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God-Tier Fog Sudoku Makes My Brain Hurt!

Cracking The Cryptic

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** TODAY'S PUZZLE **
We must have missed this fog sudoku from Kaktuslav when it was first released.... it's called Reciprocals and Mark suggested Simon have a go at it today. When Simon read the rules, he wondered if Mark was having a joke at his expense!!

Play the puzzle at the links below:

https://sudokupad.app/gf8l5f4s64

Rules:
Normal sudoku rules apply. Yin–Yang: shade some cells in the grid so that all shaded cells are orthogonally connected, all unshaded cells are orthogonally connected, and no 2×2 region is entirely shaded or entirely unshaded. Every shaded cell containing digit x has value 1/x; every unshaded cell has value equal to its digit. Product/Sum cages: each cage has a corner mark in the leftmost cell of its top row identifying its type. If it shows Σ=X, the values in the cage sum to X; if it shows Π=X, the values in the cage multiply to X (X is not necessarily an integer). Digits or values MAY repeat within a cage. The grid is partially covered in fog. Placing correct digits reveals their own cell and may also lift fog elsewhere in the grid. No guessing is required.

▶ BLUE PRINCE THIS WEEK ◀

This week's Blue Prince stream is available to watch here:
https://youtube.com/live/9r4myloEFUU

▶ ZETAMATH'S WORM: OUR BRAND NEW APP ◀
We've released a brand new collection of sudoku puzzles by Blobz. The worm features 12 connected sudokus and is a great way to enjoy more creations from this popular constructor:

Steam:
https://store.steampowered.com/app/47...
AppStore
https://apps.apple.com/us/app/crackin...
Android:
https://play.google.com/store/apps/de...

All our apps can be found at our website with links to all outlets:
https://www.crackingthecryptic.com/apps

▶ NEW MERCHANDISE INCLUDING RAT RUN GOODIES ◀
https://cracking-the-cryptric-shop.fo...

▶ SIMON & MARK ATTEMPT BLUE PRINCE: THE SUDOKU ◀
This new epic (and epically long) video is now on Patreon (as well as our monthly sudoku hunt competition by Sandra & Nala). Check it all out here:

Patreon: crackingthecryptic

▶ SVEN’S SUDOKU PAD ◀
Input classic sudoku puzzles into our software. With 12 free puzzles:
iOS: https://apps.apple.com/us/app/svens-s...
Steam: https://store.steampowered.com/app/17...
Android: https://play.google.com/store/apps/de...
Amazon: Search for “SudokuPad”

▶ CTC FAN DISCORD SERVER◀
Discord: discord

▶ OUR VIDEO CATALOGUE – CATEGORISED WITH LINKS◀
https://tinyurl.com/CTCCatalogue

▶SEND US PUZZLES TO SOLVE/CONTACT US◀
crackingthecryptic@gmail.com
Snailmail: Mark Goodliffe, Box 102, 56 Gloucester Road, London SW7 4UB

▶FOLLOW US◀
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@crypticcracking
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▶Credits◀
Music: Lucy Audrin (Piano): Mozart's Sonata no 16; and Mark’s outro music by Riffclown.
SOFTWARE: Thanks to Sam Cappleman-Lynes and Sven Neumann
Logo: Melvyn Mainini
Opening Credits: Burak Mescioğlu

▶ Contents Of This Video ◀

0:00 Intro music and puzzle introduction
2:51 New Patreon Competition
1:44 GCHQ's Puzzle Of The Year
3:50 Happy Birthdays etc
6:51 Rules
10:04 Start of Solve: Let's Get Cracking

#sudoku #puzzle #maths

20 Comments

@Kaktuslav 2 months ago Pinned

Wow, thanks so much for the feature! Fantastic solution, Simon, as always, and a great exposition of the key intuitions. Also, thanks for the recommendations and kind comments — I hope solving did and will make brains hurt only in good ways :)

I have to say I don’t remember much about setting this puzzle, but let me share the recollections I do have. Spoilers below!
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Considering lowest common denominators can be used to prove statements such as: if the sum of reciprocals of some collection of digits 1–9, including at least one 7, is again an integer, then there have to be at least seven 7s. Formally, when expressed with the lowest common denominator, which is a multiple of 7 but not of 49, all the other fractions — those not coming from 1/7s — will have numerators divisible by 7. The whole sum, expressed with this denominator, will also have numerator divisible by 7, since it is an integer. But all the expanded 1/7s have equal numerators which are not divisible by 7, so the number of such summands has to be divisible by 7.

