ronk wrote:JPF wrote:Ed Russell & Frazer Jarvis wrote:

The complete list of operations that we can perform is:

1. Relabel (i.e., permute) the 9 digits;

2. Permute the three stacks;

3. Permute the three bands;

4. Permute the three columns within a stack;

5. Permute the three rows within a band;

6. Any reflection or rotation.

and of course, any combinations of these operations.

Let's call such a combination a transformation (isomorphism) t of the grid G.

The resulting grid is tG.

Is there an equally simple statement to be sure one hasn't constructed a combination of operations that result in an identity transformation?

[edit:No answer required. I just remembered...]

Too late !

At least, for somebody else :

Actually, line 6 can be limited to : transposition = diagonal reflection.

Any reflections or rotations is a combination of some of these operations : permutations 2,3,4,5, transposition.

For example :

Let's take a 90-rotation.

Initial grid G :

- Code: Select all
` 1 2 3 4 5 6 7 8 9`

10 11 12 13 14 15 16 17 18

19 20 21 22 23 24 25 26 27

28 29 30 31 32 33 34 35 36

37 38 39 40 41 42 43 44 45

46 47 48 49 50 51 52 53 54

55 56 57 58 59 60 61 62 63

64 65 66 67 68 69 70 71 72

73 74 75 76 77 78 79 80 81

Final grid G'

- Code: Select all
` 73 64 55 46 37 28 19 10 1`

74 65 56 47 38 29 20 11 2

75 66 57 48 39 30 21 12 3

76 67 58 49 40 31 22 13 4

77 68 59 50 41 32 23 14 5

78 69 60 51 42 33 24 15 6

79 70 61 52 43 34 25 16 7

80 71 62 53 44 35 26 17 8

81 72 63 54 45 36 27 18 9

By using these succesive operations (see the notations below):

- transp
- stack(1,3)
- col(1,3)
- col(4,6)
- col(7,9)

we get G'.

which is written :

rot(90) = col(7,9).col(4,6).col(1,3).stack(1,3).transp

As far as reflections are concern, here are some possible breakdowns (which are not unique) :

RefV = col(7,9).col(4,6).col(1,3).stack(1,3)

RefAD= row(7,9).row(4,6).row(1,3).col(7,9).col(4,6).col(1,3).stack(1,3).band(1,3).transp

Notations :

rot(90) : 90 rotation

transp : transposition

thing(i,j) : permutation of thing i with thing j

RefV : vertical reflection

RefAD : antidiagonal reflection

Consequence :

Every (isomorphic) transformation on a grid G is a combination of the following operations :

1. Relabel (i.e., permute) the 9 digits;

2. Permute the three stacks;

3. Permute the three bands;

4. Permute the three columns within a stack;

5. Permute the three rows within a band;

6. transposition

which gives a number of possible transformations equal to 9!x 6^8 x 2 = 362880 x 3359232 =1,218,998,108,160

JPF