[Fwd: Metapuzzle involving NUMBER PLACE Puzzles]
Nick Baxter
each starting with 20 spaces filled. Both
were found in Puzzler magazine from Japan.
They are both good challenges, but not
horribly difficult - a must for any Number
Place fan!
+-----+-----+-----+ +-----+-----+-----+
|7 - -|- - -|- - 3| |7 4 -|- - -|- 2 -|
|- 2 -|8 - -|- - -| |- - -|- 9 -|- - 1|
|- - 9|- 6 -|1 - -| |- - 8|- - 3|- - -|
+-----+-----+-----+ +-----+-----+-----+
|- - -|- - 3|- 9 -| |- - -|7 - -|8 - -|
|4 - -|- - -|- - 6| |- 1 -|- - -|- 4 -|
|- 1 -|7 - -|- - -| |- - 3|- - 2|- - -|
+-----+-----+-----+ +-----+-----+-----+
|- - 3|- 5 -|8 - -| |- - -|8 - -|3 - -|
|- - -|- - 2|- 1 -| |1 - -|- 5 -|- - -|
|8 - -|- - -|- - 7| |- 9 -|- - -|- 7 2|
+-----+-----+-----+ +-----+-----+-----+
Nick
puz...@jte.com (Edward Jackman) wrote:
> qs...@aol.com (QSCGZ) wrote:
>
> >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?
>
Nick Baxter
record for fewest starting numbers is 18. (The best for
a symmetrical arrangement is 20.)
Here is an example of 18:
+-----+-----+-----+
|- - 5|- - -|- 4 -|
|- - -|8 - -|- - 6|
|3 - 2|- - 1|- - -|
+-----+-----+-----+
|- - -|- - 4|- 2 -|
|- - 9|- - -|5 - -|
|- 6 -|3 - -|- - -|
+-----+-----+-----+
|- - -|- - -|- - 3|
|- - -|- - 5|- - -|
|- 1 -|- - -|6 8 -|
+-----+-----+-----+
QSCGZ
Nick Baxter <nba...@inprise.com> wrote:
{ -What is the fewest number clues that must
be given in order to have a unique solution? }
>Here are two sparse Number Place puzzles,
>each starting with 20 spaces filled. Both
>were found in Puzzler magazine from Japan.
>
>They are both good challenges, but not
>horribly difficult - a must for any Number
>Place fan!
>
> +-----+-----+-----+ +-----+-----+-----+
> |7 - -|- - -|- - 3| |7 4 -|- - -|- 2 -|
> |- 2 -|8 - -|- - -| |- - -|- 9 -|- - 1|
> |- - 9|- 6 -|1 - -| |- - 8|- - 3|- - -|
> +-----+-----+-----+ +-----+-----+-----+
> |- - -|- - 3|- 9 -| |- - -|7 - -|8 - -|
> |4 - -|- - -|- - 6| |- 1 -|- - -|- 4 -|
> |- 1 -|7 - -|- - -| |- - 3|- - 2|- - -|
> +-----+-----+-----+ +-----+-----+-----+
> |- - 3|- 5 -|8 - -| |- - -|8 - -|3 - -|
> |- - -|- - 2|- 1 -| |1 - -|- 5 -|- - -|
> |8 - -|- - -|- - 7| |- 9 -|- - -|- 7 2|
> +-----+-----+-----+ +-----+-----+-----+
I can slightly improve your 2nd puzzle , which makes the
minimum <= 19 : ( that was before I read your post with the 18-example :-o )
+-----+-----+-----+
|7 - -|- - -|- 2 -|
|- - -|- 9 -|- - 1|
|- - 8|- - 3|- - -|
+-----+-----+-----+
|- - -|7 - -|8 - -|
|- 1 -|- - 2|- 4 -|
|- - 3|- - -|- - -|
+-----+-----+-----+
|- - -|8 - -|3 - -|
|1 - -|- 5 -|- - -|
|- 9 -|- - -|- 7 2|
+-----+-----+-----+
My estimate for the number of different solved number-place-puzzles
is ~ 10^16 (found numerically,Monte Carlo).
So, if every clue reduces the number of solutions by a factor of 9 ,
then at least 17 clues would be necessary . ( 9^17 ~ 10^16)
Another thing:
All these puzzles are rather easy to solve, as there is almost
always a number , which fits only into one place in a row/column/cell.
Only the 21-puzzle, that I posted had not had this property.
I'm surprised, that this is the case such often .
Has this behaviour been examined in Japan / puzzler magazine ?
qscgz
QSCGZ
Nick Baxter wrote:
>According to editors at Puzzler magazine, the Number Place
>record for fewest starting numbers is 18. (The best for
>a symmetrical arrangement is 20.)
are you allowed to put a clue in the mid-square here and use 180~ symmetry ?
