superior plus puzzles

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

Postby RW » Mon Jul 31, 2006 9:17 pm

This one has two swordfishes and one x-wing, but only requires one swordfish:

Code: Select all
 *-----------*
 |7..|.4.|..6|
 |.6.|...|.8.|
 |..4|3.2|9..|
 |---+---+---|
 |..2|9.4|1..|
 |4..|.3.|..9|
 |..7|6.8|3..|
 |---+---+---|
 |..5|1.3|6..|
 |.7.|...|.1.|
 |9..|.8.|..4|
 *-----------*


And this one has both a swordfish and a naked/hidden quad, again only the swordfish is required:

Code: Select all
 *-----------*
 |...|...|...|
 |9..|3.2|..4|
 |64.|1.8|.79|
 |---+---+---|
 |71.|...|.28|
 |..5|...|7..|
 |42.|...|.95|
 |---+---+---|
 |15.|9.4|.62|
 |2..|5.6|..1|
 |...|...|...|
 *-----------*


And three more swordfish puzzles:

Code: Select all
 *-----------*
 |.12|8..|.3.|
 |7..|3..|..1|
 |...|.61|..7|
 |---+---+---|
 |..5|7.6|.13|
 |..9|...|7..|
 |67.|1.5|9..|
 |---+---+---|
 |5..|41.|...|
 |3..|..9|..2|
 |.4.|..3|15.|
 *-----------*
 
 *-----------*
 |4..|.2.|..8|
 |.3.|...|.5.|
 |8.9|.5.|1.2|
 |---+---+---|
 |...|297|...|
 |..4|...|5..|
 |...|514|...|
 |---+---+---|
 |5.8|.4.|6.1|
 |.1.|...|.3.|
 |2..|.6.|..5|
 *-----------*

 *-----------*
 |6..|.8.|..9|
 |.15|...|28.|
 |.3.|...|.4.|
 |---+---+---|
 |...|321|...|
 |2..|4.6|..7|
 |...|798|...|
 |---+---+---|
 |.4.|...|.1.|
 |.62|...|43.|
 |3..|.4.|..5|
 *-----------*


RW
RW
2010 Supporter
 
Posts: 1000
Joined: 16 March 2006

Postby ab » Mon Jul 31, 2006 9:41 pm

nice puzzles RW, did you make them by hand??

no 1 can have clues removed by diagonal symmetry:
Code: Select all
 7 . . | . 4 . | . . .
 . 6 . | . . . | . 8 .
 . . 4 | 3 . 2 | 9 . .
 ------+-------+------
 . . 2 | . . 4 | 1 . .
 4 . . | . 3 . | . . 9
 . . 7 | 6 . . | 3 . .
 ------+-------+------
 . . . | 1 . 3 | 6 . .
 . 7 . | . . . | . 1 .
 9 . . | . 8 . | . . 4

no 2 can have clues removed by half turn symmetry:
Code: Select all
 . . . | . . . | . . .
 9 . . | 3 . 2 | . . .
 6 4 . | 1 . 8 | . 7 9
 ------+-------+------
 7 1 . | . . . | . 2 8
 . . 5 | . . . | 7 . .
 4 2 . | . . . | . 9 5
 ------+-------+------
 1 5 . | 9 . 4 | . 6 2
 . . . | 5 . 6 | . . 1
 . . . | . . . | . . .

or

 . . . | . . . | . . .
 9 . . | 3 . 2 | . . .
 6 4 . | 1 . 8 | . 7 9
 ------+-------+------
 7 1 . | . . . | . 2 8
 . . 5 | . . . | 7 . .
 4 2 . | . . . | . 9 5
 ------+-------+------
 1 5 . | 9 . 4 | . 6 2
 . . . | 5 . 6 | . . 1
 . . . | . . . | . . .


no 5 can have clues removed by antidiagonal symmetry:
Code: Select all
 3 . . | . 2 . | . . 6
 . 6 4 | . . . | 3 1 .
 . 2 . | . . . | . 5 .
 ------+-------+------
 . . . | 7 . 3 | . . .
 4 . . | 9 . . | . . 8
 . . . | . 6 1 | . . .
 ------+-------+------
 . 4 . | . . . | . 2 .
 . 3 1 | . . . | 4 8 .
 5 . . | . 7 . | . . 9


ab
 
Posts: 451
Joined: 06 September 2005

Postby RW » Tue Aug 01, 2006 8:43 am

ab wrote:nice puzzles RW, did you make them by hand??


