The New Sudoku Players' Forum

by **P.O.** » Wed May 24, 2023 6:32 pm

i found four posts referring to hamiltonian path but none with the project of finding the hamiltonian paths of a puzzle represented as a graph whose vertices are the clues and whose edges are pairs of clues that share a unit
from wikipedia: a hamiltonian path is a path that visits each vertex of the graph exactly once
i consider the different components of a puzzle as graphs in their own right
vertices are identified by cell numbering from 1 to 81
i try to build all hamiltonian paths for each vertex
when the number of paths is too high i only calculate a sample, in this case the figures reported are obviously inaccurate or may simply be wrong if the figure is zero

manageable, all paths are calculated:

Hidden Text: Show

Code: Select all: . . . . . . . . . . . . . . . . . 1 . . . . . 2 . 3 . . . . . . 4 2 5 . . . 6 . . . . . . . 1 7 . . . . 8 . . . . . 7 . . . 6 4 . . . 6 . . . . 9 . . 8 . . . 1 . .................1.....2.3......425...6.......17....8.....7...64...6....9..8...1. adjacency list: ((18) (26 63)) ((24) (26 33)) ((26) (18 24 35 53 80)) ((33) (24 34 35)) ((34) (33 35 53)) ((35) (26 33 34 53 80)) ((39) (47 48)) ((47) (39 48 53)) ((48) (39 47 53)) ((53) (26 34 35 47 48 80)) ((59) (63 68 76)) ((63) (18 59 80)) ((64) (68 73)) ((68) (59 64 76)) ((73) (64 76 80)) ((76) (59 68 73 80)) ((80) (26 35 53 63 73 76)) cell: 18 size: 32: ((18 63 59 76 68 64 73 80 26 24 33 35 34 53 47 48 39) (18 63 59 76 68 64 73 80 26 24 33 35 34 53 48 47 39) (18 63 59 76 68 64 73 80 26 24 33 34 35 53 47 48 39) (18 63 59 76 68 64 73 80 26 24 33 34 35 53 48 47 39) (18 63 59 76 68 64 73 80 35 34 33 24 26 53 47 48 39) (18 63 59 76 68 64 73 80 35 34 33 24 26 53 48 47 39) (18 63 59 76 68 64 73 80 35 26 24 33 34 53 47 48 39) (18 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several components:

