In variant sudoku, there are many constraints that are popular to add to puzzles. Some of these constraints behave exactly the same if you switch around the digits in certain ways. For example, many constraints behave the same when you map every digit x to 10-x. In cycle notation, that’s written as:
(19)(28)(37)(46)
German Whispers (resp. Dutch Whispers) are lines drawn in the grid where adjacent digits along the line must differ by 5 (resp. 4) or more.
Renbans are lines drawn in the grid, and they must have a non-repeating set of consecutive digits on them in any order.
White Kropki dots require that the digits in the two cells they touch must be consecutive.
In the non-consecutive global constraint, no two cells that touch may be consecutive.
However, some clues treat all digits differently, meaning they’re only preserved by the trivial permutation, also known as:
e
Killer Cages have to sum to the clue in the corner and may not repeat digits, digits along an Arrow add up to the attached circle, and digits along Region Sum Lines add up to the same value in each box.
Thermometers increase moving away from the bulb.
Counting Circles count how many Counting Circles have the same digit.
This is also where the full set of Kropki dots goes.
But some clues treat all digits the same. To notate this compactly, I’ll write:
[123456789]
Killer Cages without corner clues go here, as well as the Antiking and Antiknight global constraints where digits a king’s move (resp. knight’s move) apart cannot be identical.
[13579][2468]
Even Squares and Odd Circles force the digits on them to be even and odd respectively. Parity Lines are lines where the digits on them alternate between even and odd.
The next clues have quite a complicated symmetry group, which I’ll write as:
[[123][456][789]]
That contains (1435)(26)(78), (284395)(186), (123)(45), and several other permutations. As long as the sets {1, 2, 3}, {4, 5, 6}, and {7, 8, 9} are still those three sets in some order, the permutation is in that subgroup. That sounds complicated, and yet somehow there are two common variant sudoku constraints with that subgroup:
Entropic lines require that any three consecutive cells on them must have one digit from {123}, one digit from {456}, and one digit from {789}. Global Entropy requires that every 2×2 region has at least one digit from {123}, one digit from {456}, and one digit from {789}.
[[147][258][369]]
These symmetries are for the Modular Lines and Global Modularity, which are like Entropic Lines and Global Entropy mentioned earlier but using the sets {147}, {258}, and {369} instead.
Unique Subgroups
The Dutch Flatmates global constraint requires any 5 to have a 1 above it and/or a 9 below it, so its symmetries are [234678].
A given digit 3 has symmetries consisting of just [12456789], and different given digits have different symmetries.
Some clues have symmetry groups more complicated than this notation can handle, so in those cases I’ll just have to list a small set of generators for the subgroup.
Black Kropki Dots require the two digits they touch to have a ratio of 2, and their symmetries are generated by (18)(24), (36), and [579].
XV sudoku involves X clues that require the two digits touching them to add to 10, and V clues that require the two digits touching them to add to 5. Individually they have a lot of symmetries (X’s symmetries are generated by (19), (12)(89) and (1234)(9876), and V’s symmetries are generated by (14), (13)(24), and [56789]) but together their symmetries are generated by just (14)(69) and (12)(34)(67)(89).
That is all the common constraints I can think of, so that’s where this article ends.
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