Arctium
(Published on 30. December 2023, 12:22 by loerting)
Rules
- Normal Sudoku rules apply.
- Killer Cages: Digits in cages cannot repeat and must sum to the small clue in the top left corner of the cage.
No bifurcation is needed to solve this puzzle.
Play on SudokuPad
Theorem needed
Spoiler warning! Do not look any further if you are trying to solve without knowing the trick.Take all digits of the red cells, remove the digit of the blue cell and you will always have three sets of 1-9.
Proof: Add Col 4 and 6 and Row 4 and 6. The corner cells in the central box are now counted twice. Now remove the central box from our 4 sets. The corner cells are now counted only once, the central cell -1 times, and all other cells in the central box 0 times. You are left with three sets of 1-9.
Solution code: Column 2
Comments
on 31. December 2023, 09:31 by Fisherman
Permission to make a few brief remarks about this beautiful discovery by this setter, Loerting.
1. The theorem may also be proven using the white + shaped cells passing through the blue cell, and then subtract that from the five boxes. The sandwiched white cells contain two of each digit except the blue, which appears once.
2. Or, note that each each digit appears three times in the red region, except the blue, which appears four times. Which is precisely what the setter meant by three sets of 1-9 and an extra blue digit.
3.) The most wonderful thing about this awesome Loerting discovery is that the blue digit can be moved to any of the other 80 squares and the reds will follow to make the theorem hold.
4.) But that is just the start. We can have multiple blue squares, each with their red satellites, to formulate a general Loerting set theorem.
5.) Yes. I have read the comments that this is a particular case of Set. It is still a great discovery in my humble opinion.
on 30. December 2023, 14:52 by Fisherman
Very educational.