Hidden Text: Show

Code: Select all: . . . . . . . . 1 . . . . . . . 2 3 . . 2 . . 4 . . . . . . . 5 . . . . . . . 3 1 . . . . . 6 . . . . 4 . . . . . 1 2 . . . . . 7 . . . . 8 . . 6 4 . . . . 9 . . ........1.......23..2..4.......5.......31.....6....4.....12.....7....8..64....9.. adjacency list: ((9) (17 18)) ((17) (9 18)) ((18) (9 17)) ((21) (24)) ((24) (21)) ((32) (40 41 59)) ((40) (32 41 58)) ((41) (32 40 59)) ((47) (52 65 74)) ((52) (47 70 79)) ((58) (40 59)) ((59) (32 41 58)) ((65) (47 70 73 74)) ((70) (52 65 79)) ((73) (65 74 79)) ((74) (47 65 73 79)) ((79) (52 70 73 74)) cell: 9 size: 2: ((9 18 17) (9 17 18)) cell: 17 size: 2: ((17 18 9) (17 9 18)) cell: 18 size: 2: ((18 17 9) (18 9 17)) cell: 21 size: 1: ((21 24)) cell: 24 size: 1: ((24 21)) cell: 32 size: 6: ((32 41 59 58 40) (32 59 58 40 41) (32 40 58 59 41) (32 59 41 40 58) (32 40 41 59 58) (32 41 40 58 59)) cell: 40 size: 4: ((40 58 59 41 32) (40 58 59 32 41) (40 41 32 59 58) (40 32 41 59 58)) cell: 41 size: 6: ((41 59 58 40 32) (41 40 58 59 32) (41 32 59 58 40) (41 59 32 40 58) (41 40 32 59 58) (41 32 40 58 59)) cell: 47 size: 26: ((47 74 65 73 79 70 52) (47 74 73 65 70 79 52) (47 74 79 73 65 70 52) (47 65 73 74 79 70 52) (47 65 74 73 79 70 52) (47 74 73 79 52 70 65) (47 52 70 79 73 74 65) (47 52 70 79 74 73 65) (47 74 65 73 79 52 70) (47 65 73 74 79 52 70) (47 65 74 73 79 52 70) (47 52 79 74 73 65 70) (47 52 79 73 74 65 70) (47 74 65 70 52 79 73) (47 74 79 52 70 65 73) (47 65 70 52 79 74 73) (47 52 70 79 74 65 73) (47 52 70 65 74 79 73) (47 52 79 70 65 74 73) (47 65 70 52 79 73 74) (47 52 70 79 73 65 74) (47 52 70 65 73 79 74) (47 52 79 70 65 73 74) (47 74 73 65 70 52 79) (47 52 70 65 73 74 79) (47 52 70 65 74 73 79)) cell: 52 size: 22: ((52 79 70 65 73 74 47) (52 70 65 73 79 74 47) (52 70 79 74 73 65 47) (52 70 79 73 65 74 47) (52 70 79 73 74 65 47) (52 70 79 73 74 47 65) (52 47 74 73 79 70 65) (52 79 73 74 47 65 70) (52 47 65 74 73 79 70) (52 47 65 73 74 79 70) (52 47 74 79 73 65 70) (52 47 74 65 73 79 70) (52 79 70 65 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73) (79 70 52 47 74 65 73) (79 52 70 65 47 74 73) (79 73 65 70 52 47 74) (79 70 52 47 65 73 74))

no hamiltonian path:

Hidden Text: Show

too many paths, just a sample per vertex:

Hidden Text: Show

Code: Select all: . . . . . . . . . . . . . . . . . 1 . . 2 . . 3 . 4 . . . . . . . . . . . . . . 4 5 . 2 . . 6 . 1 . . . . 7 . . . . . 2 4 3 . . 8 . . . . . . . 7 1 . 6 . . . . . .................1..2..3.4..............45.2..6.1....7.....243..8.......71.6..... adjacency list: ((18) (26 54)) ((21) (24 26)) ((24) (21 26 42 60)) ((26) (18 21 24 44 62)) ((41) (42 44 49)) ((42) (24 41 44 49 60)) ((44) (26 41 42 54 62)) ((47) (49 54 65 74)) ((49) (41 42 47 54 76)) ((54) (18 44 47 49)) ((60) (24 42 61 62 76)) ((61) (60 62)) ((62) (26 44 60 61)) ((65) (47 73 74)) ((73) (65 74 76)) ((74) (47 65 73 76)) ((76) (49 60 73 74)) cell: 18 size: 25: ((18 54 47 65 73 74 76 49 41 44 42 60 61 62 26 24 21) (18 26 62 61 60 76 74 73 65 47 54 49 41 44 42 24 21) (18 54 44 42 41 49 47 65 73 74 76 60 61 62 26 24 21) (18 54 44 26 62 61 60 76 73 74 65 47 49 41 42 24 21) (18 26 62 61 60 76 73 65 74 47 54 49 41 44 42 24 21) (18 54 47 65 74 73 76 49 41 44 42 24 60 61 62 26 21) (18 54 47 65 73 74 76 49 41 44 42 60 61 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by **Leren** » Wed May 24, 2023 10:28 pm

Hamiltonian paths and circuits were investigated by myself and JPF (about 10 years ago) in respect of 5 digit primes being "connected" if they differ by exactly one digit.

After a lot of effort I managed to find a maximal Hamilton cycle of 8358 such primes, there being 5 other primes that can't be in such a circuit, being singly connected.

You can read about this work here.

