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'
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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