby 25 дней назад

Seeing that the x-sum cells cover more than half the grid (45 cells out of 81) and sum to significantly _less_ than half of that the grid total of 405, I assumed that we would need to minimise the x-sums so started with that addition. I realised straight away it would be slot machines, so c5 was a write in. Entropy rules meant that one of c4/c6 starts with a 1 and therefore the other one starts with a 7, and the 13 cage determines which way round they do, and everything then just fills in for a 3m14s finish 😎

@stevieinselby 25 дней назад

(I couldn't for the life of me follow Simon's logic with all the different columns filled in with dummy digits...)

@SirJefferE 25 дней назад

I didn't want to bother thinking about the individual triangle numbers, so I just considered them as a group. If we're minimizing X-sums then: 9 sums must include a 1. 8 sums include a 2. (All but the 1 clue) 7 sums include a 3. (All but the 1 and 2 clues) So the x-sum minimum is the sum of this pattern: 9x1, 8x2, 7x3, 6x4, 5x5, 4x6, 3x7, 2x8, 1x9 The pattern is a palindrome, so I just summed the first four (9 + 18 + 21 + 24 = 70), then doubled it and added 25 for the 5x5 in the middle. Honestly, this approach probably isn't all that different to figuring out the triangle numbers, but it seemed a lot easier to do in my head.

@alaguerre6362 18 дней назад

One of my favorite puzzles so far. I thought through the 165 sum, then figured out the numerical pattern the sum required. Once I realized how to line up the seven and use that to solve the killer cages, the rest fell into place nicely.

@M0z1ng0 24 дня назад

14:33 really happy that I was able to get this one on my own! Realizing the numbers had to be descending was a lot of fun

@francoisduez601 25 дней назад

8:26 Nice miracle 😊 A few minutes to understand how it works, afterwards it flows very quickly 🎉

@Alex6STV 21 день назад

At first, I thought there is no way I was going to be able to solve this. Once I wrote the numbers 1-9 in the first row (just to get an idea how the columns might build themselves out) and figured out that 165 was a minimum value, I was quickly able to see sequential pattern when I worked out where the 1 would be in the 2 column, where would the 2 and 1 be in the 3 column, etc.... Awesome puzzle. Loved it.

@darreljones8645 25 дней назад

Finding the maximum possible X-sums total in a row of a finished sudoku is complicated by the fact that X must be present in an X-digit X-sum (i.e., the maximum possible 2-digit total isn't 9+8=17, but 2+9-11), but still, it's not hard to figure it out: 1 + 11 + 20 + 28 + 35 + 39 + 42 + 44 + 45 = 265.

@Pathogenus 25 дней назад

now thats a proper miracle sudoku unlike the previous one with long ruleset and tricky steps to do along the way

@AngelWedge 26 дней назад

4:20 for me; about half of which was double-checking my arithmetic before starting :p

@psiphiorg 20 часов назад

I spotted the trick immediately, and was able to solve it in 1 minute, 43 seconds! 165 is a tetrahedral number, specifically the sum of the first 9 triangular numbers (1+3+6+10+15+21+28+36+45). Therefore, the X-Sum with a 1 in its first digit is free to have any numbers below it, the 2 must have a 1 below it, and the 3 must have a 2 and 1 below it. The key deduction here is that the 3 must therefore have a 2 directly below it, and a 1 directly below that. This sequence continues such that the 9 column must be 987654321. Now, working your way back down the numbers, the 8 column must start 87654321, which leave only 9 for the final digit. Then the 7 must have 8 and 9 below it, in descending order, and so on. So that means that each column is 987654321, just rotated through one way or the other. Figuring out which side is which in the fourth and sixth columns is solved with the 13 cage, since the cell in it in box 8 would be either a 3 or a 9. The 3 isn't possible there, so it's the 9, and the 3 goes into the 11 cage instead. Solving the cages shows the next columns over, and the cages in row 9 go out one more. Then there are only two columns left, the first and the ninth, which just get the remaining three digits from their boxes in descending order. I was solver number 9844.

@mscha 24 дня назад

Around 13:20, it's even easier to realize that if one of the columns in box 8 has 7/6/5, the other two columns will be 4/3/2 and 1/9/8. Everything falls in place immediately then.