Thank you ab. I did not make them by hand all the way. I had a generator make puzzles until I found some with swordfishes, then I did some manual tweaking of individual numbers to get rid of unwanted required techniques. None of the puzzles are directly out of the generator.

ab wrote:no 1 can have clues removed by diagonal symmetry:


I also noticed that, but that gets rid of one possible swordfish and the x-wing... and of course the full symmetry. Is it required that puzzles are stripped down to the absolute symmetrical minimum, according to any symmetry? I usually like puzzles with at least 4-way symmetry the most.

RW
RW
2010 Supporter
 
Posts: 1000
Joined: 16 March 2006

Postby ab » Tue Aug 01, 2006 12:17 pm

no 1 doesn't need an x-wing to solve it, just swordfish (having written that i put it back into my solver and saw that there's an swordfish that does the same as the x-wing you mention, so maybe the x-wing is required). also you still need the swordfish with my modification of the puzzle! In answer to your other question puzzles don't have to be symmetrically minimal, it's up to the composer. Sometimes , making them symmetrically minimal adds further techniques that need to be used to solve them, other times it makes no difference.
ab
 
Posts: 451
Joined: 06 September 2005

Postby ab » Tue Aug 08, 2006 12:38 am

Superior plus puzzles are quite elusive. My generator has found about a dozen more that solve with swordfish, and it seems to have a prediliction for pairing them off with pairs. Here's the pick of the bunch:
Code: Select all

 . . 7 | 2 . 8 | 9 . .
 . . . | 1 . 3 | . . .
 . . 2 | . . . | 8 . .
 ------+-------+------
 2 . . | . . . | . . 4
 9 5 . | 7 . 2 | . 1 6
 . 1 4 | . . . | 2 8 .
 ------+-------+------
 . 8 . | . . . | . 5 .
 4 . . | . . . | . . 2
 . . 1 | 9 . 5 | 6 . .

 . 7 6 | 3 . 1 | . 4 .
 9 . . | 8 . 7 | . . 5
 . . . | . . . | . . 6
 ------+-------+------
 . 1 . | 6 . 8 | . 3 .
 . . . | . . . | . . .
 . 3 . | 9 . 4 | . 7 .
 ------+-------+------
 2 . . | . . . | . . .
 1 . . | 7 . 6 | . . 3
 . 8 . | 4 . 2 | 9 6 .

 . . 9 | 5 . . | . . .
 7 . . | 9 . . | . 4 .
 . . 8 | 7 . 3 | 1 . .
 ------+-------+------
 4 2 5 | . . . | . . .
 . . . | . . . | . . .
 . . . | . . . | 7 5 6
 ------+-------+------
 . . 6 | 8 . 1 | 3 . .
 . 9 . | . . 6 | . . 1
 . . . | . . 4 | 9 . .


quads and in particular jellyfish are far more elusive. After tweaking the program a bit it found this:
Code: Select all
 . 3 5 | . . . | 4 8 .
 7 . . | 3 . 2 | . . .
 1 . . | . . . | . . 9
 ------+-------+------
 . 4 . | 9 . 1 | . 7 .
 . . . | . . . | . . .
 . 1 . | 6 . 3 | . 5 .
 ------+-------+------
 9 . . | . . . | . . 5
 . . . | 8 . 7 | . . 3
 . 8 6 | . . . | 7 2 .


Interestingly none of the puzzles submitted requires more than one superior plus technique to solve it.
ab
 
Posts: 451
Joined: 06 September 2005

Postby gsf » Tue Aug 08, 2006 1:09 am

ab wrote:Interestingly none of the puzzles submitted requires more than one superior plus technique to solve it.

I specifically limited my search to singles+1-superior-plus
gsf
2014 Supporter
 
Posts: 7306
Joined: 21 September 2005
Location: NJ USA

Postby tarek » Tue Aug 08, 2006 12:14 pm

Look what happens when I'm away:) .

I was away working on some stuff. Good thing that you started this ab, I was hoping that it would start at some stage......

