The Principle of Least Action
(Published on 22. November 2024, 14:12 by Tobias Brixner)
Puzzle link: Play on SudokuPad.
Rules: Normal Sudoku rules apply. For each pair of cages with the same clue, draw a separate, continuous path from one cage to the other through orthogonally connected cells. A path must not touch itself or other paths orthogonally, but it may touch itself or other paths diagonally. Each cell on a path, including its cages, must contain a number that is smaller than all its orthogonally or diagonally adjacent neighbors that themselves are not on any path. Cells joined by a black dot must have a ratio of 2; cells joined by a white dot must have a difference of 1.
Your feedback, ratings and comments are highly appreciated. Have fun!
Background: The principle of least action (correctly called the “principle of stationary action”, also known as “Hamilton's principle”) is a fundamental principle that can be used in practically all areas of physics to derive the dynamics of physical systems. Newton's equations of motion, the laws of refraction in optics, Einstein's equations of general relativity and more can be derived from it. In doing so, a physical quantity (the “action”) must be minimal (or more precisely “stationary”). For the example of the classical motion of a particle, this means that the path actually traveled by the particle has the smallest action. This principle is illustrated in the puzzle by the fact that the action (here the numbers in the cells) is locally smallest for the actual path that has to be found. If you “move” the path at any point to a free neighboring cell instead, the action is larger.
Solution code: All digits of row 1 (from left to right) followed by column 1 (from top to bottom) without spaces.
Comments
on 8. December 2024, 11:37 by Firebird
Wonderful new constraint! It played out very well. Hard, but I never felt stuck.
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So happy you liked it, many thanks for commenting! - TB
on 22. November 2024, 20:48 by Anisette
Brilliant brilliant brilliant! Took me hours but I was so satisfied, and it had me thinking right up to the end.
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Wow, thank you so much for the praise! Glad to hear you liked it despite the long time you had to invest. - TB
on 22. November 2024, 19:23 by itsid
That was indeed a wonderful puzzle, I usually don't trust myself enough to try a four star rated puzzle (I look at many and try only a few) but this I thought I saw something to at least get the break in; so I tried, and it was really great, I got stuck every now and again but not because of the puzzle; becaus of me being mediocre at best.
AAAAAnyways... great puzzle loved it as well.
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Many thanks for playing and commenting (again)! Nice to read you enjoyed it. - TB
on 22. November 2024, 19:01 by askaksaksask
This puzzle is EXCELLENT. Great logical flow, wonderful interactions, a great negative constraint...loved it. Perhaps 3.5-4* difficulty
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Thanks a lot! Glad you liked it. - TB
on 22. November 2024, 15:05 by Tobias Brixner
Changed the word order in one sentence following a posted comment.
on 22. November 2024, 14:45 by itsid
"Each cell on a path, including its cages, must contain a number that is smaller than all its orthogonally or diagonally adjacent neighbors that are not themselves on any path."
WHAT?!
Maybe you want to rephrase that...
reread that four times.. and I think I have an idea what you mean, but "are not themselves on any path"... is maybe easier to understand if reordered to
"that themselves are not on any path"
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Thank you for your suggestion. I am not a native English speaker but always attempt to do my best, within my abilities, when formulating new constraints (and I also have all puzzles and their rules checked by testers before posting here). It is quite possible that the Grammar is incorrect in my initial version of the rules. For anyone who likewise has trouble understanding the meaning, please change the word order as suggested. - TB