>Here is an example of 18:
>
> +-----+-----+-----+
> |- - 5|- - -|- 4 -|
> |- - -|8 - -|- - 6|
> |3 - 2|- - 1|- - -|
> +-----+-----+-----+
> |- - -|- - 4|- 2 -|
> |- - 9|- - -|5 - -|
> |- 6 -|3 - -|- - -|
> +-----+-----+-----+
> |- - -|- - -|- - 3|
> |- - -|- - 5|- - -|
> |- 1 -|- - -|6 8 -|
> +-----+-----+-----+
I could change this a little , to get an example of 17 with 2 solutions.
You can fill in the first 68 entries , and then you can choose ... :
+-----+-----+-----+
|- - 5|- - -|- - -|
|- - -|8 - -|- 1 6|
|3 - 2|- - 9|- - -|
+-----+-----+-----+
|- - -|- - 4|- 2 -|
|- - 9|- - -|5 - -|
|- 6 -|3 - -|- - -|
+-----+-----+-----+
|- - -|- - -|- - 3|
|- - -|- - 5|- - -|
|- - -|- - -|6 8 -|
+-----+-----+-----+
qscgz
Nick Baxter
in Japan. These are a bit harder than those previously posted. If you
want more, I recommend the puzzles on Hirofumi Fujiwara's page:
http://kjmcci.kct.ne.jp/~fuji/java/puzzle/numplace/book1/index-eng.html
Enjoy.
+-----+-----+-----+
|- - -|7 - -|4 - -|
|- 3 -|- 9 -|- 2 -|
|4 - -|- - 5|- - -|
+-----+-----+-----+
|- - 8|- - -|- - 5|
|- 9 -|- 3 -|- 7 -|
|6 - -|- - -|3 - -|
+-----+-----+-----+
|- - -|4 - -|- - 6|
|- 7 -|- 2 -|- 9 -|
|- - 5|- - 8|- - -|
+-----+-----+-----+
+-----+-----+-----+
|6 - -|- - -|- - 3|
|- 4 -|- 5 -|- 1 -|
|- - -|7 - 2|- - -|
+-----+-----+-----+
|- - 7|- 3 -|4 - -|
|- 3 -|8 - 4|- 2 -|
|- - 9|- 1 -|8 - -|
+-----+-----+-----+
|- - -|9 - 7|- - -|
|- 5 -|- 6 -|- 7 -|
|3 - -|- - -|- - 6|
+-----+-----+-----+
Nick
Edward Jackman
>My estimate for the number of different solved number-place-puzzles
> is ~ 10^16 (found numerically,Monte Carlo).
>So, if every clue reduces the number of solutions by a factor of 9 ,
>then at least 17 clues would be necessary . ( 9^17 ~ 10^16)
Sounds convincing to me ... if I new what
(found numerically,Monte Carlo) meant.
>Another thing:
>All these puzzles are rather easy to solve, as there is almost
>always a number , which fits only into one place in a row/column/cell.
I've always found this to be the case with the DELL puzzles, but
I've found several online, both on Hirofumi Fujiwara's page and
other places for which this did not seem to be the case. Also,
I've found, at least at my skill level, the difficulty of the
puzzle is only partially controlled by the number of starting
clues. I've breezed threw several 21 and 22's while being
nearly stumped by a 27.
Edward
Edward Jackman
>According to editors at Puzzler magazine, the Number Place
>record for fewest starting numbers is 18. (The best for
>a symmetrical arrangement is 20.)
>Here is an example of 18:
>
> +-----+-----+-----+
> |- - 5|- - -|- 4 -|
> |- - -|8 - -|- - 6|
> |3 - 2|- - 1|- - -|
> +-----+-----+-----+
> |- - -|- - 4|- 2 -|
> |- - 9|- - -|5 - -|
> |- 6 -|3 - -|- - -|
> +-----+-----+-----+
> |- - -|- - -|- - 3|
> |- - -|- - 5|- - -|
> |- 1 -|- - -|6 8 -|
> +-----+-----+-----+
I've come across a few symmetricals with 20 starting numbers.
Here's a variant:
2 2 6
+-----+-----+-----+
|1 - -|- - -|- - 3|
8 |- - -|- 8 -|- - -|
|- - 4|- - -|9 - -|
+-----+-----+-----+
2 |- - -|1 - 9|- - -|
|- 6 -|- - -|- 7 -|
4 |- - -|7 - 5|- - -|
+-----+-----+-----+
|- - 1|- - -|2 - -|
7 |- - -|- 3 -|- - -|
|4 - -|- - -|- - 7|
+-----+-----+-----+
The external numbers represent the difference between
the numbers on the ends of their row or column. That is:
2 2 6
+-----+-----+-----+
|1 w -|- - -|- - 3|
8 |x - -|- 8 -|- - y|
|- - 4|- - -|9 - -|
+-----+-----+-----+
2 |- - -|1 - 9|- - -|
|- 6 -|- - -|- 7 -|
4 |- - -|7 - 5|- - -|
+-----+-----+-----+
|- - 1|- - -|2 - -|
7 |- - -|- 3 -|- - -|
|4 z -|- - -|- - 7|
+-----+-----+-----+
|x-y| = 8, so x = 9 or 1 and y = 1 or 9.
|w-z| = 2, etc
What sort of puzzle selection does Puzzler magazine have?
