Metapuzzle involving NUMBER PLACE Puzzles
puz...@jte.com
divided into 9 3x3 areas with some of the
cells filled with 'clues', that is, a digit from
1 to 9. Typically there are between 27 and
36 clues given. Here is an example from DELL'S
BEST OF MATH PUZZLES AND LOGIC PROBLEMS, February
1998:
+-----+-----+-----+
|- 5 -|2 1 -|- 7 -|
|3 - -|- - 6|9 - 4|
|2 - 9|- 3 -|- - 5|
+-----+-----+-----+
|- 3 2|- - 1|- 6 -|
|- 9 -|4 - -|8 3 -|
|7 - -|6 2 -|- - 1|
+-----+-----+-----+
|6 2 -|- - 5|- 1 -|
|- - 1|3 - -|2 - 6|
|- - 8|- 6 2|3 - -|
+-----+-----+-----+
The goal is to fill in all the remaining cells so that each row,
column and 3x3 area has one of each digit. A puzzle should
have only one possible solution. Solution to the above example:
+-----+-----+-----+
|8 5 4|2 1 9|6 7 3|
|3 1 7|5 8 6|9 2 4|
|2 6 9|7 3 4|1 8 5|
+-----+-----+-----+
|4 3 2|8 9 1|5 6 7|
|1 9 6|4 5 7|8 3 2|
|7 8 5|6 2 3|4 9 1|
+-----+-----+-----+
|6 2 3|9 4 5|7 1 8|
|9 4 1|3 7 8|2 5 6|
|5 7 8|1 6 2|3 4 9|
+-----+-----+-----+
Although the puzzles typically use numbers, any 9 different letters
or other symbols may be used since no math is involved.
METAPUZZLE
================================================================
+
+ -What is the fewest number clues that must
+ be given in order to have a unique solution?
+
+ Give a proof and an minimum clue example.
+
================================================================
Also,
-When creating one of these puzzles, how does one
go about ensuring that only the minimum information
needed to solve the puzzle is given?
ADDITIONAL INFORMATION ABOUT N U M B E R P L A C E PUZZLES:
DELL publishes some of theses puzzles in their Math and Logic Puzzles
magazines. All of their puzzles are always the same -- 4 clues
in each of the 9 areas are given for a total of 36 clue numbers,
giving more information then is required to reach a unique
solution. All their number place puzzles have the same difficulty
level -- easy. Once you've solved a few they get repetitive.
Some Japanese magazines publish these puzzles and there is
also a website --
http://www.pro.or.jp/~fuji/java/index-eng.html
-- with 110 puzzles by a Japanese composer, complete with a
nice Java implementation. This puzzles are difficulty rated
from 1 to 10 stars. (The hard ones are really far more challenging
than the easy ones.)
The Japanese version is identical except that they have more variety
in the number and arrangement of the starting clues. Typically the
numbers are arranged in a pleasing way forming a picture or
identifiable pattern. This serves several purposes. First, it's just
nicer to work on puzzles that look different from each other. Second,
it's easier to see which one's you have worked on as they can be
recognized at a glance. Third, different arrangements lead
to different problems within the puzzle and varying difficulty.
The Japanese versions start with different numbers of clues.
I've seen some with as many as 36 and as few as 22. Obviously, in
general, the fewer clues given, the harder the puzzle. However, the
puzzle must contain enough clues so that only one solution is
possible.
Below is an example of a Japanese Number Place puzzle from a
recent Japanese puzzle magazine. (I can't tell you the name
of the magazine -- the title is in Japanese and I can't read
it.) Notice that the clues are placed symetrically about the
vertical axis and roughly form a face. The solution is at the
bottom of this post.
+-----+-----+-----+
|- 3 -|8 6 4|- 1 -|
|- 4 6|5 - 1|3 2 -|
|- 5 -|- - -|- 9 -|
+-----+-----+-----+
|5 - -|- - -|- - 1|
|3 6 -|- 7 -|- 8 9|
|- 8 -|- - -|- 7 -|
+-----+-----+-----+
|- - 5|- - -|9 - -|
|- - 4|6 - 7|8 - -|
|8 - -|9 4 5|- - 7|
+-----+-----+-----+
(this example is from a Japanese "Picture Logic Mate" aka
Paint By Numbers puzzle magazine)
Web sites with Number Place puzzles include:
-Java and Puzzle World by Hirofumi Fujiwara
http://www.pro.or.jp/~fuji/java/index-eng.html
Hirofumi has 110 Number Place puzzles here
which he rates in difficulty from 1 to 10.
The site includes Java implementations of
Cross Sums, Paint By Numbers and Sliding
Block puzzles. A very good site with great
puzzles.
-http://www.pluto.dti.ne.jp/~rcn/Java/Puzzle/np.html
This site has a Number Place SOLVER that works
very well. The site is mostly in Japanese but it's
simple enough to figure out how it works. The page
includes 11 problems. You can input your own and
use it to manually solve a problem or let it solve
it partially or completely in an instant.
The site also has covers Cross Sums in a similar
fashion.
-http://www3.osk.3web.ne.jp/~bs1484/np_p1.html
-http://www3.osk.3web.ne.jp/~bs1484/np_p2.html
Two Number Place puzzles with just 22 clues.