In a similar fashion, one can prove that whenever 1/8 appears in such an integer sum of one-digit reciprocals, there have to be at least two 8s, since only these fractions contribute odd numerators. Similarly, whenever 1/9 is one of the summands, there have to be at least two more 9s. I think this formalizes quite naturally the intuition Simon gave in the video, but to me Simon’s explanation was perfectly clear and more than enough.

The hard Yin-Yang step was definitely intended, and I was stunned that Simon decided to do YY at exactly the right moment and spotted this nontrivial deduction so quickly. Well done! I believe revealing r5c9 at that stage was meant to be a hint to look at that cell; inspecting its color can, I think, lead to conclusions similar to those in Simon’s solution.

@TheMNbassHunter 2 months ago

70 tooths! That's a lot of tooths!

@rnulfschmer7537 2 months ago

Seventyteeth ... surely!

@TheDukerino 2 months ago

40:50 "Its getting a little bit easier" ... 44:46 "this is a horrendous cage"

@hummakavula3750 2 months ago

Thinking in Base 12 will finally help me today

@sjm6280 2 months ago

Being a non-native English speaker, I get solace seeing also English people have a hard time pronouncing English fractions 😅

@flsal27 2 months ago

One underappreciated element of these videos is the feeling that we’re invited, twice a day, into their living room to hang out virtually around a Sudoku grid.

The little life moments—the kids fighting, the neighbor renovating, the morning after a party with maybe a bit too much wine, Maverick, the text message arriving mid-logic explanation, the package delivery, the weather, the bees, Maverick again—all of it contributes to that warm feeling of being invited to spend time together.

It’s not just Sudoku. It feels like a daily visit. 😊😊😊

@SmokuZnadPotoku 2 months ago

58:40 3-3-9 cage would work as well but there was a 9 in box 8 and therefore blocking the possibility of it

@werefrogofassyria6609 2 months ago

That was a crazy puzzle. Had to go away because doing the maths was causing anger. Then, came back and saw the trick of reciprocals and regulars.

@studgerbil9081 2 months ago

there is something unbearably charming about a former financial analyst using the term "twoths" and having enormous difficulty with fractions.

@helar2574 2 months ago (edited)

Number of times Simon said "72th" in first part of solving remind me famous "nori-nori solve"

@A_CC_K 2 months ago

What are those rules, Kaktuslav is a mad genius. This is going to be an amazing watch.

@wrightn9 2 months ago

I just noticed that at 40:47, r5c9 is unveiled from the fog, but I don't think that revealing it actually gives us any useful information. Instead, I think Kaktuslav did that to draw the solver's eye to that narrow gap that that purple cell would have to escape through. If that's true, then that is incredible setting.

@OptingOut 2 months ago

I like the progression of 1-product cage, 1-product cage, 1-product cage, 5-product cage! But actually that's just another 1-product cage with a random 5 thrown in.

@alexanderlori7651 12 days ago

I feel like thinking about these fractions as decimals helped me understand it far better

@stephenbeck7222 2 months ago

I made a version of this but couldn’t figure out how to get it into Sven’s sudoku pad. It was two infinitely long rows (normal sudoku rules did not apply). Kropki rules applied to digits, not values. The first row was one long cage with black kropki connecting each cell and Sigma = 1 in the top corner. The second row was one long cage with white kropki connecting each cell and Sigma = -1/12 in the top corner.

@eliana_d 2 months ago

no way i would ever attempt a puzzle this mathy even if i were capable at this ⭐ difficulty, but i am more than happy to watch simon battle it!

@rnhk6800 2 months ago

This was diabolical and I am proud I was able to understand enough to get through this one. The way you solve it with ease is so impressive.

@SiddSharma-ub8by 6 minutes ago

It could be easier to write all the fractions in decimal

@nwahally 2 months ago

"This cage is one of the most horrendous creatures in Sudoku Christendom." (45:28) That's not a sentence I thought I would ever hear, but now I do not want to imagine life without having heard it.