StrmCkr wrote:i belive all puzzles are formed form a automorphic grid in the first place....

i perhaps think that what he meant was you can get a puzzle from an automorphic grid, remove a few clues and add a few different ones - and be able to get a new puzzle for every one of the 5e^9 essentially different grids.

This boggling assertion is that all grids can be shown to be pretty close to an isomorph of every other different grid. - they might share a puzzle with n identical clues..

Heres an example derived from one extreme solution grid / puzzle.

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

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

|.56|.31|.87|

|.89|.64|.21|

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

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

|.12|.89|.46|

|.45|.23|.79|

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

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

|.31|.78|.65|

|.97|.45|.32|

+---+---+---+ pseudopuzzle with 2 solution grids [36 clues]

123897654456231987789564321978456213312789546645123879564312798231978465897645132

174852693256931487389764521963417258712589346845623179528396714431278965697145832

.2........56.31.87.89.64.21..........12.89.46.45.23.79..........31.78.65.97.45.32 - 1 sol.

.7........56.31.87.89.64.21..........12.89.46.45.23.79..........31.78.65.97.45.32 - 1 sol.

Each solution grid can have at least 10^13 essentially different puzzles.

Each one of these puzzles has 9! * 6^8 * 2 isomorphs.

There are certainly enough puzzles for this to be a possibility !

To test this:

here are two random grids

- Code: Select all
`639845712782913645514627389256139478498576123371284596827351964943768251165492837 - grid 1`

974356281321784965685219734237891546469573812158462379816927453542638197793145628 - grid 2

Do any of their isomorphs share at least one pseudopuzzle [with 2 solutions]

I would say probably ......but it might be difficult to find it !

Heres my first attempt, both grids are morphed to have these clues - easy.

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`+---+---+---+`

|123|...|...|

|456|1..|...|

|789|...|1..|

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

|.1.|...|...|

|...|.1.|...|

|...|...|.1.|

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

|..1|...|...|

|...|..1|...|

|...|...|..1|

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

- Code: Select all
`639845712782913645514627389256139478498576123371284596827351964943768251165492837 - grid 1`

123486597456197382789253164817524936564319278392678415671945823248731659935862741 - grid 1 iso

974356281321784965685219734237891546469573812158462379816927453542638197793145628 - grid 2

123586749456179328789243156214758963935612874867394215371865492698421537542937681 - grid 2 iso-1

123486597456197382789253164817524936564319278392678415671945823248731659935862741 - grid 1 iso

123586749456179328789243156214758963935612874867394215371865492698421537542937681 - grid 2 iso-1

123.86...4561..3..7892.31...1....9......1..7........15.71..5.....8..1...........1 - 28 clues - 86887 sol

next attempt

- Code: Select all
`123486597456197382789253164817524936564319278392678415671945823248731659935862741 - grid 1 iso`

123784659456193782789652143512847936947316528368925417671438295895261374234579861 - grid 2 iso-4

123.8....45619..82789.5.1...1....936...31...83.....41.671...........1....3......1 - 32 clues - 2471 sol

We have 2 random grids - they have 32 common clues !

How far can we take this ?