-http://altair.prec.kyoto-u.ac.jp/~chihara/java/NumberPlace.html
-http://altair.prec.kyoto-u.ac.jp/~chihara/java/NumberPlace1.html
Two puzzles, one with 27 and another with 28 clues.
Please post information on where to find published Number Place
puzzles, both on and off the web.
+-----+-----+-----+
|2 3 9|8 6 4|7 1 5|
|7 4 6|5 9 1|3 2 8|
|1 5 8|7 2 3|4 9 6|
+-----+-----+-----+
|5 9 7|3 8 6|2 4 1|
|3 6 1|4 7 2|5 8 9|
|4 8 2|1 5 9|6 7 3|
+-----+-----+-----+
|6 7 5|2 1 8|9 3 4|
|9 1 4|6 3 7|8 5 2|
|8 2 3|9 4 5|1 6 7|
+-----+-----+-----+
(Solution of puzzle from a Japanese "picture logic mate" magazine)
-------------------------------
Edward
Courtenay Footman
>ADDITIONAL INFORMATION ABOUT N U M B E R P L A C E PUZZLES:
>
>DELL publishes some of theses puzzles in their Math and Logic Puzzles
>magazines. All of their puzzles are always the same -- 4 clues
>in each of the 9 areas are given for a total of 36 clue numbers,
>giving more information then is required to reach a unique
>solution. All their number place puzzles have the same difficulty
>level -- easy. Once you've solved a few they get repetitive.
>
sixteen by sixteen puzzle (with the additional constraint that there be
no duplications on the main diagonal) that are wonderful (that is, hard).
These magazines will also have upwards of twenty cross sums, so are well
worth keeping an eye open for.
--
Courtenay Footman I have again gotten back on the net, and
c...@lightlink.com again I will never get anything done.
(All mail from non-valid addresses is automatically deleted by my system.)
QSCGZ
>A NUMBER PLACE puzzle starts with a 9x9 grid
>divided into 9 3x3 areas with some of the
>cells filled with 'clues', that is, a digit from
>1 to 9. Typically there are between 27 and
>36 clues given.
>
>+ -What is the fewest number clues that must
>+ be given in order to have a unique solution?
The least I found was 21 (found by deleting entries from your 2nd solution).
If I randomly delete 54 entries (27 clues remaining ) from your second
solution, then I get a unique-solution-puzzle with probability 1:1000 .
> -When creating one of these puzzles, how does one
> go about ensuring that only the minimum information
> needed to solve the puzzle is given?
for each clue , let the computer count the number of solutions with
that clue removed.
If you get 1 all the time , then the puzzle is minimal.
---------------------
how many different (solved) NUMBER PLACE puzzles are there ?
I know, that when disregarding the 9 3x3 cells , it is
~ 10^28 .
Disregarding the 9 columns too , it is ~10^50 .
Disregarding the 9 rows too , it is ~10^77 .
what , if we take (\,/)hyperdiagonals instead of 3x3 cells ?
or hyperdiagonals _and_ 3x3 cells ?
I suppose , this has already been investigated somewhere.
qscgz
Edward Jackman
>puz...@jte.com wrote:
>
> >A NUMBER PLACE puzzle starts with a 9x9 grid
> >divided into 9 3x3 areas with some of the
> >cells filled with 'clues', that is, a digit from
> >1 to 9. Typically there are between 27 and
> >36 clues given.
> >
> >+ -What is the fewest number clues that must
> >+ be given in order to have a unique solution?
>
>The least I found was 21 (found by deleting entries from your 2nd solution).
Could you post this?
>If I randomly delete 54 entries (27 clues remaining ) from your second
>solution, then I get a unique-solution-puzzle with probability 1:1000 .
How did you arrive at this figure?
> > -When creating one of these puzzles, how does one
> > go about ensuring that only the minimum information
> > needed to solve the puzzle is given?
>
>for each clue , let the computer count the number of solutions with
>that clue removed.
>If you get 1 all the time , then the puzzle is minimal.
What software are you using to do this?
>what , if we take (\,/)hyperdiagonals instead of 3x3 cells ?
>or hyperdiagonals _and_ 3x3 cells ?
Which do you mean by hyperdiagonals, I, II or III?
I II III
+-----+-----+-----+ +-----+-----+-----+ +-----+-----+-----+
|x - -|- - -|- - -| |- - -|- - -|- x -| |- - -|- x -|- - -|
|- x -|- - -|- - -| |- - -|- - -|- - x| |x - -|- - -|- - -|
|- - x|- - -|- - -| |x - -|- - -|- - -| |- - -|- - x|- - -|
+-----+-----+-----+ +-----+-----+-----+ +-----+-----+-----+
|- - -|x - -|- - -| |- x -|- - -|- - -| |- x -|- - -|- - -|
|- - -|- x -|- - -| |- - x|- - -|- - -| |- - -|- - -|x - -|
|- - -|- - x|- - -| |- - -|x - -|- - -| |- - x|- - -|- - -|
+-----+-----+-----+ +-----+-----+-----+ +-----+-----+-----+
|- - -|- - -|x - -| |- - -|- x -|- - -| |- - -|- - -|- x -|
|- - -|- - -|- x -| |- - -|- - x|- - -| |- - -|x - -|- - -|
|- - -|- - -|- - x| |- - -|- - -|x - -| |- - -|- - -|- - x|
+-----+-----+-----+ +-----+-----+-----+ +-----+-----+-----+
Puzzles