The Force of Attraction
(Published on 14. September 2025, 12:41 by Tobias Brixner)
Puzzle link: Play on SudokuPad.
Rules: Normal Sudoku rules apply. Each cage is attracted to exactly one other cage, forming an attractive pair. The force of attraction, F, marked as a clue in the top left corners of both cages, is the product of the digits in the two cages divided by the square of their distance, F = m1 m2 / r2. Here, m1 and m2 are the two digits and r is their distance in units of cells as measured between the respective cage centers. For a horizontal separation x and a vertical separation y, the distance is calculated using the Pythagorean theorem, r2 = x2 + y2. For example, if row 3 column 3 and row 6 column 7 form an attractive pair, their distance is 5 because 52 = (7-3)2 + (6-3)2. There is no force of attraction between cages from different attractive pairs.
Your feedback, ratings and comments are highly appreciated. Have fun!
Background: The puzzle exemplifies Newton's law of gravitation, which states that the gravitational force of attraction is F = G m1 m2 / r2 between two masses m1 and m2, with gravitational constant G. For an appropriate choice of physical units (here G = 1), this is exactly the equation from the puzzle rules. In reality, however, attraction occurs pairwise between all masses, not just to one selected partner. Similarly, the attraction or repulsion of charged particles, i.e., the Coulomb force, follows the same law when considering m1 and m2 to represent charges instead of masses and using 1/(4πε0) instead of G as the proportionality constant.
Example: The following image provides a fully solved example on a 4x4 grid. You may solve the example for yourself here on SudokuPad. Two cages that are part of the same attractive pair are marked with matching colors.
Solution code: All digits of row 5 (from left to right) followed by column 9 (from top to bottom) without spaces.
Comments
on 17. September 2025, 20:32 by dzamie
That worked surprisingly well! There were just enough options to keep it from being trivial, but limited enough that the main difficulty came from keeping a few possibilities in mind.
Really, my only "complaint" is that the puzzle is basically only finding the cage values, since you can do them all at the start and then filling in the rest is trivial. And even then, that's more of a "it would be really cool to see something like this in conjunction with other variants."
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Many thanks for playing and commenting! I wanted to keep the first implementation of this rule set "pure", but certainly there is the option for combinations with other variants in the future. - TB
on 14. September 2025, 23:36 by MattYDdraig
Very nice concept and executed well.
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Glad to hear that, thank you! - TB
