Symmetrically-clued 17s

Everything about Sudoku that doesn't fit in one of the other sections

Symmetrically-clued 17s

Postby Red Ed » Sat Mar 18, 2006 3:03 pm

We all know that no 17s have been found whose clue positions are 180-degree rotationally symmetric. Well, it turns out that you can devise 17s whose clue positions have reflection symmetry across a diagonal. Here are ten non-isomorphic examples:
Code: Select all
+---+---+---+    +---+---+---+    +---+---+---+    +---+---+---+
|...|...|...|    |...|...|...|    |...|...|..1|    |...|...|..1|
|...|...|.12|    |...|...|.12|    |...|...|.23|    |...|...|.23|
|..3|.45|...|    |..3|.45|...|    |...|.45|...|    |..4|..5|...|
+---+---+---+    +---+---+---+    +---+---+---+    +---+---+---+
|...|6.1|.7.|    |...|6.1|.7.|    |...|2.1|...|    |...|1..|...|
|..4|...|6..|    |..4|...|6..|    |..4|...|...|    |...|.3.|6..|
|..5|8..|...|    |..5|8..|...|    |..6|7..|5..|    |..7|...|58.|
+---+---+---+    +---+---+---+    +---+---+---+    +---+---+---+
|...|.3.|4..|    |...|.3.|5..|    |...|..6|8..|    |...|.67|...|
|.1.|2..|...|    |.1.|2..|...|    |.1.|...|...|    |.1.|..4|...|
|.7.|...|...|    |.7.|...|...|    |32.|...|..9|    |52.|...|...|
+---+---+---+    +---+---+---+    +---+---+---+    +---+---+---+


+---+---+---+    +---+---+---+    +---+---+---+    +---+---+---+
|...|...|..1|    |...|...|..1|    |...|...|..1|    |...|...|..1|
|...|...|.23|    |...|...|.23|    |...|...|.23|    |...|...|.23|
|..4|..5|...|    |..4|..5|...|    |..4|..5|...|    |..4|..5|...|
+---+---+---+    +---+---+---+    +---+---+---+    +---+---+---+
|...|1..|...|    |...|1..|...|    |...|1..|...|    |...|1..|...|
|...|.3.|6..|    |...|.4.|6..|    |...|.4.|6..|    |...|.4.|6..|
|..7|...|58.|    |..6|...|4.5|    |..6|...|4.5|    |..6|...|4.5|
+---+---+---+    +---+---+---+    +---+---+---+    +---+---+---+
|...|.97|...|    |...|.76|...|    |...|.76|...|    |...|.76|...|
|.1.|..4|...|    |.1.|...|...|    |.1.|...|...|    |.1.|...|...|
|52.|...|...|    |32.|..8|...|    |38.|..9|...|    |82.|..9|...|
+---+---+---+    +---+---+---+    +---+---+---+    +---+---+---+


+---+---+---+    +---+---+---+
|...|...|..1|    |...|...|..1|
|...|..2|...|    |...|..2|.3.|
|..1|...|.34|    |..4|..5|...|
+---+---+---+    +---+---+---+
|...|.1.|...|    |...|...|.5.|
|...|35.|...|    |...|..4|6..|
|.6.|...|7..|    |.17|.8.|...|
+---+---+---+    +---+---+---+
|...|..7|26.|    |...|.1.|..7|
|..5|...|8..|    |.2.|9..|...|
|4.3|...|...|    |5..|...|4..|
+---+---+---+    +---+---+---+
These are all isomorphic to grids in Gordon's big list of 17s. In fact, that's how I found them: I searched for any of Gordon's grids whose clue positions had any symmetry. Turns out the only ones to do so were equivalent to those listed above.

None of these is automorphic in the sense of there also being a consistent relabelling of the digits but, hey, you can't have everything.
Last edited by Red Ed on Sat Mar 18, 2006 2:50 pm, edited 1 time in total.
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Re: Symmetrically-clued 17s

Postby Ocean » Sat Mar 18, 2006 5:41 pm

Red Ed wrote:..17s whose clue positions have reflection symmetry across a diagonal.

That's a beautiful symmetry - I actually prefer reflection symmetry before rotational (and it's about just as hard to achive/find).

