Sudoku Miracle

[Lorriane Nasuti]

I've been attempting to construct a miracle sudoku puzzle for a while now, with a little twist. Instead of one sudoku grid, there are two combined grids, as a new-ish idea. The grid I've been using to construct is shown below:

However, a very common problem I faced in my fruitless attempts of creating a solvable miracle was that I would end up with numbers that I couldn't put into the grid anymore, making me feel that this grid makes the puzzle impossible to create.

Without any means of finding out whether there exists a grid by myself (due to the lack of coding effort or skill), I cannot continue further in my pursuits of an elegant puzzle. As such, I'm asking for a list of possible solutions (where the solution to both sudoku grids are uniquely different from each other) for this grid with the miracle sudoku restraints, so that I know I'm not going down a rabbit hole trying to produce a puzzle. Help would be appreciated.

The first step was to write a solver that creates all miracle sudokus for an initially blank grid. I didn't bother about symmetrical solutions, because the condition that no two directly adjacent cells can have adjacent numbers rules out the possibility that you can create n sudokus with a 1 in the first cell and then generate 9! n variations by shuffling the numbers around. I think it is unlikely that condition 3 will survive such a shuffling.

Anyway, it turns out that there are only 72 possible miracle sudokus. The next step is now to save these sudokus as they are generated. Then try to find pairs by comparing the southwest and northeast regions. Again, I wasn't very subtle about it: I used a nested loop that tries all 7272 possibilities.

It turns out that there are again 72 possible double miracle sudokus. (The extra care described above does not seem to have an effect, probably because equal numbers are already arranged in a spaced out N-queens-pattern with (3, 1) distances between them.)

I hope I haven't made any mistakes. I've replaced single numbers and stretches of adjacent numbers with hash marks to check whether the conditions are all met. Seems to look okay, but I've only spot-checked.

... according to my program, there are only 72 possible double miracle sudokus. So once such a sudoku is published, it might be easy to find the answer by just checking the known numbers against this short list.

I think that's generally a problem with sudokus, which are easy to solve mechanically. Still, many people enjoy solving them by hand.

This was generated with an online sudoku solver which supports anti-knight, anti-king and anti-consecutive rules. I generated the top-right grid first by inserting random givens until the grid was unique, and then generated the second grid by using the green cells as givens. (As it happens, this one seems to also obey anti-knight and anti-king constraints across grid borders, although it wasn't guaranteed to.)

A few days ago, I kept seeing a tweet regarding some kind of impressive sudoku puzzle. I was mildly interested, but not enough to trigger me to watch a 25 minute video. Then I saw Jason Kottke write about it, and that was then linked to by John Gruber. And as I had some free time tonight, I decided to give it a watch.

After about 4 minutes into the video, the rules were explained, and I just had to try this myself. So I paused the video, and found the link to the puzzle. And after around 45 minutes (A rough estimate as I was doing other things at the same time), I finally did it. It truly is a fantastic sudoku.

I still had to watch the rest of the video though, and it was totally worth it. You watch him start with a seemingly impossible puzzle, which he starts to make progress on, and then suddenly it clicks. A fascinating thing to watch.

Doing mathematical proofs is kinda fun. Unfortunately they only make you do a few fun ones in school, then they get frustrating and tedious. So I have long been looking for a game that is about doing mathematical proofs. Euclidea was good, but eventually runs into the same problem as the hard proofs you do in school, so I never finished the game. But recently a lot of hard Sudoku variants have come along that feel exactly like doing a mathematical proof, but are designed to be fun.

Normal sudoku rules apply. The grey line in the central box is a thermometer. Digits along that thermometer must increase from the bulb to the tip, but the positions of the bulb and the tip are hidden. In the outer three rings, the sum of the digits along a ring between two 9s must always be the same throughout the puzzle. For the avoidance of doubt, those two 9s could be the same 9 if a ring has only a single 9 in it.