Leren

by **P.O.** » Thu May 25, 2023 5:44 am

i know, i read the thread, they are also interesting in the context of puzzles.

by **coloin** » Fri May 26, 2023 12:56 pm

Interesting that the finite world of 5 digit primes are connected so ....

Of course the finite world of sudoku puzzles are also connected ... if not rather more

Your hamiltonian path of that first 17C puzzle doesnt actually have a valid puzzle within {-1+1}

There exists a valid path {-1+1} for most puzzles - especially when the clue count exceeds 19

Those puzzles with no [valid] hamilton path were described as "untouchable" and many 17s are thus. here

There exists a set of 17C puzzles all connected by {-1+1} gsf calculated this ... 18950 out of 41324 17C known puzzles at that time ...[? is this the closed set ?} here

There also would exist a set of around a closed group of 900 million 18C puzzles connected by {-1+1} which an anonymous contributor [anon17] valiantly processed !

As clue count increases the proportion of puzzles with no path decreases, somehow a long time ago I stumbled upon a puzzle with 21 clues which had no exchangeable clues ...
its difficult to imagine this puzzle not having near relatives even within the puzzle solution ...

Code: Select all: +---+---+---+ |...|...|...| |12.|3..|8..| |34.|12.|...| +---+---+---+ |...|.6.|..7| |..9|..7|..8| |6..|...|5..| +---+---+---+ |..5|..3|..4| |.1.|...|...| |...|8..|.9.| +---+---+---+ 21C - no puzzle within {-1+1}

Of course there exists puzzles where all clues are exchangeable and or mutable - see first referenced post

After a determined search Mauricio found several puzzles where 1 locus had 9 options to provide a valid puzzle ...

by **P.O.** » Fri May 26, 2023 6:06 pm

thank you Coloin for your information, it seems really interesting, i will read them carefully.
on the other hand i do not understand very well the relationship between hamiltonian path and the notion of ’untouchable’
i understand it as follows: for some puzzles with a hamiltonian path you can't get a valid puzzle with the mechanism (-1+1)

by **coloin** » Sun May 28, 2023 12:09 pm

Indeed ... I was trying to understand what a hamilton path is in terms of valid puzzles.
Re-reading your initial post it seems you are making a "path" out of possible patterns [ not puzzles]
But then i dont understand how there is a "no path".

The hamilton path when applied to the prime numbers seemed pretty clear however ... and a well achieved objective too.

A repeat analysis of the 17C connectivity is now possible as we now know this group is complete.

by **P.O.** » Sun May 28, 2023 1:52 pm

you are right with this representation the 'puzzle' disappears, because the color information is not coded only the pattern remains
so some patterns do not have a hamiltonian path, a path that visits every vertex in the graph only once
i don't know if it really has much use but find it interesting to write the algorithms