@katiekawaii 25 дней назад

18:07! I don't know how long I would have been starting at the puzzle if I hadn't gotten Simon's suggestion to consider if the sum clue was minimized or maximized, but once I figured that out (with a calculator, not impressively in my head like Simon 😅) I was able to do the whole thing in 18:07. Pretty proud of that one. Doubt I could have done that a few months ago. Thanks for making my puzzle brain sharper, Simon!

@Xiuhtec 25 дней назад

Finished this in about 7 minutes, much of which was "typing" in the values on a phone. Figured out the patterns pretty quickly after a minute or so realizing 165 was just the minimum possible total. Very cool setting!

@PathOfShrines 25 дней назад

Haha, very clever idea. Immediately saw the top half of the induction, then figured out the bottom half a few minutes later. Finally convinced myself the starts had to be not just in separate high/middle/low per box but actually modularly related, and then it was done. 13:46

@srwapo 26 дней назад

15:38, for some reason I spent a few minutes thinking about this as an X-sums puzzle, then figured out how it went but made a mistake, then solved it real fast (and super appreciated the given clues) on the 3rd go (my time is the total sum including my mistakes).

@steve470 25 дней назад

3:52 for me, including doing the mental arithmetic and proving that the columns must indeed do what I thought they were going to do. Since some others are having trouble understanding why the columns must do that, here's my stab at an approachable proof: One column has a 1 in the top row. Another column will have a 2 in the top row, and to keep the sum of sums down to 165, that column must have a 1 in the second row. Another column will have a 3 in the top row, which needs a 1 and a 2 immediately below it. There's already a 1 in the second row, so the 1 can only go in the third row, and the 2 must go in the second row. Continue this logic through all nine digits, and you've proven that, starting from the top row, every column descends sequentially from its topmost digit down to 1. Now, look at it from the other side. Whichever column has an 8 in its top row, has used every digit except 9, so the 9 must be in the bottom row. (By the earlier logic, this puts 9 directly below that column's 1.) Whichever column has a 7 in its top row, has used every digit except 8 and 9. There's already a 9 in the bottom row, so the 9 can only go in the second-to-bottom row (directly below that column's 1), and the 8 must go in the bottom row. Whichever column has a 6 in its top row, has used every digit except 7, 8, and 9. These can all be placed by row logic (9 below the 1, then in descending order). Continuing this logic, every column must descend sequentially from its topmost digit down to 1, then have a 9 below it, and continue sequentially downward from there. QED.

@jillyapple1 22 дня назад

I spent a couple minute confirming 165 was the sum of triangular sums of 1 through 9, then realized that meant roping descending from the X. The 7 gave me my first column, then my first box, and the cages gave me the rest. 5:49.

@frankjiang1857 25 дней назад

Finished in 7:09. Fairly straightforward sudoku where if you know the trick it solves itself. Fun puzzle!

@davidhughes7174 25 дней назад

The first columns comedy! thank you.

@martinepstein9826 25 дней назад

The sum of the first n numbers, AKA the n'th triangular number, can be found on Pascal's triangle. On the left side of the triangle the n'th triangular number is down and to the right from n itself. The sum of the first n triangular numbers, AKA the n'th tetrahedral number, can also be found on Pascal's triangle. On the left side of the triangle the n'th tetrahedral number is down and to the right from the n'th triangular number.

@johnbell9074 25 дней назад

Figured out the pattern in 5 mins, then reset and finished in 3 mins. Thanks.

@Eckbert1410 25 дней назад

Nearly got a stroke staring at the lonely remaining blue-colored cell 🤯

@Caladryus 25 дней назад

11:03 My reasoning after getting the centre column was slightly different than the reasoning Simon had. I recognised the cycling nature of the columns, meaning that columns 4, 5 and 6 would, in each box, have the numbers 4-3-2, 1-9-8, 7-6-5 in those orders in their boxes. This would, in box 8, put a 3 or a 9 in the cages. Since a 3 cannot be in the 13 cage, that one has a 9 in box 8 and the 11 cage gets a 3 in box 8, and after that it solves itself by using the cycle and remaining cages

@kurtu5 22 дня назад

The total number of gifts on the 12 days of Christmas is the "Tetrahedral" number for 12. You are stacking cannon balls in a tetrahedron and the geometric proof is copying the tetrahedron, stacking them so you get a rhombus(skewed box) and then calculating the side lengths.

