here is an exotic deduction i've become fascinated by recently, first explored (as far as i'm aware) in these puzzles: Antidote, Poison, and Serum

it is an extension off of Gurths Symmetrical Placement which you can read about here and here, but i will also provide a brief rundown of the logic and show how its similar to anti-GSP

GSP states that:

- Code: Select all
`if`

- a puzzle has 1 solution

- the given information is entirely symmetrical in a particular way

then

- the solution must be symmetric in the same way

anti-GSP on the other hand says:

- Code: Select all
`if`

- a puzzle has 1 solution

- the solution cannot be symmetric in a particular way

then

- the given information cannot be entirely symmetrical in the same way

you can think about it from the perspective of creating a sudoku, if you're only ever adding in symmetric clues, how can you ever reach an asymmetric solution? you either will end up with asymmetric clues, a symmetric solution, or no solution

i've gone and made a bunch of examples to show off different ways we can apply this

in each one i've made 8r1c1 the asymmetric digit, just so that it's clearer to see the symmetry in the remaining clues

- Code: Select all
`+-------+-------+-------+`

| 8 . 1 | 2 3 . | . 9 . |

| . 7 . | . . . | . . 8 |

| 6 . . | . 8 4 | . . . |

+-------+-------+-------+

| 9 . . | . . . | 5 . . |

| 5 2 . | 4 . . | 9 . 3 |

| . . . | . 5 . | . . 2 |

+-------+-------+-------+

| . . 3 | . . . | . . 1 |

| 7 . . | . 2 . | . 6 . |

| . 6 . | . 4 8 | 7 . . |

+-------+-------+-------+

Serum

estimated rating: 9.0

8.123..9..7......86...84...9.....5..52.4..9.3....5...2..3.....17...2..6..6..487..

lets start simple, how could we determine a solution cannot be symmetric for a given puzzle? i think the easiest way to do so would be if a particular cell that is undisturbed after the symmetry operation cannot be a self-mapping digit, which is whats going on in this one

(ignoring 8r1c1,) we have positive diagonal symmetry with the mappings: 1-1, 2-2, 3-3, 4-5, 6-7, 8-9

however, r1c9 is not able to be 1, 2 or 3, so it has to be a paired digit, this implies that the solution to this puzzle is asymmetrical. note that this would not apply if r1c9 was instead given as a 4 for example, because then the given information is not symmetric

our conclusion is that r9c9 cannot be 9, it is the symmetrical counterpart to r1c1 and placing it in would lock the puzzle into having either no solution, or multiple

so -9r9c9, solves with a locked pair

- Code: Select all
`+-------+-------+-------+`

| 8 3 . | . 7 . | . 5 6 |

| 1 . . | . 4 . | . . . |

| . . 2 | . . 8 | . . . |

+-------+-------+-------+

| . . . | . . . | 9 1 4 |

| . . . | . . . | . . . |

| 5 1 8 | . . . | . . . |

+-------+-------+-------+

| . . . | 9 . . | 2 . . |

| . . . | . 5 . | . . 1 |

| 7 4 . | . 6 . | . 3 . |

+-------+-------+-------+

FMK

estimated rating: 9.1

83..7..561...4......2..8.........914.........518.........9..2......5...174..6..3.

another way we could determine the solution is asymmetrical is if the mapping is incompatible with the symmetry type

in this puzzle, we nearly have rotational symmetry with the mapping 1-1, 2-2, 3-3, 4-5, 6-7, 8-9

this is incompatible because there are too many self-mapping digits, only one of 123 could occupy the undisturbed cell r5c5, leaving the other two to repeat in r5, c5, and b5

knowing this, we use our only asymmetrical given to conclude -9r9c9 again, this time giving stte

- Code: Select all
`+-------+-------+-------+`

| 8 . 3 | . . 5 | . . 1 |

| . 9 . | . 1 . | . 8 . |

| 1 . . | 4 . . | 3 . . |

+-------+-------+-------+

| . . 6 | . . . | . 9 . |

| 4 . . | . 3 . | . . 5 |

| . 8 . | . . . | 7 . . |

+-------+-------+-------+

| . . 5 | . . 8 | . . 2 |

| . 7 . | . 2 . | . 6 . |

| 2 . . | 9 . . | 4 . . |

+-------+-------+-------+

Needle Pusher

estimated rating: 8.9

8.3..5..1.9..1..8.1..4..3....6....9.4...3...5.8....7....5..8..2.7..2..6.2..9..4..

now so far i've only shown symmetries that have ways of working under traditional GSP, but theres actually nothing to stop us from using "doomed" symmetries! those being, any symmetry type that wouldnt work under sudoku rules, stuff along the lines of a vertical flip

in this puzzle, the morph we will perform is swapping the rows to be 321654987, and swapping the columns to be 987654321 (using the shortcuts from elevens mapping categories it would be CxRxSx i believe)

or alternatively we can subdivide the grid into 3 rotationally symmetric regions, those being each band of the sudoku. this way of parsing it i first heard about in mith's puzzle Thunderstorm

from this our mapping is 1-1, 2-2, 3-3, 4-5, 6-7, 8-9

the undisturbed cells (r258c5) are fine, however the placements for the remaining self-mapping digits in rows 258 and boxes 258 will always result in a repeat, similar to the last puzzle

8r1c1 is breaking the symmetry here so therefore -9r3c9 stte

- Code: Select all
`+-------+-------+-------+`

| 8 . 4 | . 3 . | 7 . . |

| . 9 . | . . . | . 3 . |

| 1 . . | . 9 . | . . 4 |

+-------+-------+-------+

| . . 7 | . . . | 5 . . |

| . 8 . | . . . | . 2 . |

| 5 . . | . . . | . . 1 |

+-------+-------+-------+

| . . 8 | 1 2 3 | 9 . . |

| . 2 . | 4 5 6 | . 8 . |

| 3 . . | 7 8 9 | . . 2 |

+-------+-------+-------+

Antithesis

estimated rating: 9.0

8.4.3.7...9.....3.1...9...4..7...5...8.....2.5.......1..81239...2.456.8.3..789..2

one last one to wrap up, although there are certainly more ways i can think to explore this, which i will likely continue to do so in this thread

in this puzzle we nearly have sticks symmetry (in rows) with the mapping 1-7, 2-8, 3-9, 4-4, 5-5, 6-6

and at first blush, it seems there isnt a way to disprove possible symmetric solutions, instead what is key to notice is that the undisturbed cells r25c456 form a DP if limited to only self-mapping digits. so we know for certain that if this puzzle was symmetric then it would have either multiple solutions or none! this particular deduction is very useful for sticks anti-GSP, as there are certain required givens needed to disambiguate the undisturbed cells of those puzzles

from this, -2r3c7 stte

let me know of any other ways you can think to apply this! i had a lot of fun exploring the logic and creating this set of examples, i'm very interested in this logic and would love to know your thoughts. also, if this has had documentation in the past that i'm unaware of, please leave that below