I am surprised that in most puzzles changing one clue in an appropriate cell (and only in this cell) gives a new valid puzzle.

Furthermore, in some cases the nature of the puzzle changes dramatically.

By "nature", I mean the way of solving it, its difficulty, its rating, etc...

As far as I'm concerned, the best example I know is Gurth's G910:

- Code: Select all

. 4 2 | . . 9 | 1 . .

. 1 . | . 8 . | . 3 .

5 . . | 1 . . | . . 4

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

6 . . | . . . | . . .

. 9 . | . . 5 | . 7 .

. . 7 | . . . | . . 8

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

4 . . | . . 1 | . . 6

. 3 . | . 6 . | . . .

. . . | 7 . . | 2 4 .

rated 8.9 by Sudoku Explainer solved by a bunch of chains...

Try the same puzzle with r3c4=6 and you get an easy singles-only puzzle (11 stepper).

In a sense, the puzzle is not stable.

But some puzzles don't have this problem.

For those "untouchable" puzzles, you can't find a cell such that by modifying it only, you get a valid puzzle.

Here is one :

- Code: Select all
`. . 7 | . . . | . . 2`

. 1 . | . 5 4 | . . .

4 . . | . 6 2 | 8 . .

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

7 . . | . . . | 1 . .

. . 6 | . 3 . | . 4 .

2 . . | 5 . . | . . .

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

. 8 . | . . . | . . .

. . 9 | . . 7 | 6 . 8

. . . | 8 . 5 | . . .

Here are more examples :

- Code: Select all
`001000000000000579860090002000000740009000008200007000450980000030700000000420300`

000000040800050000600300008056080020709600500000030004008006072400000006001005000

090000705070860000100500090003000000006000000200048900040300809000000400001400026

007000002050900076000300000000530000630090000100070080000000004310600000024000309

020600100050080040000007000009001050002700090000850200001030060030020000000406003

000004070080700906003000000000002000097000020500000003008100700002009050039840010

000500000030900200200004706000641320000000000900030001005096000004800000827000000

000400200006000007091000300085037000060000004007000090009600400050000901000700056

gsf wrote:what is the 95% relative to?

I generated randomly 1000 minimal puzzles. I got 33 untouchables (3.3%)

I did it again but with 10000 minimal puzzles ; I had 339 unt. (3.4%)

So, my guess is that the average number of minimal untouchables in a random* sample of minimal puzzles is 3 %+, say 5%.

May be gsf could work this percentage out with a large sample of minimal puzzles ?

Red Ed wrote:"Untouchables" might be rare in the grand scheme of things, but there are lots of examples amongst Gordon's list of 17-clue puzzles. (It seems obvious that low-clue puzzles should be more likely to be untouchable.)

sure. The % for the Gordon's list would be interesting to know.

Generalization

If a puzzle is untouchable at the level 1 (changing 1 cell at a time), we can continue the test at the level 2 :

are there 2 cells (Ci,Cj) such that by changing their numbers (Ci->C'i ; Cj->C'j) we get a valid puzzle ?

If not, we have an untouchable level 2.

And so on, if necessary : level 3,4,....

In any case, the level is limited by the number of clues of the puzzle (cyclic permutation of the digits).

Any takers to produce untouchables level 2 or more ?

(the Gordon's list could be appropriate)

What are their properties ?

Chameleons

At the opposite, some puzzles can easily accept to change lots of their clues and remain valid.

We had a good example of that, given by coloin in the Megaclue thread.

In this minimal 34 clues-puzzle :

- Code: Select all
`1 . . | . . . | 7 . 9`

. 5 . | . 8 9 | 3 2 .

6 . . | 3 . . | . 5 .

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

2 . 1 | . . 5 | . . .

. . . | 8 9 . | . 3 .

8 . . | 7 3 2 | 5 . .

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

. 1 . | 2 . 4 | 9 . 5

. 7 4 | 9 1 . | . . .

. . 2 | . . 3 | 4 . .

27 cells can be changed (1 at a time) and still leave the puzzle valid.

Here are the numbers of possible changes for each clue :

- Code: Select all
`1,0,0,1,1,1,1,1,2,1,2,1,0,0,1,1,2,1,1,2,1,1,0,1,1,0,2,0,1,1,1,1,1,1`

Only 7 faithful cells !

Is there a minimal puzzle (chameleon) which doesn't have any faithful cell ?

JPF

* what does that mean exactly ?