@rickwoods5274 24 дня назад

I solved this almost entirely in a notepad. Wrote out my logic up to the point where I knew the pattern each column must conform to. Spent an *embarrassingly* long time looking at the grid before realizing the given 7 determines a whole column, then wrote that in. Then worked out the possibilities for the cages, figured out there was only 1, and wrote the remaining 72 digits directly into the grid

@angec9908 26 дней назад

I had to use a calculator to prove 165 😂

@user-hj9pc9oo4h 25 дней назад

me too :)

@katiekawaii 25 дней назад

Same!

@chrishaynes7425 24 дня назад

Very satisfying. Cracked it by working out the min and max x-values (in Excel!)

@brianarsuaga5008 24 дня назад

Proud of my performance here! I spotted the 165 constraint immediately, then fooled around to prove the low-middle-high snookering from 1-5-9 puzzles, then figured out the descent. Really fun how it just completely falls apart after that, I've never filled so many squares back to back!

@emptyset1312 25 дней назад

Very neat puzzle. I was able to work out that everywhere throughout the puzzle, digits must ascend in order upwards, e.g. if you place a 3, it must have a 4 above it. Once I knew that fact, it was quick and easy to fill everything in starting from the given 7 and then the cages. Very very satisfying.

@antosandras 25 дней назад

15:00 Once cyclicity where found, r8c6 can't be 4, since the r7c6 would be 5 clashing r9c5. Telling differently: cyclicity implies that in any row, anything in the same box has to be separated by at least 3, that force r8c6 to be 9 given the clue 13.

@chocolateboy300 25 дней назад

I finishes in 29 minutes. Seeing the value, I quickly counted up the minimums of 1-9 as they are all being used. This gave me the magic number of 165. I had trouble implementing, until I saw the pattern of 9 followed by its descending numbers for each of the X Sums. That's where the given 7 was amazing and gave me my first digit of 4 in the top row. After that, it was easy just to cycle through the other numbers with the cage limitations. Great Puzzle!

@geezerama 25 дней назад

Good grief! Somewhere between 8:15 and 9:10 I was overcome with a very personal nostalgia whack. That's how I felt many years ago, in another far younger life, within the confines of my mind most of the time. Always trying to draw it all together to make the old model conform to the new. Sorry if I texted that out loud. Great puzzle, thanks to setter and CTC.

@HolyChez 24 дня назад

9:09 time for me. Once I figured out that 1-9 cycles must be in the same descending order from the top-down, I immediately can fill in from the 7 and the rest just falls in place through the sums.

@penningmeestercgkdelft9159 25 дней назад

Solved it in 9:14. :-) A true miracle sudoku indeed, and it fills itself in immediately as soon as you see the trick!

@user-kt9vr1wj6y 25 дней назад

6:13 for me, nice puzzle maybe we could include R7C5 in a vertical 2-cell cage sums to 15 or 13 to make it no digits given at the start.

@janeflett4971 17 дней назад

Cool puzzle!

@longwaytotipperary 26 дней назад

❤❤❤❤❤ this puzzle!! I ❤ magical puzzles!!

@seanb7807 25 дней назад

Solved in 8:15, first time I've solved faster than Simon. Can't say my solution was the cleanest, but I noticed 165 was pretty small relative to the total sum of 405, and 240 was a lot to cram into 36 digits since 45 digits would have to be in the X-sum per the secret, so I figured the only way it could solve was if the 9 was left out of the 8, the 8 and 9 out of the 7, and so on. Did the math and confirmed it which made the rest just a matter of setting the orders. Fun puzzle!

@JalebJay 25 дней назад

5:50 I read the clues upside down at first. But knew the strategy immediately.

@TheMeanderingduck6 25 дней назад

2:58 - I realized the trick immediately and the 7 made everything click.

