Some Balancing Product
(Published on 9. November 2025, 12:36 by Manta-Ray)
Normal sudoku rules apply.
Yin Yang: Shade some cells in the grid such that all shaded cells are orthogonally connected, all unshaded cells are orthogonally connected, and no 2x2 region within the puzzle is entirely shaded or unshaded.
Cages: Within a cage, digits do not repeat and must sum to the value in the top-left corner of the cage (if given). Every cage within the puzzle contains at least one cell of each shading. Within a cage, the sum of the unshaded cells is equal to the product of the shaded cells. For clarity, if only one shaded cell exists within a cage then the cage's product value is that cell's digit. No two cages have the same sum/product value.
Solve on SudokuPad
Watch Mark's solve on CTC here
Solution code: Row 3 (read left to right) with a '-' representing a change in shading, e.g. 12-3456-78-9
Comments
on 18. November 2025, 10:28 by Manta-Ray
Added Mark's solve on CTC
on 15. November 2025, 17:55 by deltameth
What an engaging puzzle! It is rather difficult but I never felt that I had stuck. It unfolds so smoothly.
Wonderful puzzle, 98% rating is well deserved.
on 11. November 2025, 17:02 by Daniblitz
I thought the puzzle was fun with varied and interesting logic, flowing nicely up until the ending. But that fell a bit flat for me, as I didn't really need to use yin yang/cage rules for the last cage, except for reusing an earlier deduction to quickly eliminate a candidate, which made the rest solve by regular sudoku. Not as satisfying when I finish a yin yang puzzle suddenly with a lot left uncolered. Other than that, great puzzle!
on 11. November 2025, 09:30 by XuanZzz
Excellent puzzle! I used a python program to help with the enumeration lol lots of fun!
on 11. November 2025, 00:58 by JPlay
I was kind of intimidated by the yin yang as I'm generally scared of path puzzles, but this flowed really nicely and I never felt particularly stumped for where to go next. Good fun!
One rules note: "Every cage within the puzzle contains at least one cell of each shading" is an intuitive consequence of the next sentence of the rules so could be cut for a slightly tighter ruleset without increasing difficulty in my opinion.
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Hi JPlay, thank you for solving my puzzle! I agree that the extra rule might be overkill, but in a previous YY puzzle I created (where Arrows only counted cells of the same shading as the circled cell) there were some comments that if the rule didn’t explicitly say that there was at least one cell of the same shading then it could be ambiguous
on 10. November 2025, 20:46 by Onkel_Dagobert
How can a sum and a product be equal and add to 15 or 11? Wouldn‘t that mean that 15 and 11 are divisible by 2? Quite confused by these rules…
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Hi Onkel_Dagobert! So the rules state that the sum of the digits within the cage must be the value given, but that within the cage the sum of some digits must be equal to the product of the other digits (as separated by shading).
So as an example, if a 6-cell cage contained 135679 then the cage as a whole would sum to 31 - but the cage could contain a shaded 3/7 pair (which multiply together to give 21) and an unshaded 1/5/6/9 quad (which all sum to 21).
I hope this helps!
on 10. November 2025, 20:21 by henrypijames
@CitrusGremlin: The eight-cage doesn't need to be brute-forced on its own - as the solve progresses, other factors eventually and clearly narrows it down.
on 10. November 2025, 11:56 by CitrusGremlin
I'm not sure if I actually proved the one option available for the 8 cell cage or not. Maybe 80% certain. Still a very fun puzzle even if I made a faulty leap in logic.
on 9. November 2025, 14:38 by MathGuy_12
As a self proclaimed math nerd, this is an absolutely AMAZING puzzle!
on 9. November 2025, 14:03 by Snookerfan
Fabulous puzzle! Original and very hard, just as I like it. Certain disambiguations were very satisfying. Thank you!
