Almost two years ago, I posted an article to this blog about KenKen and CalcuDoku. In that article, I compared the 9x9 puzzles made available in the web sites kenken.com (® Nextoy LLC), calcudoku.org, and zambon.com.au (my own). Like last time, I will use K to identify the puzzles provided by
kenken.com, C to identify those provided by calcudocu.org, and Z to
identify mine.

Of the three web sites, only the K site provides 9x9 puzzles of
different levels of difficulty, but I have only ever solved the "tough"
ones.

After solving more than 500 9x9 puzzles, I have a much better feel about
the differences between the puzzles generated by the three sites.

K and Z are comparable in terms of difficulty and general "feel". The C puzzles are different.

First of all, the C puzzles include many more single-cell cages.
Patrick (the developer of the C site) told me that he uses singles to
remove ambiguities. I understand why the singles are there, but I have
always found it annoying that their average number exceeds 9. K puzzles
include very few singles (0 to 3), and I can configure my generator to
limit the number of singles to whatever I like. I normally set the
maximum number of singles to 3, but I could also set it to 0 (I have
tried it out) and systematically generate puzzles without singles at
all, although it requires on average a longer time to generate each
puzzle.

I don't think that Patrick's decision to "single out" the ambiguities
was a good one. I commented on the C web site that I didn't like to go
through the chore of filling up so many singles when starting with a new
puzzle, but other users replied that it didn't bother them. Fair
enough.

Another problem I have with C puzzles is that the difficulty of solving
them is not uniform as you progress: invariably, it is easy to fill in a
third of the cages or so without much need for reflection. Then, you
hit a wall and the cages suddenly become very difficult. It might be
connected to the fact that there are many single cages used for
disambiguation, but I am not sure.

I have lived for almost two years with the two problems I have just
mentioned, because 9x9 C puzzles also have an interesting feature: they
can have divisions and subtractions in cages with more
than two cells. For example, an angled three-cell cage with "2:" as
target admits 841, 822, 631, 421, and 211
as possible solutions.

But there is another problem that has finally convinced me to stop
solving C puzzles for good: some of them are not solvable analytically,
and I refuse to solve a puzzle by trial and error. I find it very
frustrating when I encounter a puzzle that cannot be solved without
guessing.

I know what you are thinking: perhaps I'm not good enough. It's possible, but, perhaps not surprisingly, I don't think so.

I believe that the problem is due to the fact that C puzzles sometimes
have too many large[r] cages with sums and subtractions. This can
result in untameable combinatorial explosions. Those angle cages with
targets ranging between 13+ and 20+ and between -0 and -4 sometimes
combines in ways that leave too many alternatives open.

Patrick calculates for each puzzle a difficulty level, which for 9x9
puzzles is in the range 100±30 (i.e., I haven't seen anything outside
that interval). I asked him how he arrives to those figures, but he
refused to divulge his algorithm. I understand that when he started his
web site he went though several tests, but his calculations are not
right. Last June, I solved a puzzle with an alleged difficulty of
129.3, but in September I didn't succeed in solving a puzzle with a
difficulty of 70.2.

Here are the statistics for the June puzzle:

#1-cell: 8

#2-cell: 16 (3+, 9-, 2x, 2:)

#4-cell: 4 (1+, 2x, 1:) all T-shaped

#5-cell: 5 (2+, 3x) 1 cross-shaped, 4 angled

And here are those for the September puzzle:

#1-cell: 9

#2-cell: 13 (2+, 3-, 8x)

#3-cell: 6 (1+, 4-, 1x) all angled

#4-cell: 4 (3+, 1-) 1 T-shaped, 2 squares, 1 lightning-shaped

#5-cell angled: 2 (1+, 1x) 1 cross-shaped, 1 tap/fawcett-shaped

Do you see what I mean? The June puzzle, which was supposed to be among
the most difficult ones, had no 3-cell cages, and of the 9 cages with 4
and 5 cells, only 3 had sums. The September puzzle, which was supposed
to be among the easiest 9x9s, had only two multiplications among the 12
cages with at least 3 cells. In two adjacent columns, there were three
3-cell cages with 4-, 1-, and 0-, and arranged in such a way that they
took up six cells of one column.

4- can be 941, 932, 831, 822, 721, 611; 1- can be 971, 962, 953, 944,
861, 852, 843, 751, 742, 733, 641, 632, 531, 522, 421, 311; and 0- can
be 981, 972, 963, 954, 871, 862, 853, 844, 761, 752, 743, 651, 642, 633,
541, 532, 431, 422, 321, 211. Yes, there were some crossings that
reduced the possibilities, but the same puzzle also included two 4-cell
15+, a 4-cell 16+, a 4-cell 0-, and a 5-cell 32+. No way that it could
have been solved analytically!

All in all, I finally got fed up with the C puzzles. That's why a couple of days ago I gave up on them.

The C web site has the best application to solve the puzzles online, but
for the 9x9 puzzles (and I am not interested in the smaller ones), the
best way is to print them out and use pencil and eraser. Therefore, no
loss there...

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