I will be back to analyse & post some more in about a week or 2. Things in my opinion that should be taken into account when creating these puzzles are shortcuts:

If a swordfish can be bypassed using a type 1 UR or (in my opinion) a finned x-wing then it probably should score less.

Also a non- 2*2*2 configuration should score higher ...

in the mean time I'm not sure how this would score:
Code: Select all
 9 3 . | . 2 1 | . 4 7 
 . . . | . . . | 8 . . 
 6 . . | . 8 . | . 3 . 
-------+-------+------
 . . . | 2 . . | . 6 . 
 1 . . | . . . | . . . 
 . 9 . | . . 4 | . . 2 
-------+-------+------
 . 1 . | . 5 . | . . 3 
 . . 3 | . . . | . . . 
 7 4 . | 8 6 . | . 5 1 


Tarek
User avatar
tarek
 
Posts: 2622
Joined: 05 January 2006

Postby ab » Tue Aug 08, 2006 9:22 pm

Tarek wrote:in the mean time I'm not sure how this would score:

That's probably the best puzzle by far Tarek. Quads and a swordfish and hidden pair and locked candidates to boot.

gsf wrote:I specifically limited my search to singles+1-superior-plus

I'm not sure why you chose to look for puzzles with only one superior plus technique and singles gsf, my first two puzzles require pairs as well as swordfish. Anyway maybe you'd like to ammend your search?
ab
 
Posts: 451
Joined: 06 September 2005

Postby gsf » Tue Aug 08, 2006 9:45 pm

ab wrote:I'm not sure why you chose to look for puzzles with only one superior plus technique and singles gsf, my first two puzzles require pairs as well as swordfish. Anyway maybe you'd like to ammend your search?

the restricted search criterion was way to contribute to this thread and also build up a taxonomy
of puzzles that illustrate a technique exactly when singles fail to progress the solution

I also did it this way to refute any claims of alternate (lesser) techniques solving the puzzle
i.e., for a puzzle with 1 jellyfish singles lead up to the jellyfish, and at that point no other
superior plus technique but jellyfish advances the solution

if you want to see the other end of the spectrum (how many superior plus techniques can
be crammed into one puzzle) I'd be glad to oblige, but that would require a technique ordering
for comparison with other puzzles, e.g., something like
Code: Select all
singles pairs box-line x-wing triples swordfish quads jellyfish

and possibly a step/batch constraint like for the inferior/ulterior threads
gsf
2014 Supporter
 
Posts: 7306
Joined: 21 September 2005
Location: NJ USA

Postby udosuk » Wed Aug 09, 2006 7:54 am

gsf wrote:something like
Code: Select all
singles pairs box-line x-wing triples swordfish quads jellyfish

To me, hidden pairs are much harder than x-wing or naked triples, and hidden triples are much harder than swordfish and naked quads...

BTW fabulous puzzle from tarek... Even with a program it is a great challenge...:)
udosuk
 
Posts: 2698
Joined: 17 July 2005

Postby gsf » Thu Aug 10, 2006 6:26 pm

a search for quads { naked hidden jellyfish } on gordons 36K 17's found 1
and on gordons 22M 18's found 501; only one puzzle with 2 quads was found

the search was done in step mode -- once a technique is identified at a position
it is applied, and the technique scan restarts at the top of the order

I also have a background generator filtering quads
the search is biased against order N x-wing because order N naked and hidden
tuples are applied first and in many cases are in a race with order N x-wing

here are the ones with quads and one or more plus techniques
the 17's and 18's are gordon royle's
the .'d triple in the comment is #quads.#triples.#pairs
m:minimal M:symmetric-minimal
Code: Select all
# 2.0.1 C18.m/F12.40/N6.23/B2.4.2.2/T3.9.1.0.2

. 7 . | . 4 . | . . .
. . . | . 3 . | . . 5
. . . | 8 . . | 1 . .
------+-------+------
4 . 3 | . . . | . 2 .
. . . | 5 . 1 | 9 . .
6 . . | . . . | . . .
------+-------+------
. 5 . | 9 . . | 7 . .
2 . . | . . . | . 4 .
. . . | . . . | . . .

