## Gurth's Puzzles

Everything about Sudoku that doesn't fit in one of the other sections
GC53 : 4th CHALLENGE

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`#GC53SE 7.2 (11 forcing chains)  9 . . 7 . . 1 . .  5 . . . . . . 8 .  . 6 8 . 9 . . . .  7 . . . . 1 . 9 3  . 2 . . . . . . .  . . . . . . . 4 6  2 . . 4 6 . 5 . .  . 5 . 9 3 . . 7 .  . . 9 . . . . . .`

Quick solution via ST wanted. There IS something new here.

udosuk, in case you are wondering where GC52 is, the answer is that not all my saved puzzles get posted. Not yet anyway.
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gurth

Posts: 358
Joined: 11 February 2006
Location: Cape Town, South Africa

gurth wrote:GC53 : 4th CHALLENGE
...
Quick solution via ST wanted. There IS something new here.

I have no idea what STs to apply in this one (yet), but using some educated guessing I found the solution to this puzzle in less than 90 seconds...

Here is the initial grid:
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`9..|7..|1..5..|...|.8..68|.9.|...---+---+---7..|..1|.93.2.|...|......|...|.46---+---+---2..|46.|5...5.|93.|.7...9|...|...`

Notice the similarity in b6 & b8... Then look at b1, we can duplicate that pattern by setting r1c2=3 and r3c1=4... Also, seeing the [468] in r3c123, we can duplicate them in b6 & b8 by setting r6c7=r7c6=8... Hence, the grid becomes:
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`93.|7..|1..5..|...|.8.468|.9.|...---+---+---7..|..1|.93.2.|...|......|...|846---+---+---2..|468|5...5.|93.|.7...9|...|...`

And bingo! Singles solved it all...

For once I'm happy about being a shameless cheater...

PS: Still no idea about GC51 (3rd challenge)...
udosuk

Posts: 2698
Joined: 17 July 2005

Seems that you got the trick.

Morphing the puzzle here
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` +-------+-------+-------+ | 9 . . | 7 . . | . . 1 | | . 6 8 | . 9 . | . . . | | 5 . . | . . . | 8 . . | +-------+-------+-------+ | 7 . . | . . 1 | 9 3 . | | . . . | . . . | 4 6 . | | . 2 . | . . . | . . . | +-------+-------+-------+ | . 5 . | 9 3 . | 7 . . | | 2 . . | 4 6 . | . . 5 | | . . 9 | . . . | . . . | +-------+-------+-------+`
the solution has 3 identical boxes each.
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` +-------+-------+-------+ | 9 3 2 | 7 8 4 | 6 5 1 | | 4 6 8 | 1 9 5 | 2 7 3 | | 5 1 7 | 3 2 6 | 8 4 9 | +-------+-------+-------+ | 7 8 4 | 6 5 1 | 9 3 2 | | 1 9 5 | 2 7 3 | 4 6 8 | | 3 2 6 | 8 4 9 | 5 1 7 | +-------+-------+-------+ | 6 5 1 | 9 3 2 | 7 8 4 | | 2 7 3 | 4 6 8 | 1 9 5 | | 8 4 9 | 5 1 7 | 3 2 6 | +-------+-------+-------+`
ravel

Posts: 998
Joined: 21 February 2006

ravel wrote:Seems that you got the trick.

Thanks... I suppose the property is more obvious once you get to the final solution and look at it... But it remains to be proved rigidly why the initial clues imply the existence of that property (3 sets of identical boxes) in the grid...

BTW, any idea on the 3rd challenge?
udosuk

Posts: 2698
Joined: 17 July 2005

udosuk wrote:BTW, any idea on the 3rd challenge?

This morphed grid seems to have some symmetry:
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` +-------+-------+-------+ | 9 7 2 | 5 4 8 | 1 3 6 | | 4 3 5 | 6 1 2 | 7 8 9 | | 8 1 6 | 3 9 7 | 2 5 4 | +-------+-------+-------+ | 5 4 8 | 2 6 3 | 9 1 7 | | 3 2 7 | 1 8 9 | 4 6 5 | | 6 9 1 | 7 5 4 | 8 2 3 | +-------+-------+-------+ | 7 6 3 | 9 2 1 | 5 4 8 | | 2 8 9 | 4 3 5 | 6 7 1 | | 1 5 4 | 8 7 6 | 3 9 2 | +-------+-------+-------+`
ravel

Posts: 998
Joined: 21 February 2006

udosuk wrote:But it remains to be proved rigidly why the initial clues imply the existence of that property (3 sets of identical boxes) in the grid...
E.g. take those transformations:
swap band and 2
swap stacks 1 and 3
swap stacks 1 and 2
swap bands 1 and 3

Then the boxes
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`123456789`
are mapped to
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`897231564`

But applying this (where number n is mapped to itself) to the morphed sudoku does NOT map it to itself.