Hidden Text: Show

Code: Select all: . . . 6 . 1 3 . . 8 5 . . . . . . . . . . . . . 4 . . . . . . 7 . . 5 2 3 . 1 . . . . . . . . . . . . . 8 . . 2 . . . . . 7 . . . . 1 . . 6 . . . . . 9 . . . . . ...6.13..85.............4......7..523.1.............8..2.....7....1..6.....9..... adjacency list: ((4) (6 7 67 76)) ((6) (4 7)) ((7) (4 6 25 70)) ((10) (11 37)) ((11) (10 56)) ((25) (7 70)) ((32) (35 36)) ((35) (32 36 53 62)) ((36) (32 35 53)) ((37) (10 39)) ((39) (37)) ((53) (35 36 62)) ((56) (11 62)) ((62) (35 53 56 70)) ((67) (4 70 76)) ((70) (7 25 62 67)) ((76) (4 67)) order: 17 size: 23 degre: (1 2 2 2 2 2 2 2 2 3 3 3 4 4 4 4 4) cell: 4 size: 0: NIL maxpath: size: 1 length: 11: (4 6 7 25 70 62 56 11 10 37 39) cell: 6 size: 0: NIL maxpath: size: 1 length: 12: (6 7 4 76 67 70 62 56 11 10 37 39) cell: 7 size: 0: NIL maxpath: size: 1 length: 12: (7 6 4 76 67 70 62 56 11 10 37 39) cell: 10 size: 0: NIL maxpath: size: 3 length: 11: (10 11 56 62 70 25 7 6 4 67 76) cell: 11 size: 0: NIL maxpath: size: 3 length: 10: (11 56 62 70 25 7 6 4 67 76) cell: 25 size: 0: NIL maxpath: size: 1 length: 13: (25 7 6 4 76 67 70 62 56 11 10 37 39) cell: 32 size: 0: NIL maxpath: size: 9 length: 12: (32 36 35 53 62 70 25 7 6 4 67 76) cell: 35 size: 0: NIL maxpath: size: 3 length: 12: (35 32 36 53 62 70 25 7 6 4 67 76) cell: 36 size: 0: NIL maxpath: size: 3 length: 12: (36 32 35 53 62 70 25 7 6 4 67 76) cell: 37 size: 0: NIL maxpath: size: 3 length: 12: (37 10 11 56 62 70 25 7 6 4 67 76) cell: 39 size: 0: NIL maxpath: size: 3 length: 13: (39 37 10 11 56 62 70 25 7 6 4 67 76) cell: 53 size: 0: NIL maxpath: size: 3 length: 12: (53 36 32 35 62 70 25 7 6 4 67 76) cell: 56 size: 0: NIL maxpath: size: 3 length: 9: (56 62 70 25 7 6 4 67 76) cell: 62 size: 0: NIL maxpath: size: 3 length: 8: (62 70 25 7 6 4 67 76) cell: 67 size: 0: NIL maxpath: size: 1 length: 13: (67 76 4 6 7 25 70 62 56 11 10 37 39) cell: 70 size: 0: NIL maxpath: size: 4 length: 7: (70 25 7 6 4 67 76) cell: 76 size: 0: NIL maxpath: size: 1 length: 13: (76 67 4 6 7 25 70 62 56 11 10 37 39)

by **Pat** » Sun May 28, 2023 4:02 pm

the puzzles with 18 givens
were explored by ano1

(who vanished after the great crash)

by **Pat** » Sun May 28, 2023 4:23 pm

coloin wrote:
Mauricio found several puzzles where 1 locus had 9 options to provide a valid puzzle

i remember that!

but lost my bookmarks

- could you please point to the discussion?

by **coloin** » Sun May 28, 2023 8:58 pm

Pat wrote:- could you please point to the discussion?

9-mutable

and here is {-1+1} valid puzzle path on a random 21C which changes 19 of the clues..

Code: Select all: 1..542..6.56....7...4......5...9......3.....4....78......3.........6..9...71....2 #01 1...42..6.56....7...4...5..5...9......3.....4....78......3.........6..9...71....2 1....2..6.56...47...4...5..5...9......3.....4....78......3.........6..9...71....2 1....2....56...47...4...5..5...9......3.....4....78......3...8.....6..9...71....2 1....2....56...47.......5..5...9......3.....4....78......3...8...4.6..9...71....2 1....2....56...47.......5..5...9......3.....4....78......3...8...4.6..9.9.7.....2 1....2....56...47.......5..5...96...........4....78......3...8...4.6..9.9.7.....2 1....2....56...4........5.75...96......5....4.....8......3...8...4.6..9.9.7.....2 1....2....5....4........5.75...96......5....4.....8..6...3...8...4.6..9.9.7.....2 1.8..2.........4........5.75...96......5....4.....8..6...3...8...4.6..9.9.7.....2 1.8..2.........4........5.75...96......5...74.....8..6...3...8...4.6....9.7.....2 1.8..2.........4........5.75..9.6......5...74.....8..6...3...8...4.6....9.7.....2 1.8..2.........4......9.5.75..9.6......5...74........6...3...8...4.6....9.7.....2 1.8..2.........4......9.5.75..9.6....3.5...74........6...3...8...4.6....9.7...... 1.8..2.........4......9.5.754.9.6....3.5...7.........6...3...8...4.6....9.7...... ..8..2.........4......9.5.754.9.6....3.5...7.........6...3...8...4.61...9.7...... .18..2.........4......9.5.754.9.6....3.5...7.........6.......8...4.61...9.7...... .18..2.........4......9.5.754.9.6....3.5...7.........6.......8...42.1...9.7...... .18..2.........4......9.5.754.9.6....3.5...7.........6..7....8...42.1...9........ #19 X X

