How many empty units?

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How many empty units?

Theoretically there could be 10 empty units in a valid puzzle (3 rows, 3 columns and 4 boxes), right? It seems to me a puzzle with 7 empty units is quite easy to make and I found one 8:

Code: Select all
` *-----------* |...|4.6|7.8| |...|7.9|1.3| |...|2.1|5.6| |---+---+---| |2.5|...|4.1| |6.1|...|3.9| |...|...|...| |---+---+---| |3.4|1.8|...| |...|...|...| |8.2|5.4|...| *-----------*`

Can anybody construct the 9 and 10?

RW
RW
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Posts: 1000
Joined: 16 March 2006

Re: How many empty units?

RW wrote:Theoretically there could be 10 empty units in a valid puzzle (3 rows, 3 columns and 4 boxes), right? It seems to me a puzzle with 7 empty units is quite easy to make and I found one 8:

Can anybody construct the 9 and 10?
RW

Yes, tso made one 9 :
(3 boxes & 6 rows/cols)
Code: Select all
` . . . | . 2 5 | . 3 9 . . . | . 3 7 | . 4 1 . . . | . . . | . . .-------+-------+------- . . . | . . . | . . . 4 7 . | . . . | . 6 8 1 9 . | . . . | . 5 2-------+-------+------- . . . | . . . | . . . 6 4 . | . 5 3 | . . . 7 8 . | . 1 2 | . . .`
See here

I made a 8 (4 boxes & 4 rows/cols) here :

Code: Select all
` . . . | . . . | 9 . 6 . . . | . . . | 3 . 5 . . . | . . . | 8 . 4-------+-------+------- 5 . 4 | . 7 2 | . . . 3 . 6 | . 5 9 | . . . 9 . 7 | . 8 1 | . . .-------+-------+------- . . . | . . . | . . . 6 . 9 | . 4 3 | . . . 2 . 3 | . 1 6 | . . .`

To make a 10 empty units, we need 4 boxes and 6 rows/cols empty
(as 7 rows/cols or 5 boxes is impossible).

The only valid 4 empty boxes shape is :

00X
XX0
XX0

So, it means that a 10 empty units pattern is isomorphic to :

Code: Select all
` . . . | . . . | . x x . . . | . . . | . x x . . . | . . . | . . .-------+-------+------- x x . | . x x | . . . x x . | . x x | . . . . . . | . . . | . . .-------+-------+------- x x . | . x x | . . . x x . | . x x | . . . . . . | . . . | . . .`

JPF
JPF
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Posts: 3755
Joined: 06 December 2005
Location: Paris, France

Interesting, tso posted a 9 almost on the day 1 year ago... Someone should make a nice index of all topics already discussed in the General/puzzle section...

I also think 9 is the most empty units there can be, a 10 would have to be equivalent to:

Code: Select all
`X.X ... X.X... ... ...X.X ... X.X... X.X ...... ... ...... X.X ...X.X ... X.X... ... ...X.X ... X.X`

and I cannot find that in the fully symmetrical thread.

JPF wrote:I made a 8 (4 boxes & 4 rows/cols)

Could a 9 with 4 boxes & 5 rows/columns be possible?

RW
RW
2010 Supporter

Posts: 1000
Joined: 16 March 2006

RW wrote:Someone should make a nice index of all topics already discussed in the General/puzzle section...
Hard work, but you're right : "someone" should make it...

RW wrote:I cannot find that in the fully symmetrical thread.
right, no puzzle has been posted for this F.S. pattern [20-0-4-2-0] or for its 2 equivalents [0-12-4-2-0] & [24-8-0-2-0]

In the Fractal Sudoku thread I posted this puzzle (+5 clues) with 2 solutions :
Code: Select all
` 6 . 2 | . . . | 8 . 7 . 3 . | . . . | . 1 . 7 . 8 | . . . | 9 . 2-------+-------+------- . . . | 2 . 9 | . . . . . . | . 6 . | . . . . . . | 8 . 3 | . . .-------+-------+------- 9 . 1 | . . . | 3 . 4 . 7 . | . . . | . 8 . 2 . 4 | . . . | 7 . 6`

Ocean, posted one with 3 solutions.
Even this pattern seems difficult to match !

RW wrote:Could a 9 with 4 boxes & 5 rows/columns be possible?
doubt it, but who knows ?

JPF
JPF
2017 Supporter

Posts: 3755
Joined: 06 December 2005
Location: Paris, France