Just counting the areas between nines, I count 8 areas. This makes sense because the outer three rings cover 8 boxes, so there must be 8 nines. I also tried the extreme case: If I have just one nine in a ring, there are still 8 areas between them:

The next-easiest options are on the outer ring. We can pretty easily rule out that there are two or three 9s on that outer ring. The sums always ends up being too big because we have to use big digits on the outer ring. So there have to be four 9s on the outer ring:

Now after this we can eliminate the 3s from the other rings because our total has to be 8. So if we have 4 on the outer ring, we can only have 3 on another ring if we had 1 on the last ring, and we already ruled that out:

How would we prove this case? If we could eliminate the 2 from any of the two rings, that would also eliminate the 2 from the other ring, so there would have to two 4s and a 0. We can do that on the middle ring. If we tried to place the smallest possible digits there, we find that we can never put digits small enough that would add to 36. The smallest possible digits, after placing two 9s, are the digits from 1 to 6 twice and the digits from 1 to 5 twice, which add up to 21+21+15+15=72=2*36. This seems to barely work, but the pre-placed 3 messes up that plan because it rules out 3 from one row. We have to use a larger digit at least once, so our sum will be too big.

I used Mathematica to create the following puzzle. A playable version is here. Normal sudoku rules apply. Digits cannot repeat along diagonals except when they appear on an arrow. In which case, they must repeat but only in the cell indicated by the arrow and digit (ie giving the direction and distance of the repeated cell). All of the numbers 1 to 8 must appear on (at least) one arrow.

A set of values in a sudoku can be considered a permutation. We can find all the permutations that would pass the boxes test. Then we can find all the permutations that would satisfy a queen's position, and then find all 7 cliques. FindClique[queengraph, 8, All] reveals that no 8-cliques exist.

That's as far as I got for awhile. I don't recall making any good sudoku puzzles from the 7-cliques, though there may be a method I missed. I looked at the 6-cliques without anything amazing occuring to me. A few weeks ago I came back to the problem and pondered making a Super-X sudoku, where the 5 main diagonals each way would not have repeats. I don't think I found a solution. Then I tried reversing it -- what if all the repeats were just on a few diagonals. I found a position where only 4 diagonals had repeats, and created the following puzzle.

Duplicate Bridge! What if all the repeats were set up as bridges? A number on an arrow would give the distance and direction to the duplicate number. After a search, that led to the following puzzle.

That's more elegant. Unfortunately, when restricted to just one duplication per number, there were only three basic configurations to choose from. Two of them were used in the puzzle immediately above. Then it occured to me that I could allow more than a single 1-bridge. A 9-bridge would be impossible, but there could be solutions with an 8-bridge and all smaller.

If I could find a configuration that had 1 to 8 bridges, then a puzzle with no givens might be possible. If any were missed, say 8, then there would be no way to distinguish 8 and 9 on the grid without some extra information being given.

Running the code, it turns out there there is only a single configuration that works! It's unique. So from there, it then needs to be set up as a puzzle, and hopefully it's human solvable. We have a computer solvable puzzle already. It's unique! Go find it, computer. But that's not a fun solving experience for a human.

There are 2^12 = 4096 ways the arrows could be set up, so I decided to go for an easy arrow configuration to see if I could solve it. After a few tweaks I came up with a configuration that had a fairly good set of solving logic. Done!

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Miracle Sudoku features beautiful puzzles that at first glance look impossible to solve!! Indeed, some of our Miracle puzzles have as few as two givens digits (!) but you will find them all to be solvable with a bit of clever logic! At launch you will be able to play through 4 variants all of which make use of a non-consecutive restraint (ie neighboring cells cannot have two consecutive numbers in them). They are Non-Consecutive Sudoku, classic Miracle Sudoku (non-consecutive + knight + king restrictions), Non-Consecutive Thermo Sudoku and Non-Consecutive Sandwich Sudoku. Each of these variants has a unique feel which refreshes the logic you may be used to, especially in thermo and sandwich sudoku!

Cracking the Cryptic has released 7 puzzle apps, available on Steam, iOS and Android. All seven apps are sudoku apps, developed by Studio Goya. The first released app was Sandwich Sudoku, released on August 19, 2019, followed by Classic Sudoku, Chess Sudoku,Thermo Sudoku, Miracle Sudoku, Killer Sudoku and Arrow Sudoku.

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