@RoderickEtheria 25 дней назад

6:39. The largest part of this puzzle was checking whether all the triangular numbers summed properly.

@Michael-hs7pr 25 дней назад

2:44 for me. Just had to check that the sum of triangular numbers is 165 and immediately knew it was a roping puzzle. Rest is just filling in the digits.

@AugustoValentini 25 дней назад

9:15 to solve the puzzle itself, I did the 165 sum while taking a shower so the timer wasn't on yet 😅 A simple and fun mathematical puzzle, just the kind of sudoku I love

@10prozenthimmel 25 дней назад

This very strongly reminded me of amother puzzle that Simon solved maybe a year ago or so. It had the same rotational symmetry. Probably just had a different break-in but resulted in the same pattern or a permutation of it.

@evetheeevee2977 25 дней назад

4:24 woah. A miracle!

@Gonzalo_Garcia_ 26 дней назад

3:25 for me. I think I remember a very similar puzzle being featured not that long ago.

@deadlybee111 25 дней назад

the sum of Triangular numbers are tetrahedral numbers and you can get them just like triangular numbers with pascals triangle, its the "diagonal" right underneath triangular numbers exactly the same as triangular numbers are in relation to natural numbers

@compiling 25 дней назад

6:10. I tried just filling in the numbers ignoring boxes and cages to see how the columns work, then once I saw the specific sequences are forced the rest of the puzzle was easy. There are only 2 values that can go next to the 4, which the 13 cage disambiguates. The formula you were looking for is 11C3. I remembered that the sum of consecutive triangular numbers is the 4th diagonal of Pascal's Triangle.

@colemanjamie1 25 дней назад

Very fun quick puzzle, 11:33 for, this is my very first time getting a Simon puzzle faster than his video time!!

@mhelvens 25 дней назад

I did this one in 13:47. Can't complain!

@michaelhoffman2011 23 дня назад

Took me 4 minutes.... Once i figured out the total for all the x-sums... and having done A LOT of CTC i could see there was a pattern forming.... Really fun puzzle, but ultra suprised i figured it out so quickly

@DennisR219 26 дней назад

I actually regrets having seen this video. This is such an approachable puzzle, I feel I could have solved it, yet didn't test myself on it hehe. Anyway, fun as always to have Simon solve it for us!

@jdyerjdyer 22 дня назад

Took a minute to work out what I suspected that the 165 sum meant, and sure enough, it meant it. Then I had all the digits in their strips and from there it was just placing which column each strip went into so as to not break the cages or the bottom boxes. Fast and fun "magic"! Amazing!

@jdyerjdyer 22 дня назад

Hint: 45 + 45 - 9 = 81, pretty high sum already just from row 1 and the column with all 9 digits. So I asked how much was left from the 165 and started subtracting the minimums. 165-45-36-28-21-15-10-6-3-1 and found out that there was nothing left. This meant that wherever the x-sums were, they took the minimum digits leaving the higher digits to sit outside the x-sums in the bottom rows. So you get a 9 in the last cell for the 8 x-sum, an 89 pair for the 7 x-sum, but the order is forced by the 8 x-sum...all the way until you end up with the 9 x-sum just counting down to 1 at the bottom. From there, each strip gets the remaining digits filled in such that they count down as well to meet the 9 in each column. From there, you can just place the 7 strip by the given 7, there is only one strip that will work next to that column with the 13 cage. Then it continues just matching which strip is forced by a cage sum or the remaining digits in the lower three boxes. Done!

@Paolo_De_Leva 25 дней назад

I can remember a very similar but more interesting puzzle in which the *1-9* cyclical sequences were horizontal, rather than vertical as in this case, and the ruleset was slightly more interesting. I believe it was published in one of the volumes of *CTC Greatest Hits.*

@MythosHB 25 дней назад

You probably have already seen the latest Numberphile video including tetrahedral numbers. Not exacltly answering your question for a formula, but it fits the topic.

@daniellucas5522 25 дней назад

4:59 for me - and about half of that was finding that 165 is the minimum.

