Mod Prod in the Fog
(Published on 13. April 2025, 13:52 by DrWraith)
This uses a new kind of constraint I have been thinking about for a while. It's my first puzzle, I hope you like it!
Click the image below to play the puzzle on SudokuPad.
Rules: Normal sudoku rules apply.
(Modulo) Product Lines: Digits along a line of any colour multiply (mod 10) to the digit in the attached circle of the same colour. "Mod 10" means we just take the last digit of the product. For example, 3 * 9 * 4 = 8. Digits may repeat on a line if allowed by other rules. Product lines do not revisit cells or cross themselves.
Weak Kropki: Cells separated by a grey dot have a difference of 1 or 2.
Dynamic fog: Most of the grid is covered by a grey fog. Placing the correct digit in a cell will reveal that cell and sometimes other cells. Guessing is never necessary.
Hint: You will never need to calculate large multiplications in your head to solve this puzzle; you can take the last digit at each substep. For example, for 3*9*4*2, you can calculate 3*9=7, then 7*4=8, then 8*2=6.
Solution code: Row 2, left to right
Comments
on 29. April 2025, 09:26 by Nylimb
For some reason I especially like sudoku puzzles that involve multiplication, and this is a fine example of that. Thanks!
When I saw the setter's name, I decided to solve it while listening to music by one of my favorite composers, who just happens (?) to have the same name.
on 29. April 2025, 08:55 by voldemortensen
Wonderful puzzle! It looks a lot more intimidating than it turned out to be. It was fun to figure out the quirks of the module product lines.
on 29. April 2025, 00:44 by Jackler
27min. Knowing how odd and even numbers work with multiplication is crucial, otherwise this one might take a long time. Nice overall.
on 15. April 2025, 02:21 by DrWraith
I just made a clarity adjustment to the description (specifying that a line does not cross itself).
@TheNineElements thank you for the thorough review! Your tips are great!
on 14. April 2025, 09:37 by TheNineElements
Solve Time: 33:41
I rated this 2/5 difficulty
Very nice puzzle! The solve was very enjoyable and there were a lot of really great ideas throughout the solve.
The rules were a bit tricky to wrap my head around at first, but once I "realized" how to approach the solve, the solve was very smooth and the logic was approachable and fun. I enjoyed the dFoW element as well, and the reveals did a good job of directing my attention (and prevented me from going down complicated rabbitholes of [Z_(10), ×] logic). I did also enjoy the final disambiguation step, which was a neat idea.
On the whole, the puzzle was well-set and reasonably approachable. I think the difficulty was somewhere around 2-3*, but I landed on 2* (though definitely a harder 2*) because the required underlying logic really wasn't too difficult on its own. I think the "most difficult" aspect of the solve was in determining which types of digits were "important" and figuring out "where" to focus my attention (fog certainly helped with this).
Even though I decided on 2*, modular product lines are still a bit tricky to understand if you aren't familiar with the rule or haven't seen similar variants. I definitely can see why some people may feel this is closer to 3* difficulty. If you are inexperienced with this type of rule, there is definitely an added level of "difficulty" when figuring out how the rules work.
Either way, great puzzle & thanks for sharing
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Also, here's some general suggestions/tips to anyone who is stuck or having trouble with the math:
Firstly, use the multiplication trick that @DrWraith explained in the description.
Second, try not to focus too much on the exact values of digits that can appear on a specific line. Instead, try to think about broader concepts that apply to ALL lines. Consider how different "types" of digits on a line influence the modular (bulb) value. Ask yourself: what sort of "types" of digits may be useful to consider?
In this puzzle, there isn't any logical step that requires "chaining" logic across multiple lines that overlap. For these overlapping lines, you don't need to exhaustively test candidates on a line and then work out how those candidates interact with the intersecting line.
Instead, there will be a simpler "idea" or "question" you can ask that will allow you to limit a digit (or type of digit) to a couple of cells. You really shouldn't spend too much time working on the math unless you have only two or three candidates on a line cell.
on 13. April 2025, 18:11 by DrWraith
Due to some feedback, I have changed one of the final steps, so the solution has now changed. There was an alternative solve path that somebody found that I wanted to eliminate. I don't know how LMD works but I hope this doesn't invalidate previous solves!
on 13. April 2025, 16:32 by wuc
Great puzzle. You found sth I haven't seen so far. Works great, especially if you like math. Great fun thx.
