The New Sudoku Players' Forum

[Red Ed]

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.

[Ocean]

..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).

[Red Ed]

(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.

[udosuk]

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..

[claudiarabia]

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


Thank you I always looked for such sweet 17s

Claudia

[ronk]

... 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]

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