although incidentally #01 and a morphed #19 share 10 clues which is normal

Code: Select all: 1..542..6.56....7...4......5...9......3.....4....78......3.........6..9...71....2 #01 6..54..........3...8....72.5.6.3........2...4.....8...9.83.7..........9...71....2 #19 morphed

by **coloin** » Tue May 30, 2023 2:02 pm

Code: Select all: +---+---+---+ |...|6.1|3..| |85.|...|...| |...|...|4..| +---+---+---+ |...|.7.|.52| |3.1|...|...| |...|...|.8.| +---+---+---+ |.2.|...|.7.| |...|1..|6..| |...|9..|...| +---+---+---+

as I undestand ityou are saying that the pattern of the 17C above is not connected to 17 other patterns by {-1+1) ... and no doubt therefore you must have all the patterns of the 17C puzzles..

looking at many of the 17C patterns ... they all look very similar and as under half of the puzzles are {-1+1 connected ... the patterns must be connected evn more

that pattern which doesnt have relatives must be quite rare

by **P.O.** » Tue May 30, 2023 2:43 pm

here what i think you are doing
you consider the whole collection of some N-clues puzzles as a graph whose vertices are the puzzles
you define a distance with the mechanism (-n+n)
with that for each vertex you can get what in graph theory is called a neighborhood of vertex v at distance d
https://mathworld.wolfram.com/GraphNeighborhood.html
i dont speak of that at all!
i simply take one puzzle, convert its pattern into a graph and compute its hamiltonian paths

by **coloin** » Wed May 31, 2023 5:58 pm

Point taken ..... and thinking about it some more...
Despite the fact that there are many more non-valid patterns than valid patterns at the 17C and 18C level ....

It seems I have interpreted gsf's table wrong

Code: Select all: === 41324 {1}* === 18950 1 4306 2 1095 3 645 4 259 5 170 6 97 7 59 8 47 9 30 10 26 11 18 12 16 13 8 14 3 15 9 16 5 17 5 18 4 19 4 20 1 21 3 22 1 23 5 24 5 25 2 27 1 28 1 30 2 31 2 32 1 33 1 37 1 38 2 41 1 42 1 50 1 55 1 87 1 173 1 560 1 616

I was able to identify the closed {-1+1} groups with 616 and 560 puzzles .... which means that 18950 of the 41324 17C are untouchable.

will look into the patterns of these two largest groups... and the 16C patterns...
each 17C will have ~ 17 16C patterns and 17 x [ 81-17 ] = 1088 17C patterns within {-1+1}

by **P.O.** » Thu Jun 01, 2023 9:50 am

as far as i understand what gsf did the (-n+n) operator acts as an accumulator, starting from one puzzle it collects all puzzles that are at (-n+n) of at least one puzzle already in the class
so i interpret gsf'table like you: at the time in the collection of 17-clues puzzles
with (-1+1) there were 18950 classes with only 1 member, the untouchables, 4306 classes with 2 members etc.
with (-2+2) there were 7734 classes with only 1 member, 2214 classes with 2 members etc.

i find this kind of investigation very interesting but besides the fact that it is sometimes difficult to fully understand what it is about it is necessary to have relatively powerful computer resources

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hamiltonian path

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