Heat Flow
(Published on 25. October 2025, 14:36 by Tobias Brixner)
Puzzle link: Play on SudokuPad.
Rules: Normal Sudoku rules apply. A cage contains its “temperature” T as a digit. If two cages with temperatures T1 and T2 (>T1) are connected by a direct, straight (red) line, this results in a “heat flow” q along the line. The length d of a line is measured in cells from end point to end point; for example, the line between cages in row 1 column 3 and row 1 column 6 has a length of d = 2. Each line has a “thermal conductivity” of k = 3. The heat flow is equal to the thermal conductivity times the temperature difference at the end points divided by the length, q = k(T2-T1)/d. A digit in a circle indicates the heat flow along the line passing through. If two lines cross in a circle, both have the same heat flow.
Your feedback, ratings and comments are highly appreciated. Have fun!
Background: This puzzle illustrates thermal conductivity as described by Fourier's law, just as defined in the puzzle rules. The customary minus sign signifying that heat flows from regions of hot to cold temperatures is omitted above because we deal only with absolute values.
Example: The following image provides a fully solved example on a 4x4 grid. You may solve the example for yourself here on SudokuPad. The rules are analogous to the full puzzle, just with a thermal conductivity of k = 2.
Solution code: All digits of row 8 (from left to right) without spaces.
Comments
on 28. October 2025, 18:31 by dzamie
Relatively simple, but pretty fun! I appreciate that there's at least one circle that doesn't have a multiple of three, rather than taking the "easy way out" and having all the differences divisible by the distances.
As KillerCard mentioned, it'd be pretty cool to have a followup puzzle that features lines of different conductivities, probably distinguished by color (a particularly clever setter might be able to have the different constants player-discoverable, but I myself can't figure out how that would be done).
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Thank you for your feedback! Indeed, that was another thought that I followed for a while when experimenting: to let the solver figure out the heat conductivity, but then at one point I opted for this simpler version. Maybe for a follow-up... - TB
on 26. October 2025, 13:23 by KillerCard10
There’s some nice discoveries to make with the thermal conductivity formula and its relationship to the line length, and a couple nice interactions between heat flows on the grid. One of the last heat flow deductions was very surprising and I loved that helped unlock the solution! It had very nice density of heat flows, so that the sudoku required to finish out the puzzle went smoothly and efficiently.
I found myself wanting the thermal conductivity to not be a constant throughout the puzzle. That made the algebra more straightforward, and I did find myself slipping up in a couple places. Although that would wrench up the difficulty, that or some other sudoku variant rule to help interact with the multiples would be interesting. Actually had elements of doing arrow sudoku on the low-distant heat flows and German or Dutch Whispers on the longer ones. I like the concept I just found myself wanting it to be developed further. Great setting Mr. Brixner!
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Thank you so much for playing and your detailed comments! Glad you enjoyed it. I indeed considered making the heat conductivity vary. However, I ultimately decided against it for this first demonstration of the ruleset because I did not want to overcrowd the grid with another type of clues regarding the heat conductivity. But of course there is always the option to set a more complex variant in the future. All the best - TB