# 1.3.0 C18.m/F12.52/N3.11/B3.5.3.2/T3.9.0.2.1/H1.3.0.1

4 . . | . 9 . | 1 . .
. . . | . . . | . 5 .
. . . | . . . | . . .
------+-------+------
1 . 9 | 3 . . | 6 . .
. . . | 5 . 7 | . 3 .
. . . | 2 . . | . . .
------+-------+------
6 . . | . . . | 8 . 9
. 5 . | . 4 . | . . .
. 7 . | . . . | . . .

# 1.3.0 C18.m/F13.38/N5.25/B3.5.3.2/T3.9.0.2.1/H1.3.0.1

6 . . | 5 . . | 3 . .
. . . | . 1 4 | . . .
2 . . | . . . | . . .
------+-------+------
. 1 7 | . 9 . | . . .
5 4 . | . . . | . . .
. . . | . . . | 2 . .
------+-------+------
3 . . | 9 . . | 6 . .
. 8 . | . . . | . 1 .
. . . | 2 . . | . . .

# 1.0.1 C17.m/F8.20/N9.44/B9.9.9/T2.5.1.0.1

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

# 0.4.2 C29.M/S2.d/F7.49/N1.3/B4.6.2.4/T3.7.1.2/H1.2.1/W2.7.0.2

. 7 . | . 8 . | 4 . .
. 4 . | 7 . 2 | . . .
2 . 5 | . 3 . | . . 7
------+-------+------
. . 8 | . . . | . 1 .
. . 6 | . 5 . | 7 . 8
7 9 . | . . . | . 3 .
------+-------+------
9 . . | . 6 5 | 8 . .
. . . | 8 . . | . 7 9
. . 1 | 2 . . | 3 . .

# 1.1.4 C25.m/S2.d/F7.44/N5.12/B8.13.7.6/T2.6.2/H2.3.2/W2.8.0.1.1

. . . | . 4 1 | . . .
. 8 . | . 9 . | . . .
2 . 4 | 5 . 3 | . . .
------+-------+------
. . . | . 1 . | 6 . 4
. . . | 9 . 2 | . 5 7
. . . | . 7 . | 1 . .
------+-------+------
9 6 1 | . . . | 4 . .
. . 7 | . . . | . 9 .
. . 5 | . . . | 2 . .

# 1.1.0 C29.M/S8.f/F8.35/N4.17/B1.2.1.1/W2.8.0.1.1

2 . . | . 6 . | . . 9
. 1 . | . . . | . 2 .
. . 9 | 1 . 8 | 4 . .
------+-------+------
. . 4 | 2 . 7 | 5 . .
5 . . | . 9 . | . . 4
. . 6 | 5 . 1 | 9 . .
------+-------+------
. . 7 | 4 . 2 | 6 . .
. 5 . | . . . | . 4 .
9 . . | . 7 . | . . 3

# 1.1.1 C30.M/S2.h/F7.45/N3.6/B5.7.3.4/H1.2.1/W2.13.0.1.1

. 4 7 | 5 . 6 | . 9 .
6 . . | . . . | . . 4
1 . 3 | . . . | 5 . 2
------+-------+------
. . 8 | 6 . 7 | . 3 .
. . . | . . . | . . .
. . 4 | 9 . 2 | . 5 .
------+-------+------
4 . 2 | . . . | 8 . 5
5 . . | . . . | . . 3
. 3 6 | 4 . 5 | . 2 .
gsf
2014 Supporter
 
Posts: 7306
Joined: 21 September 2005
Location: NJ USA

Postby ab » Fri Aug 11, 2006 2:40 pm

another puzzle from me. still 'only' a swordfish puzzle, but it also requires locked candidates, a naked pair and an x-wing:
Code: Select all
 7 . . | . 4 . | . 5 9
 . . 3 | 9 . . | 2 . 8
 . 2 . | . . . | . 3 .
 ------+-------+------
 . . . | . 8 1 | . . .
 3 . . | 6 . 5 | . . 4
 . . . | 4 7 . | . . .
 ------+-------+------
 . 4 . | . . . | . 8 .
 8 . 1 | . . 9 | 4 . .
 9 6 . | . 3 . | . . 7

you can remove the 8s in r2c9 and r8c1 to get a puzzle that also requires an xyz-wing
ab
 