Are most or all solvable without knowing Japanese? Are
subscriptions available in the US?
I just picked up a couple of other Japanese puzzle magazines. Both
have several Number Place puzzles including a few 16x16. Even better,
both have some overlapping puzzles like this from the
September issue of PUZZLE PARADISE:
+-----+-----+-----+ +-----+-----+-----+
|- - -|- - -|- - -| |- - -|- - -|- - -|
|- - -|1 8 2|- - -| |- - -|7 5 3|- - -|
|- - 5|6 - 4|8 - -| |- - 8|9 - 6|7 - -|
+-----+-----+-----+ +-----+-----+-----+
|- 2 9|- - -|5 8 -| |- 6 1|- - -|4 2 -|
|- 3 -|- 9 -|- 1 -| |- 7 -|- 1 -|- 9 -|
|- 8 6|- - -|2 7 -| |- 9 5|- - -|6 7 -|
+-----+-----+-----+-----+-----+-----+-----+
|- - 4|3 - 6|- - -|- - -|- - -|4 - 1|8 - -|
|- - -|8 4 9|- - -|3 9 2|- - -|3 6 7|- - -|
|- - -|- - -|- - -|8 - 5|- - -|- - -|- - -|
+-----+-----+-----+-----+-----+-----+-----+
|- 8 1|- - -|6 7 -|
|- 7 -|- 5 -|- 4 -|
|- 4 3|- - -|9 1 -|
+-----+-----+-----+-----+-----+-----+-----+
|- - -|- - -|- - -|9 - 8|- - -|- - -|- - -|
|- - -|7 1 4|- - -|7 3 1|- - -|1 3 8|- - -|
|- - 1|8 - 6|- - -|- - -|- - -|7 - 2|3 - -|
+-----+-----+-----+-----+-----+-----+-----+
|- 9 4|- - -|7 6 -| |- 2 9|- - -|7 8 -|
|- 5 -|- 7 -|- 4 -| |- 4 -|- 6 -|- 1 -|
|- 7 6|- - -|2 8 -| |- 1 8|- - -|4 3 -|
+-----+-----+-----+ +-----+-----+-----+
|- - 7|5 - 2|6 - -| |- - 4|5 - 9|8 - -|
|- - -|4 3 1|- - -| |- - -|4 7 3|- - -|
|- - -|- - -|- - -| |- - -|- - -|- - -|
+-----+-----+-----+ +-----+-----+-----+
Each 9x9 will have a normal solution.
Edward
QSCGZ
edw...@jte.com (Edward Jackman) wrote:
>qs...@aol.com (QSCGZ) wrote:
>>My estimate for the number of different solved number-place-puzzles
>> is ~ 10^16 (found numerically,Monte Carlo).
>>So, if every clue reduces the number of solutions by a factor of 9 ,
I'm no longer glad with this. It might be true for the first clues
that you put into the square , as long as no two clues are in the
same row/column/cell.
But the last clues usually give rise to much bigger factors
So the mid clues presumably give rise to smaller factors.
>>then at least 17 clues would be necessary . ( 9^17 ~ 10^16)
>
>Sounds convincing to me ... if I new what
>(found numerically,Monte Carlo) meant.
I found 46656 solutions ("rcc9s") , to place 9 queens
(or whatever) into a 9x9 square , such that no two queens are in
the same row,column or cell.
Two such rcc9s are called disjoint , if all their 18 queens are
on different squares.
A solved NPP (number place puzzle) is nothing but a set of 9 mutually
disjoint rcc9s.
Now I wrote a program that tried to generate solved NPPs in 9 steps:
On each step s :
count the number n(s) of rcc9s that are disjoint to all the other
previously selected rcc9s ,
and select one of those disjoint rcc9s at random.
some typical outcomes for the n(s) were:
46656,17972,6121,1848,443,96,24,2,1
46656,17972,6200,1716,470,81,10,0,-
46656,17972,6121,1848,443,96,24,2,1
46656,17972,6190,1879,426,96,18,4,1
46656,17972,6021,1784,359,63,4,0,-
46656,17972,6022,1688,383,82,10,0,-
46656,17972,6046,1748,420,82,11,0,-
46656,17972,6096,1680,392,88,14,2,1
46656,17972,6021,1712,306,72,14,0,-
46656,17972,6254,1942,528,122,11,0,-
Now I multiplied the n(s) , divided by 9! since the order doesn't
matter and got 1.5*10^16 in average.
qscgz