@shortfatboy 25 дней назад

Many has pointed out the formula for the sum of the first n triangular numbers aka n-th tetrahedral number. Perhaps, here is a quick way to rederive it. Essentially, the n-th triangular number is a binomial coefficient with lower entry two and the n-th tetrahedral number is a binomial coefficient with lower entry three. To see why this is true, visualize the pascal triangle and recall how the entries are generated (details are grossly omitted). This explains the lower entry of the binomial coefficients and with some more work, you can figure out the top entry of the binomial coefficient.

@praematura 25 дней назад

Oh dear, I got this one pretty fast as well, and completed it in just 3:23 (conflict checker off). 😅 I recall a puzzle with a similar setup previous which did help, admittedly! Many thanks to Leonhard for a wonderfully fun puzzle!

@user-gj5uc4yx3i 25 дней назад

This statement appears to be true of every cell in the puzzle. Digit X in row Y tells you in which row (X) The digit Y can be found in the same column. For each column, X and Y are the same exactly once.

@iuriikononenko9238 25 дней назад

After working out the sum, just tried to complete with random digit position. the result wa that the grid was just had to be filled in based on provided digit and sums. Other way to work out modularity and place some more digits in row 9.

@Rach881101 25 дней назад

16:45 for me. Beautiful puzzle!

@alanscott8245 25 дней назад

9:33 for me. This was a fun one.

@elizabethshort9632 8 дней назад

I managed to do this one (mostly) on my own so I'd say it's only a 3 difficulty. I got tripped up by a few mistakes which meant I had to come back to the puzzle but the logic itself is possible to spot compared to others you've done where I've hardly been able to do anything

@LarkyLuna 25 дней назад

A proof of sum of triangular numbers for those mathematically inclined, using the formulas for the sum of n numbers and the sum of n squares My trick was realizing that 1 will appear in all n triangular numbers, 2 in n-1 of them (will miss the first), etc, until we get to the nth triangular number where n will appear there and only there, so we get: 1*(n- 0) + 2*(n-1) + 3*(n-2) + ... + n*( n - (n - 1)) = expanding the multiplications: n*(1 + 2 + 3 + ... + n) - (1*2 + 2*3 + ... + (n-1)*n) = expanding the negative part as 2 = 1+1, 3 = 2+1, 4 = 3 + 1, ..., n = n-1 + 1 then multiplying n*n(n+1)/2 - (1² + 2² + ... + (n-1)²) - (1 + 2 + 3 + 4 + 5 + ... + n-1) = square formulas and sums again n*n(n+1)/2 - (n-1)(n)(2n-1)/6 - (n-1)(n)/2 = n(n+1)(n+2)/6 wah, good math warm-up

@MarushiaDark316 25 дней назад

Puzzles like this make me wonder how fast Simon could solve if he didn't have to explain things.

@trainzack 25 дней назад

I generally take about 3 times as long to solve these, so I'm super proud of my 11:48. It's the first Sudoku I've ever solved where I figured out what all of the columns were before I placed any of the digits. I only had to figure out which order they went in!

@eddieharwood7788 25 дней назад

I thought this looked impossible and was tempted to watch, but I got it in 17 minutes. I liked it.

@richardfarrer5616 25 дней назад

n(n+1)(n+2)/6 is the tetrahedral formula (sum of triangular numbers). And 9x10x11/6 = 165.

@redstonekid2222 25 дней назад

Got it in around 15 minutes. Very fun puzzle, quite similar to the "121" puzzle featured before.

@phyjob 25 дней назад

Wow ! I finished in 4'49". Once the trick is understood, everything follows...

@reubenmckay 19 дней назад

I've rewatched this video 5 times and I still don't understand how Simon arrived at the conclusion that the sequence of numbers in the columns is the slot machine/wrapping around thing that he uses at the start of the solve.

@charliemichael4052 24 дня назад

The sum of the first n triangular numbers are called the tetrahedral numbers named after the 3 dimensional equivalent of a triangle for future reference

@Kinada 25 дней назад

Yeah, the puzzle literally just filled itself in. It's interesting that the starting digits follow mod3 to keep the lines separated.

@EdDrow 23 дня назад

The 7 indicates that the middle column is a 4 which means column 4 and 6 is either 1 or 7 what you can find out with the 13cage and from then on the cages only give you the start of the next cycle.