T_8 = 6^2
(Eingestellt am 1. Juni 2026, 06:08 Uhr von 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.
Lösungscode: Row 2.
Kommentare
am 6. Juni 2026, 10:36 Uhr von 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.
am 5. Juni 2026, 00:27 Uhr von 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.
am 2. Juni 2026, 11:44 Uhr von 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
am 2. Juni 2026, 03:19 Uhr von 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.