Posts: 451
Joined: 06 September 2005

Postby ab » Sun Aug 20, 2006 9:34 am

My generator finally coughed up a jellyfish puzzle and it's fully symmeric too!
Code: Select all
 . . . | . 6 . | . . .
 . . 6 | 7 . 2 | 5 . .
 . . 9 | 8 . 5 | 1 . .
 ------+-------+------
 . 6 3 | . . . | 8 4 .
 8 . . | . . . | . . 3
 . 5 7 | . . . | 9 2 .
 ------+-------+------
 . . 8 | 5 . 7 | 3 . .
 . . 5 | 6 . 9 | 2 . .
 . . . | . 2 . | . . .

here's another puzzle. only requires one swordfish but I like the start!
Code: Select all
 . . . | . . . | . . .
 . 3 8 | 9 . 2 | 6 5 .
 . 5 . | 6 . 4 | . 2 .
 ------+-------+------
 . 9 1 | . . . | 8 7 .
 . . . | . . . | . . .
 . 4 5 | . . . | 9 3 .
 ------+-------+------
 . 6 . | 7 . 3 | . 4 .
 . 7 2 | 5 . 8 | 1 9 .
 . . . | . . . | . . .
ab
 
Posts: 451
Joined: 06 September 2005

Postby gsf » Tue Aug 22, 2006 4:01 pm

here are the results of a background search for puzzles that require one or more
{ pair (naked hidden xwing) triple (naked hidden swordfish) quad (naked hidden jellyfish) } techniques (in that order)
techniques counted stepwise as in the inferior and superior threads
the "q.t.p" numbers are number of quads, triples, pairs (order 4 3 2) steps required
Code: Select all
"0.0.5" ....69..2..5..3....6.2..7.3..1.3.8..9..6....187.........25...98......1..6.8.4.2..
"0.0.6" 1...4.2.6.94...........9..59...7.5...6.2.4..7....3.....81....5..32.9..8......8..4
"0.0.7" .2..9..8.6...38..7.....7..6.....19..26..5.....394........1..3.87.........53...4.2
"0.0.8" ...7.1..2..9....4..7..2..5.9..3..287..8......4.....39....2.74...435.6...6..9.....
"0.0.9" .4..3.98...69....4...4.5..6.54..92..2.76....3....5.41..1..63.7...3..2..1....9....
"0.0.10" ..7...85......81.61..5...7...32.6..85..7.9..22..4.39...6...1..43.48......12...3..
"0.0.11" ...2....6.2.14.57...7..5.12.53....9.6.........49....8...4..1.63.1.45.72....8....5
"0.0.12" ..52..4.6.2.14..7...7..5..2.53....9.6.........49....8...4..1..3.1.45..2...28..1.5
"0.1.5" 7.......2..38.15...9..7..6.6..2.8..9..........2.7.5.1..........4..1.9..33.7...9.6
"0.1.6" .3...1.68.1.7..52.6...5......5......78.....54......1......6...9.29..5.7.87.1...3.
"0.1.7" ....9....39.2....576...81...4..7.23.8.......7.27.4..5...67...241....9.86....3....
"0.1.8" 1...4...6.54...32..6..2..1..8.5.1.4...58.97...........32.....59...7.3.....6...1..
"0.1.9" .7.4.....61..95.4...9.6...7.37..8..5.........4..1..38.8...7.1...5.38..72.....6.5.
"0.2.5" 6..5.7..8.95.8.....7....2..9...6.3...2.7.49..7...1..2...135..4......861.3........
"0.2.6" ....95.4..5..823.1.......6..8.....369..5...171...4....67........34..1.9...135....
"0.2.7" .....6.1...413...6.31.4.28..9........87....5.3.....6....8..5.3.2.3.8.7...7......2
"0.2.8" ...58.3.49....7.....5...1.9.7..9...34....1....6..3...5..6...5.82....3......86.4.1
"0.2.9" .9.3.....1.2.46..7.6..79...9..1......13.....6.75..89.......7.42......1.5.5..3.76.
"0.3.0" ..3.2..6..26..7..9.7.9......47....1..1......88.....9.......6.41.3..7965.7..1.....
"0.3.1" ....16...26.9.8.....97.....6......4239......1..4...57.4..8..7......4..5...3.75.8.
"0.3.2" .2.4...8.714.8....3...7.........75.9....1.34......91.84...9....195.3.....7.6...9.