(I think #9 and #10 are identical to #5 and #6 - maybe you picked from the wrong place).
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Re: Symmetrically-clued 17s

Postby Red Ed » Sat Mar 18, 2006 6:51 pm

Ocean wrote:(I think #9 and #10 are identical to #5 and #6 - maybe you picked from the wrong place).
Oops, well spotted. Bad copy, as you guessed ... corrected now.
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Postby udosuk » Sat Mar 18, 2006 10:02 pm

Great work! Presentation not too nice though... here is the "easy to copy & paste each grid" version...

Code: Select all
................12..3.45......6.1.7...4...6....58.........3.4...1.2......7.......
................12..3.45......6.1.7...4...6....58.........3.5...1.2......7.......
........1.......23....45......2.1.....4........67..5.......68...1.......32......9
........1.......23..4..5......1.........3.6....7...58.....67....1...4...52.......
........1.......23..4..5......1.........3.6....7...58.....97....1...4...52.......
........1.......23..4..5......1.........4.6....6...4.5....76....1.......32...8...
........1.......23..4..5......1.........4.6....6...4.5....76....1.......38...9...
........1.......23..4..5......1.........4.6....6...4.5....76....1.......82...9...
........1.....2.....1....34....1.......35.....6....7.......726...5...8..4.3......
........1.....2.3...4..5..........5......46...17.8........1...7.2.9.....5.....4..
udosuk
 
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Re: Symmetrically-clued 17s

Postby claudiarabia » Sat Jul 01, 2006 11:26 pm

[quote="Red Ed"] 17s whose clue positions have reflection symmetry across a diagonal. Here are ten non-isomorphic examples:


Thank you:D I always looked for such sweet 17s

Claudia
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Re: Symmetrically-clued 17s

Postby ronk » Tue Aug 17, 2010 4:47 pm

Red Ed wrote:... you can devise 17s whose clue positions have reflection symmetry across a diagonal. Here are ten non-isomorphic examples:
Code: Select all
+---+---+---+    +---+---+---+    +---+---+---+    +---+---+---+
|...|...|...|    |...|...|...|    |...|...|..1|    |...|...|..1|
|...|...|.12|    |...|...|.12|    |...|...|.23|    |...|...|.23|
|..3|.45|...|    |..3|.45|...|    |...|.45|...|    |..4|..5|...|
+---+---+---+    +---+---+---+    +---+---+---+    +---+---+---+
|...|6.1|.7.|    |...|6.1|.7.|    |...|2.1|...|    |...|1..|...|
|..4|...|6..|    |..4|...|6..|    |..4|...|...|    |...|.3.|6..|
|..5|8..|...|    |..5|8..|...|    |..6|7..|5..|    |..7|...|58.|
+---+---+---+    +---+---+---+    +---+---+---+    +---+---+---+
|...|.3.|4..|    |...|.3.|5..|    |...|..6|8..|    |...|.67|...|
|.1.|2..|...|    |.1.|2..|...|    |.1.|...|...|    |.1.|..4|...|
|.7.|...|...|    |.7.|...|...|    |32.|...|..9|    |52.|...|...|
+---+---+---+    +---+---+---+    +---+---+---+    +---+---+---+


+---+---+---+    +---+---+---+    +---+---+---+    +---+---+---+
|...|...|..1|    |...|...|..1|    |...|...|..1|    |...|...|..1|
|...|...|.23|    |...|...|.23|    |...|...|.23|    |...|...|.23|
|..4|..5|...|    |..4|..5|...|    |..4|..5|...|    |..4|..5|...|
+---+---+---+    +---+---+---+    +---+---+---+    +---+---+---+
|...|1..|...|    |...|1..|...|    |...|1..|...|    |...|1..|...|
|...|.3.|6..|    |...|.4.|6..|    |...|.4.|6..|    |...|.4.|6..|
|..7|...|58.|    |..6|...|4.5|    |..6|...|4.5|    |..6|...|4.5|
+---+---+---+    +---+---+---+    +---+---+---+    +---+---+---+
|...|.97|...|    |...|.76|...|    |...|.76|...|    |...|.76|...|
|.1.|..4|...|    |.1.|...|...|    |.1.|...|...|    |.1.|...|...|
|52.|...|...|    |32.|..8|...|    |38.|..9|...|    |82.|..9|...|
+---+---+---+    +---+---+---+    +---+---+---+    +---+---+---+


