We have long investigated puzzles and their solution grids, and there are a small proportion of solution grids with > 1 automorphisms.
A stab at looking at the number of minimal puzzles per pattern was made here .
But really the numbers are too big and it is only at the edges C<21 , C>28 we can get close to an estimate by generating till puzzles dwindle with a neighbourhood search although the remote puzzles tend not to be found .....
Each puzzle can be morphed by exchanging the digits [9!] or swapping the rows colums boxes [2 x 6^8]
Each pattern can also be morphed up to 2 x 6^8 ways
The number of essentially different patterns has been worked out here
However .... its probably true that symmetry - eg diagonal symmetry must reduce the number of puzzles per pattern by half
Here are a sample pattern of each of the 15 27C puzzles with 3 diagonal/not in line clues -all Essentially Different patterns. [ED]
The #1 pattern had a surprizingly reduced number of puzzles in a run of the pattern [2 x 6^8 ways] over a large sample [100,000] of random grids
- Code: Select all
Highest rated puzzle in patterns game Estimate of number of minimal puzzles with this pattern [x50000]
# 1 ..1..2..3.2..1..4.5..6..1....2..4..1.5..6..7.7..2..3....7..5..8.8..7..9.9..3..2.. # C27/S4.da/M1.16.3 Patterns Game 16 ED = 10.6/10.6/9.8 2411
# 2 2...4...6..4..2.3..7.8..1....24...1.1...3...2.3...54....5..9.7..1.6..2..6...7...9 # C27.M/S4.da/M1.31.2 Patterns Game 150 ED = 10.7/10.7/8.9 109551
# 3 ..2..3.1..3..1.2..5..6....4..5..7.2..9..4.8..6..9....7.8..9..5.2..5..4....3..8..9 # C27/S2.d/M1.33.1 98063
# 4 ..1.2.3...2...1..43..5...6...42...8.9...3...2.3...67...4...3..75..8...4...2.6.1.. # C27.M/S2.p/M1.13.4 610993
# 5 1....2..3.9..4..1...73..5....21....9.6..2..5.9....56....3..41...7..8..2.2..6....8 # C27.M/S4.da/M1.11.6 86740
# 6 2...3.1....14...2..5...6..3.6.2..4..3...5..7...7..3..65..8..7...9..1..4...6..2..9 # C27/S2.d 701514
# 7 .2..7...84..6..9....5..4.7..1.2..3..2...1...4..3..5.6..9.5..6....1..7..37...3..5. # C27.M/S4.da/M1.16.1 641463
# 8 1..3..2...2..4..1...3..6..52...8..7..7.5..3....9..1..67...9.5...4.1....8..8..2.6. # C27/S2.d Patterns Game 410 ? 1393902
# 9 ..2.1.3...1...2.4.5..3....2..5..7.8.6...3...4.4.6..9..7....4..6.3.2...5...4.9.7.. # C27.M/S4.da 695457
#10 ..1..2..3.2..1..4.3..5..6....51..2...4..7...12....8.3...96...7..6..4.3..8....7..4 # C27 [asymmetric] [no symetric pattern] 1468539
#11 ..1..3..2.2..4..3.4..6..5....23...1..4...57..5...9...8..6.8..7..9.1..6..7....2..1 # C27/S2.d 257856
#12 2..1....3..1..2.4..5..3.6..5..7...9...6.8.2...8...4..1..7.6..8..6.4..1..3....7..5 # C27/S4.da 761411
#13 2...3.1....14...2..5...6..3.6.2..4..3....4.7...7.8...65..8..7...9..1..4...6..2..9 # C27/S2.d 389077
#14 6....2.7...8.1...5.9.5..3....2..1..3.1..2..4.5..3..2....4..8.9.1...9.8...3.4....7 # C27.M/S4.da 765426
#15 ..1.2...3.2...34..5..1...6...4..1.7.2...9...6.6.4..8...9...2..8..69...1.3...5.7.. # C27.M/S4.da 254596
The #1 pattern has the most reductions it would appear, they all have diagonal clues in the boxes and all have 27 clues
+---+---+---+
|..1|..2|..3|
|.2.|.1.|.4.|
|5..|6..|1..|
+---+---+---+
|..2|..4|..1|
|.5.|.6.|.7.|
|7..|2..|3..|
+---+---+---+
|..7|..5|..8|
|.8.|.7.|.9.|
|9..|3..|2..|
+---+---+---+
By my reckoning this pattern has ? 432 automorphs per pattern and there are 7776 different looking formations !! [432 x 7776 = 2 x 6^8]
[ 9 boxes,6 ways in a box, 2 x 2 ways for each band and chute, and one x 2 reflection] = 9x6x8 = 432
And this goes to explain why there were so many fewer puzzles compared with the other 27C puzzles
Apart from trivial 1, 81 and 9-template patterns ! this pattern may well be the pattern with the most automorphs
Maybe if we can confirm the automorphisms of these 15 ED [and there are only 15 ED ] patterns we can confirm the association
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