## 0-hint sudoku puzzles

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

### 0-hint sudoku puzzles

The following 0-clue puzzle was originated from tso in another thread:

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

Place the digits in this Sudoku such that the resulting 81 digit number formed by stringing the rows one after the other in order from top to bottom is minimum.

This one isn't too hard that a normal player could probably solve in a few minutes. But it intrigued me to create some harder puzzles which are quite enjoyable to solve...

We can change the rule that the 9 rows must form the smallest increasing sequence, which is equivalent to the following 8-clue puzzle with tso's rule:

Code: Select all
`1 . . | . . . | . . .2 . . | . . . | . . .3 . . | . . . | . . .------+-------+------4 . . | . . . | . . .5 . . | . . . | . . .6 . . | . . . | . . .------+-------+------7 . . | . . . | . . .8 . . | . . . | . . .. . . | . . . | . . .Place the digits in this Sudoku such that the resulting 81 digit number formed by stringing the rows one after the other in order from top to bottom is minimum.`

Or we can require that the 162-digit number formed by the 1st row followed by the 1st column followed by the 2nd row then the 2nd column... must be minimum. The solving starts out like this:

Code: Select all
`1 2 3 | 4 5 6 | 7 8 94 . . | . . . | . . .5 . . | . . . | . . .------+-------+------2 . . | . . . | . . .3 . . | . . . | . . .6 . . | . . . | . . .------+-------+------7 . . | . . . | . . .8 . . | . . . | . . .9 . . | . . . | . . .`

Or, we can define certain paths that run through all 81 numbers, and require that the 81-digit number formed by that path must be minimum. For example, the following zig-zag path is of the same difficulty of tso's original puzzle:

Code: Select all
`0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0                                |0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0|0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0                                |0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0|0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0                                |0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0|0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0                                |0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0|0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0`

But if we use a "spiral" path instead, the puzzles are much more fun to solve:

Code: Select all
`0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0                                |0 - 0 - 0 - 0 - 0 - 0 - 0 - 0   0|                           |   |0   0 - 0 - 0 - 0 - 0 - 0   0   0|   |                   |   |   |0   0   0 - 0 - 0 - 0   0   0   0|   |   |           |   |   |   |0   0   0   0 - 0   0   0   0   0|   |   |   |       |   |   |   |0   0   0   0 - 0 - 0   0   0   0|   |   |               |   |   |0   0   0 - 0 - 0 - 0 - 0   0   0|   |                       |   |0   0 - 0 - 0 - 0 - 0 - 0 - 0   0|                               |0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0`

And there are 2 different approaches: inbound spiral (from top-left corner to centre) and outbound spiral (from center to top-left corner).

Of course, by introducing the idea of "minimum" and "comparison of numbers" the puzzle is no longer a pure logical one, but a bit mathematical as well...
Last edited by udosuk on Fri Aug 12, 2005 2:53 pm, edited 1 time in total.
udosuk

Posts: 2698
Joined: 17 July 2005

### Re: "Sudoku" with the fewest clues possible

All of these could just as easily be the maximum possible.

For the first one I found

Code: Select all
`9 8 7   6 5 4   3 2 16 5 4   3 2 1   9 8 73 2 1   9 8 7   6 5 48 9 6   7 4 5   2 1 37 4 5   2 1 3   8 9 62 1 3   8 9 6   7 4 55 7 9   4 6 8   1 3 24 6 8   1 3 2   5 7 91 3 2   5 7 9   4 6 8`

But we must be careful or this could 'spiral' out of contol.
Condor

Posts: 62
Joined: 19 June 2005

So Condor, I take it you've solved the "alternating rows-columns" and the 2 spiral puzzles with relelative ease then? I actually thought they were quite challenging, especially if you try to solve them by hand... without making pencil marks...

For example, when you solve the inbound spiral, everything seems so trival until you arrive at the following situation:

Code: Select all
`1 2 3 4 5 6 7 8 9. . . . . . . . 1. . . . . . . . 2. . . . . . . . 3. . . . . . . . 44 . . . . . . . 53 . . . . . . . 62 . . . . . . . 79 7 6 5 4 3 2 1 8`

One careless mistake right here and you'll run into trouble... Of course you'll encounter many similar situations later in the puzzle...

BTW, You could PM me to check your solutions if you don't want to show spoilers to the public...
udosuk

Posts: 2698
Joined: 17 July 2005

### Re: "Sudoku" with the fewest clues possible

udosuk wrote:But if we use a "spiral" path instead, the puzzles are much more fun to solve:

Solved the inbound spiral for minimum. A fun little exercise. nice to try these as a diversion.

udosuk wrote:we can define certain paths that run through all 81 numbers, and require that the 81-digit number formed by that path must be minimum.

The thought occured to me that there are 81! (5.797125 x10^120) paths. Of course only a few will be interresting, but it also means that each sudoku is the result of many paths.
Condor

Posts: 62
Joined: 19 June 2005

For anyone who liked the spirals, here is another path:

Code: Select all
`0--0--0--0--0--0--0--0--0                        |0--0  0--0--0--0--0--0  0|  |  |              |  |0  0  0--0--0--0--0  0  0|  |              |  |  |0  0  0--0  0--0  0  0  0|  |  |  |  |  |  |  |  |0  0  0  0  0  0  0  0  0|  |  |  |     |  |  |  |0  0  0  0--0--0  0  0  0|  |  |           |  |  |0  0  0--0--0--0--0  0  0|  |                 |  |0  0--0--0--0--0--0--0  0|                       |0--0--0--0--0--0--0--0--0`

Again 2-puzzles-in-1, inbound and outbound. PM solutions-checking service is available for those who don't like programs...

Condor, finished the outbound spiral already?
udosuk

Posts: 2698
Joined: 17 July 2005