Modropic Lines 2
(Published on 20. December 2025, 12:03 by GraphZahl81)
I have decided to use modropic lines for another puzzle. Not sure yet whether it will turn into a mini series.
Standard Sudoku rules apply: Every row, column, and 3x3-box contains the digits 1-9 exactly once.
Every line in this puzzle is both entropic and modular: Any set of three consecutive cells on a line contains a small digit (1,2,3), a medium digit (4,5,6), a large digit (7,8,9), and a digit from each of the sets (1,4,7), (2,5,8), and (3,6,9).
No two lines in this puzzle contain the same set of digits (i.e. there is at least one digit that lies on one of the lines, but not on the other). Furthermore, all lines are maximal in the sense that they cannot be extended without branching or meeting another line.
The letter V between two cells indicates that their digits have sum 5. For the letter X, their sum is 10. Not all possible letters are necessarily given.
Solution code: The digits of row 5 (9 digits)
Comments
on 16. February 2026, 18:45 by GraphZahl81
@firespire:
I had another look at it. The rule that no two lines contain the same set of digits is indeed not necessary to solve the puzzle, but can make some arguments slightly easier.
I don't quite get the sentence "the lines will often be able to break one of the rules but not both combined". If one of the rules "modular" or "entropic" is broken, the combined rule "both modular and entropic" is broken as well, so breaking one rule but not the combined rule is impossible.
If any cell is added to the end of any line, the extended line will no longer be modropic, which means that it is
(a) still modular, but no longer entropic,
(b) still entropic, but no longer modular, or
(c) neither modular nor entropic.
That's what "at least one of the two properties is violated" means: One property fails, or maybe even both fail.
on 16. February 2026, 18:11 by firespire
Did the other one but it wasn't fresh in my mind and rushed into this maybe not taking time to think the correct way. I suspect I solved it in a different way the authour intended because the rule "No two lines in this puzzle contain the same set of digits" was never used.
I was fixated on entropic forgot about modular till the end but did remember to sequence from the first puzzle.
Lines need to be prevented being able to be extended breaking BOTH the modular and entropic rule combined. The comment mentioning "at least one" makes no sense to me as my completed correctly marked grid the lines will often be able to break one of the rules but not both combined.
on 25. December 2025, 12:24 by GraphZahl81
@L00ping007: Sorry for the late reply, I don't check my notifications as often during the holidays.
Your interpretation is exactly what that sentence means. For every line in the puzzle, if one were to continue in any direction from either of its ends while avoiding all other lines, the resulting line would violate at least one of the entropic/modular properties.
on 21. December 2025, 07:40 by GraphZahl81
Clarified the description of the solution code.
on 21. December 2025, 01:46 by Fisherman
Excellent puzzle.
