## conditions precedent

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

### conditions precedent

1. what is the minimum number of initial numbers required to have a unique solution on 9X9 grid?

2. can you have a unique solution without every number being present at the onset? in other words, can you have a unique solution without the number 9 as an initial clue with minimum clues?
dahwa

Posts: 3
Joined: 31 March 2006

### Re: conditions precedent

dahwa wrote:1. what is the minimum number of initial numbers required to have a unique solution on 9X9 grid?

The minimum number of clues is currently 17 for a non-symmetric puzzle and 18 for a 180-degree rotational symmetric one. The search is on for a 16-clue one, but it might not exist at all.

2. can you have a unique solution without every number being present at the onset? in other words, can you have a unique solution without the number 9 as an initial clue with minimum clues?

Yes. You need at least one of each of eight of the numbers, though, so only one can be missing. (Otherwise, after finding one solution, you could switch the two missing numbers around and have another one.)

Vidar
vidarino

Posts: 295
Joined: 02 January 2006

### Re: conditions precedent

vidarino wrote:
dahwa wrote:1. what is the minimum number of initial numbers required to have a unique solution on 9X9 grid?

The minimum number of clues is currently 17 for a non-symmetric puzzle and 18 for a 180-degree rotational symmetric one. The search is on for a 16-clue one, but it might not exist at all.

2. can you have a unique solution without every number being present at the onset? in other words, can you have a unique solution without the number 9 as an initial clue with minimum clues?

Yes. You need at least one of each of eight of the numbers, though, so only one can be missing. (Otherwise, after finding one solution, you could switch the two missing numbers around and have another one.)

Vidar

thanks for your answers. I assume there is a proof for the 17/18 answer. If i gave you the positions of all of the 8's and 9's which is one more than the 17/18 would one be able to come up with a unique solution?
dahwa

Posts: 3
Joined: 31 March 2006

### How many initial numbers present one time only by omitting 1

dahwa wrote:1. what is the minimum number of initial numbers required to have a unique solution on 9X9 grid?
2. can you have a unique solution without every number being present at the onset? in other words, can you have a unique solution without the number 9 as an initial clue with minimum clues?

Dahwas question brought me to another question. Constructing sudokus I found out that sometimes it's easier to omit one initial number, take Dahwas number 9, so that the other ones are stronger present in the puzzle and can define the hidden 9 even more.

How many initial numbers can be present in a puzzle one time only by omitting one completely, to get a good sudoku. Or is it to become too easy then?

Claudia
claudiarabia

Posts: 288
Joined: 14 May 2006

It is easy to prove your 2nd question by imagining a grid which was full appart from the number 9's. This would leave one blank cell in each row, collum and box.
Pi

Posts: 389
Joined: 27 May 2005

### Numbers 1-9 present one time only

Pi wrote:It is easy to prove your 2nd question by imagining a grid which was full appart from the number 9's. This would leave one blank cell in each row, collum and box.

Maybe I used the wrong terms. I meant the number 9 is omitted completely and from the other numbers most of them have to be present one time only.
In other words I would like to see a sudoku with the maximum of the numbers 1-9 to be present one time only and one number is omitted completely. Maybe the amount will differ when all the numbers are present in the puzzle.

Claudia
claudiarabia

Posts: 288
Joined: 14 May 2006

Here are 7 puzzles with 17 initial clues, among which one number is completely absent, 3 numbers appearing once only, 2 appearing twice, 2 thrice and one 4 times...

......13.....8...542.......6......2...5.1..........4...7.4.2......6..2..........1
....316..2.......7......1...5.2...4..36........1......8..71...........2.......3..
...4.12..8..6...1.3.....5...4.1..7..2...3.................2..8..1..............3.
.4..3...........52......67..8....3.....2........7.....2.3...1..7..5.2.......8....
16.5..........87..5........45.1...6...7...3......2...........51....73............
6..3............81..........18....5...56........7.3...2...1....4.....6...6....3..
83..2....2..5..1............7.....82...6.1.............2..8..3...1...6..4........

And for interest, here are 10 which 5 numbers appear once, 4 numbers appear thrice (none absent)...

....1.3..52........6.....7.4.....13....8.6......5........6....51.9......3........
....6..3....8..2..2..5........29.5...76............1.......7.645...............7.
....81...5.....2............84.....7...2..5...........25.6...4.....4..389........
.7..3.....2.....6........39...2..7..3.6............1.....7.24..6......5....8.....
.8..3.....2.....6........39...2..8..3.6............1.....7.24..6......5....8.....
2...6.......7...1.......3......2..96..1.......7........5.4..7..6.....8.2...1.....
23..4.5......81.4.7...........3.62...14......................913..2..............
3..2.....2.....8..........1..8.3...7...9...3..1...........185..4....6.2..........
5...2.......3...4.1...........4..57..3.9.....6.....1......5.8...4......3.....1...
6..1...........25.1.........2..57..........6.....2.....59...4.....6.1..8...3.....

And 5 which 8 numbers appear twice and 1 number appears once:

.1......9...3..8........6......124..7.3......5........8..6.........4..2....7...5.
.3.6...8..19..........2....7.....45.....31...2........4..8......6.5...........9..
.524.........7.1..............8.2...3.....6...9.5.....1.6.3...........897........
.923.........8.1...........1.7.4...........658.........6.5.2...4.....7.....9.....
8.......1...95.................7.42.3.16...............4....57.6..3.8.........2..

Thanks for Ocean who posted me these lists (ages ago), and Gordon (gfroyle) who created the whole database of 17-clue puzzles...

Don't know if there are more than 3 once-appearing numbers (with 1 absentee number) when there are more than 17 initial clues...
Last edited by udosuk on Mon Jul 24, 2006 9:54 am, edited 4 times in total.
udosuk

Posts: 2698
Joined: 17 July 2005

For the 17-clues also see the statistic here.
ravel

Posts: 998
Joined: 21 February 2006

Based on the line of answers it appears that one needs a minimum of 18 initial clues to develope a unique solution.

Does this answer apply if the initial clues were all the nines and eights for example (which would total 18)?
dahwa

Posts: 3
Joined: 31 March 2006

dahwa wrote:Does this answer apply if the initial clues were all the nines and eights for example (which would total 18)?

Such a puzzle would not be unique ... since you could interchange the eighteen positions of any two digits not given as clues.
ronk
2012 Supporter

Posts: 4764
Joined: 02 November 2005
Location: Southeastern USA

### Maximal amount of numbers appearing only once

udosuk wrote:Here are 7 puzzles with 17 initial clues, among which one number is completely absent, 3 numbers appearing once only, 2 appearing twice, 2 thrice and one 4 times...
.....Don't know if there are more than 3 once-appearing numbers (with 1 absentee number) when there are more than 17 initial clues...

Now I have a feeling about that. Thank you very much.

Claudia
claudiarabia

Posts: 288
Joined: 14 May 2006