+---+---+---+    +---+---+---+
|...|...|..1|    |...|...|..1|
|...|..2|...|    |...|..2|.3.|
|..1|...|.34|    |..4|..5|...|
+---+---+---+    +---+---+---+
|...|.1.|...|    |...|...|.5.|
|...|35.|...|    |...|..4|6..|
|.6.|...|7..|    |.17|.8.|...|
+---+---+---+    +---+---+---+
|...|..7|26.|    |...|.1.|..7|
|..5|...|8..|    |.2.|9..|...|
|4.3|...|...|    |5..|...|4..|
+---+---+---+    +---+---+---+
These are all isomorphic to grids in Gordon's big list of 17s. In fact, that's how I found them: I searched for any of Gordon's grids whose clue positions had any symmetry. Turns out the only ones to do so were equivalent to those listed above.

None of these is automorphic in the sense of there also being a consistent relabelling of the digits but, hey, you can't have everything.

I re-examined the symmetry of 17s in these same five classes (of clues-per-box). Surprisingly, after four years with the count of 17s now at 49151, there are only two new symmetric puzzles, and still with diagonal symmetry only.

Code: Select all
 CPB-Class    Puzzles
 011123333    5
 011222333    1
 111113333    1
 112222223    4 (gained two since 2006)
 122222222    1

 *-----------*
 |...|..1|..2|
 |..3|...|...|
 |.4.|...|.1.|
 |---+---+---|
 |...|.5.|...|
 |...|64.|...|
 |2..|...|3.7|
 |---+---+---|
 |...|..8|.6.|
 |..5|...|4..|
 |7..|..3|...|
 *-----------*

 *-----------*
 |...|...|.12|
 |..3|...|...|
 |.45|..6|...|
 |---+---+---|
 |...|.7.|...|
 |...|8..|...|
 |..9|...|5.4|
 |---+---+---|
 |...|..9|4..|
 |8..|...|.7.|
 |2..|..5|...|
 *-----------*
ronk
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Re: Symmetrically-clued 17s

Postby ronk » Tue Aug 24, 2010 11:11 am

A search of the entire collection of 49,151 puzzles with 17 clues uncovered 2 more.

Code: Select all
 *-----------*
 |...|...|..1|
 |...|...|.23|
 |..4|..2|...|
 |---+---+---|
 |...|.5.|4..|
 |...|16.|...|
 |..7|...|8..|
 |---+---+---|
 |...|4.8|5..|
 |.3.|...|...|
 |61.|...|...|
 *-----------*

 *-----------*
 |...|...|...|
 |...|...|.12|
 |..3|..4|...|
 |---+---+---|
 |...|.2.|...|
 |...|15.|..3|
 |..6|...|7..|
 |---+---+---|
 |...|..6|.7.|
 |.2.|...|48.|
 |.5.|.3.|...|
 *-----------*

In summary, there are currently 14 known symmetric 17s, all with diagonal symmetry in 10 different patterns.

Code: Select all
 CPB-Class    Puzzles
 011123333    5
 011222333    1
 111113333    1
 111122333    1 (new) (since 2006)
 111222233    1 (new)
 112222223    4 (two new)
 122222222    1

................12..3..4.......2.......15...3..6...7.......6.7..2....48..5..3.... # d - new (since 2006)
................12..3.45......6.7..8..4...3....52.........6.5...8........2.1..... # d
................12..3.45......6.7..8..4...3....52.........6.4...8........2.1..... # d
........1.......23..4..5......1.........3.6....7...58.....67....1...4...52....... # d
........1.......23..4..5......1.........3.6....7...58.....97....1...4...52....... # d
........1.....2.3...4..5..........5......46...17.8........1...7.2.9.....5.....4.. # d
........1.......23..4..2.......5.4.....16......7...8.....4.85...3.......61....... # d - new
........1.......23..4..5......2.........4.6....7...5.8....54....3.......21...7... # d
........1.......23..4..5......2.........4.6....7...5.8....54....3.......29...7... # d
........1.......23..4..5......2.........4.6....7...5.8....54....9.......21...7... # d
........1.....2.....3....45....5.......34.....6....7.......768...5...2..3.1...... # d
........1.......23....45......2.1.....4........67..5.......68...1.......32......9 # d
.......12..3.......45..6.......1.......7.......8...6.5.....53..9......7.2....3... # d - new
.....1..2..3.......4.....5.....6.......53....7.....8.4.....4.3...7...6..1....2... # d - new
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