"0.3.3" ..8...1...4....52.3.5.8........21..4..48...6....5..3..67...3.9..9..6.2.....1.....
"0.3.4" .6.7...343...2.5.......8.2.6..9..1..1......587..4..3.......9.7.9...7.2...5.2...19
"0.3.5" ..9.4...5.6..3..7.7....9..3...9....624....5....5....17....2..8..5...73..3.14.6...
"0.3.6" ....2...895.6..2...3...8.1.......692..1.4..........841.4...2.8.19.5..3......7...6
"0.3.7" 67..428...3.......5..3....12....46.5..4....9.7....14.24..9....8.1.......86..172..
"0.3.8" 9....26.3..861...7.6.9.5....27....41.5.......1.4...2..3....9......4...8.41.5....6
"0.4.0" .9..5....4.5.23.7..8..4...3....1....2369......7...5.6........37.4...82.5..7...18.
"0.4.1" .6.417.5.............6.5...8.......4.7.8.9.6..........4..2.3..1.93...72..2.7.6.9.
"0.4.2" .7..8.4...4.7.2...2.5.3...7..8....1...6.5.7.879.....3.9...658.....8...79..12..3..
"0.4.3" 3..........81....5.65.2.4...5...6.8...1.4...7...3.5..1..6....7....9..1.3.9..32.5.
"0.4.5" ..3.7.9....15.96..5..6.3..7..........18...25.4..1.5..91..9.6..4..........5.7.1.8.
"0.4.7" .4.5.83..5..79...42....4.8.1.4....6..2.....1..3....4.2.5.1....88...56..7..78.2.5.
"0.5.1" ...6..5..8.2...96.4..79..83..5......24....7....9...3.4..89.7.5..2..4........2.83.
"0.6.0" ..........3.....8.7..3.2..65..649..29..8.3..5..85.71..2.......8..9...4....74.19..
"1.0.5" 9..5.372..4.1.9.3............7..5..63..9.......2..8..9..........5.8.4.6.4..3.128.
"1.0.6" ...5....7..6..195..9....1.62.....6.......2.81.7..4..2..472......2..13.6.6.9.8....
"1.0.7" .....26..2......94.7.95.3.2.1768..............5872.....8.17.5.37......26.....37..
"1.0.8" ..9.....5...5.76...6.3...743.1.8...9.........6.8.1...3.7.2...46...4.68....6.....7
"1.1.5" 89..4..73.........54.2.7.8115.9.3.6468..2..35.........41.8.9.52...1.2...9.......8
"1.1.6" ..26.3..8..9..12......7..3.6....89...5.....7...37....4.3..5......51..8..2..9.61..
"1.1.8" 29.8..13.7....398.........5.1.7.9.....96......6.3.5...........43....679.97.1..26.
"1.2.5" .......7...43....538..69....2...41..5.....3..7.18...2...94...1......18.....25.7..
"1.3.0" .79.3.65...87.92....3...8...6..7..8..9.....2.3..1.2..9...8.1.......2....1..4.3..8
"1.3.1" ...2.3....6.....1..749.682..2.....7.9.......8.57...96.7..6.5..2.8..9..4...1.8.5..
"1.3.6" 2.....8....9.712...4..6..1.97..14.3..........81..36.9..3..5..4...8.937..6.....3..
"2.0.0" ..8.....7.1....5..9......34.....2.......384.....94.825.6..25.1...2..36.93.5..6.4.
"2.0.1" .128....79.....4..6.83..1..2.4617......9........2..7...26..9.18......2.57.....69.
"2.0.2" .7.6.....2..........6.98.524...6..9...913.8....2....46....7..8...52.196...1..9...
"2.0.3" .....35...1..5.76....2....9.48.7.....9....3.8.71.9.......5....7.2..4.81......64..
"2.1.0" 4.8.15.....36.....5..9.3...1.....3.8.8......7..7...26.9..2...35....7......6..91.4
gsf
2014 Supporter
 
Posts: 7306
Joined: 21 September 2005
Location: NJ USA

Postby ab » Tue Aug 22, 2006 5:40 pm

your list shows the techniques that are in the puzzles but they're not all necessarily required. For instance the last puzzle solves with one quad only and the third last puzzle only requires the pairs and one of the swordfish.
ab
 
Posts: 451
Joined: 06 September 2005

PreviousNext

Return to General