T_8 = 6^2
(Published on 1. June 2026, 06:08 by Nylimb)
This puzzle is based on the fact that the 8th triangular number equals the 6th square number: 1+2+...+8 = 6^2 = 36.
Fill the grid with digits from 1 to 8, so that each digit D occurs exactly D times.
Anti-king rule: Two digits which are adjacent, either horizontally, vertically, or diagonally, cannot be equal.
Kropki dots: If a white dot is between two digits then they are consecutive. If a black dot is between two digits then one is double the other. All possible dots are given. (If a 1 is next to a 2, the dot between them may be either black or white.)
You can solve this in SudokuPad.
Solution code: Row 2.
Comments
on 6. June 2026, 10:36 by JustinTucker
All configurations with 4 dots which allow a solution are equal to my first example (modulo rotations and reflections). And therefore non-unique.
Anyway, a very interesting puzzle!
@JustinTucker: Thanks again. As a mathematician, it's nice to know the limits on how few dots a puzzle like this can have. As a puzzle creator, I'm happy with my 8-dot version that can be solved without bifurcation.
on 5. June 2026, 00:27 by Nylimb
In a (formerly) hidden comment, JustinTucker gave an example of a puzzle with only 4 dots (but 2 solutions), and stated that there's one with 5 dots that's uniquely solvable. Using my solving program to look 'near' his 4-dot puzzle, I've found this one with 5: https://sudokupad.app/c3io81itb3
I was able to solve it by hand, but only with some unpleasant bifurcations.
I don't know if 4 dots can give a unique solution.
on 2. June 2026, 11:44 by JustinTucker
How many dots did you use in the other arrangements? You need at least 4.
@JustinTucker: It sound like you've figured out something that I haven't. I can only show that there must be at least 3 dots, namely 3 white dots touching a 7. How do you show there are at least 4?
The minimum number of dots that I've found in any grid is 6. I found 6 different arrangements of dots like that, each of which has a unique solution. But none of them seem to make good puzzles; I can't solve them without a lot of bifurcation.
@Nylimb I used a commercial MILP Solver to prove that you need a minimum of 4 dots to allow a solution. For example white dots at R12C2, R12C4, R4C56, R6C56, all connecting a 7 and an 8. The solution is only "nearly unique" since 1 and 2 may be swapped.
There is a configuration with 5 dots which has a unique solution. But as you remarked the manual solving won't be a lot of fun
@JustinTucker: Thanks for the information. By modifying your 4-dot example, I've found some 5-dot puzzles with unique solutions. I was able to solve this one by hand, but only with a couple of unpleasant bifurcations: https://sudokupad.app/c3io81itb3
on 2. June 2026, 03:19 by Playmaker6174
Incredibly cool and satisfying puzzle!
The puzzle has this theory feeling to it and once indulging myself into having the right thoughts, the solve really unfolded itself wonderfully :)
@Playmaker6174: I'm glad you liked it! I tried quite a few other arrangements of dots before choosing this one. The others had the same "theory" part, but didn't resolve so nicely after that.
