Entropy of Parity
(Published on 22. October 2025, 17:46 by Tobias Brixner)
Puzzle link: Play on SudokuPad.
Rules: Normal Sudoku rules apply. Digits must not repeat within a cage. An arrow counts, for the neighboring cage it points at, in how many ways the locations of cells containing an odd digit could be arranged within the cage, assuming there were no Sudoku restrictions. If there are no odd digits in the cage, this number is 1. Cells connected by an X sum to 10.
Your feedback, ratings and comments are highly appreciated. Have fun!
Background: The cage logic can be alternatively defined as follows, which justifies the title of the puzzle: Let us define the “parity pattern” of a cage as a record of which of its cells contains an even and which an odd digit. A cage's “entropy” is the number of its possible different parity patterns assuming that the digits that the cage contains could be arranged within the cage without restrictions. An arrow cell marks the entropy of the neighboring cage at which the arrow points.
This formulation makes a connection to the concept of entropy in information theory, where entropy is usually defined with an additional logarithm, however. Since logarithms generally do not result in integer numbers, that calculation step was not implemented in the puzzle. The quantity in an arrow cell corresponds to the number of “microstates” (i.e., the number of arrangements of positions of odd and even digits) for a given “macrostate” (i.e., the total number of odd and even digits), a concept also relevant for calculating entropy in statistical thermodynamics (again with an additional logarithm).
Example: The following image provides a fully solved example on a 4x4 grid. You may solve the example for yourself here on SudokuPad. The three-cell cage has an entropy of three because the two odd digits could in principle occupy any of the three following cell pairs: (R1C2,R2C2), (R1C2,R2C3), or (R2C2,R2C3) even though only the third option is realized in the solved grid; it is irrelevant for the entropy count that the two odd digits might exchange positions because that would leave the parity pattern itself unchanged.
Solution code: All digits of row 2 (from left to right) followed by column 8 (from top to bottom) without spaces.
Comments
yesterday, 00:24 by dzamie
Oh cool, another semi-applied math puzzle. Break out your Pascal's Triangles, everyone, cuz we're getting into combinations!
I think this is my favorite of yours I've solved so far. Once you understand what the rule means, it restricts the possibilities of the arrow cells greatly, BUT even knowing what goes in the arrow doesn't tell you much about the box it points to (4 choose 1 = 4 choose 3, after all).
Edit: but yeah, as Nicko said, solution codes are usually just 9 digits - it's slightly quicker for people who use the sudokupad copy function, and MUCH faster for people who don't.
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Glad you liked the puzzle, many thanks for your comment!
Concerning the definition of solution code, I copied the scheme of asking for one row and one column from other puzzles on this site when I constructed my first Sudoku, and since then I just stuck with it. (Compared to the overall time typically needed for solving a puzzle, I thought entering a few numbers in the end is a minor time investment.) But OK, I will shorten the procedure for my next releases. - TB
on 22. October 2025, 20:26 by GorgeousNicko
An interesting idea ... the answer code not so much :D
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Thanks for playing and commenting. - TB