And in the grid for the 3rd challenge i have an additional problem with the numbers 3 and 6. The other mappings would work:
1->7->2->1, 4->4, 5->5, 9->9
ravel

Posts: 998
Joined: 21 February 2006

ravel wrote:Morphing the puzzle here
(...)
the solution has 3 identical boxes each.
Code: Select all
` +-------+-------+-------+ | 9 3 2 | 7 8 4 | 6 5 1 | | 4 6 8 | 1 9 5 | 2 7 3 | | 5 1 7 | 3 2 6 | 8 4 9 | +-------+-------+-------+ | 7 8 4 | 6 5 1 | 9 3 2 | | 1 9 5 | 2 7 3 | 4 6 8 | | 3 2 6 | 8 4 9 | 5 1 7 | +-------+-------+-------+ | 6 5 1 | 9 3 2 | 7 8 4 | | 2 7 3 | 4 6 8 | 1 9 5 | | 8 4 9 | 5 1 7 | 3 2 6 | +-------+-------+-------+`

And consequently your isomorph is "symmetric" about both the diagonal and anti-diagonal.
ronk
2012 Supporter

Posts: 4764
Joined: 02 November 2005
Location: Southeastern USA

ronk wrote:And consequently your isomorph is "symmetric" about both the diagonal and anti-diagonal.
Yes, but you neither cant use these symmetries in gurths puzzle.

[Edit:] Grid deleted, what i had in mind, is not possible
Last edited by ravel on Tue Jan 16, 2007 3:46 pm, edited 1 time in total.
ravel

Posts: 998
Joined: 21 February 2006

ravel wrote:Here is a grid with a mapping to itself ...

What do you mean by "a grid with a mapping to itself?"
ronk
2012 Supporter

Posts: 4764
Joined: 02 November 2005
Location: Southeastern USA

ronk wrote:What do you mean by "a grid with a mapping to itself?"
An automorphism. But i have to apologize. Today i cannot reconstruct it myself, must have made a mistake last night
ravel

Posts: 998
Joined: 21 February 2006

GC53 (4th CHALLENGE)

udosuk, an excellent 90-second job! (I didn't ban guessing this time).

ravel, congratulations on discovering the 3 identical sets of boxes... (I didn't ban computers).

But major congratulations to ronk for discovering the DOUBLE DIAGONAL SYMMETRY, the main point of this puzzle. (All the clues conform to DDS, and every clue has a DDS partner.)(Clues on the diagonals partner themselves.)

Note: this Sudoku also has Rotational Symmetry.

Questions:
Is this the first published Sudoku to exhibit DDS?
Is this the first Sudoku to combine the box symmetry, the DDS and the Rotational Summetry?
Is GC53 then the most symmetrical of all Sudokus?
Mirror, mirror on the wall...
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GC56 : with UR type 4 and BUG type 1.

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`#GC56#SE 7.3  . 2 . . . . . 7 .  4 . . . . 8 3 . .  . . 9 . 3 . . . 5  . . 8 . . . . . 4  . 1 . . . 6 . . .  6 . . . 8 . . 3 .  . 6 . 9 . . . 2 .  8 . . . . 2 4 . .  . . 1 . 4 . . . 9`

Ends with a very pretty BUG of 25 cells, with cells in every box and column.( The BUG in GC30 had 33 cells, but not in every box.) But note that SS, which knows nothing of BUGs, solves the BUG situation quite easily using Simple Colouring.

There are not too many chains if you want to do it without computer.

Diagram of BUG available on request.
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gurth

Posts: 358
Joined: 11 February 2006
Location: Cape Town, South Africa

gurth wrote:ravel, congratulations on discovering the 3 identical sets of boxes... (I didn't ban computers).
I neither have nor know a program, that discovers symmetries (but i saw the box symmetry in Mauricio's samples).
Is this the first published Sudoku to exhibit DDS?
Is this the first Sudoku to combine the box symmetry, the DDS and the Rotational Summetry?
Is GC53 then the most symmetrical of all Sudokus?
AFAIK yes, yes and yes.
ravel

Posts: 998
Joined: 21 February 2006

gurth wrote:But major congratulations to ronk for discovering the DOUBLE DIAGONAL SYMMETRY, the main point of this puzzle.

Thanks, but by illustrating the isomorph of the original, ravel did all the heavy lifting.

gurth wrote:Is this the first Sudoku to combine the box symmetry, the DDS and the Rotational Summetry?

AFAIK yes. The Sudoku has double rotational symmetry (DRS), that is, both 90-degree and 180-degree. But it's not like the box, DDS and DRS symmetries are independent. If one exists, the other two must also.
ronk
2012 Supporter

Posts: 4764
Joined: 02 November 2005
Location: Southeastern USA

I cannot see a 90-degree symmetry in this grid (like in Mauricio's 3rd puzzle here).
90-degree symmetry trivially implies 180-degree symmetry.
DDS implies rotational symmetry. You get to the opposite cell by mirroring 2 times, e.g first /, then \. Both are symmetrical, so also the combination.
I dont know, if DDS implies box symmetry. Any proof or counter-example?
ravel

Posts: 998
Joined: 21 February 2006

I assume that the box symmetry you're referring to is the horizontal & vertical symmetries

before the paper was published & the symmetris unified I used the following
1. Diagonal (or triangle diagonal) or Anti diagonal (or triangle anti diagonal)
2. Horozontal (or Box horizontal) or Vertical (or Box vertical)
3. 180 degrees rotational
4. 90 degrees rotaional (implies 180 also)
the rest is combination of the above
6. double diagonal (1+3 ) implies 180 also
7. double box (2+3) implies 180 also
8. Full

box & triangle symmetris will only meet in the Full symmety
box & 90 symmetris as well as triangle & 90 symmetries will only meet in the Full symmetry

tarek

tarek

Posts: 2644
Joined: 05 January 2006

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