## Multiple solutions?

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

### Multiple solutions?

I got a Bicycle deck of Sudoku cards as a gift, and find it very fun and addicting. Its my first Sudoku experience, and I seem to do well. The problem I'm having is that its my understanding that there is only supposed to be 1 solution to these puzzles, but out of the 7 cards I've done so far, 5 had multiple possible solutions. Is this normal? Am I missing something? Below is the most recent puzzle I did.

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` 6 . 4 | . 9 2 | 5 . 1  5 . 9 | . 1 . | . . .  . 2 . | . . . | 7 . . - - - + - - - + - - -  4 . 2 | . 3 . | . . .  . . . | 2 . 1 | . 4 .  8 1 3 | . . 5 | 9 2 .  - - - + - - - + - - -  . 4 . | 3 8 . | 6 . 5  3 . . | 1 . . | 2 . 8  . . 7 | . 2 . | 3 . . `

Thats the puzzle where dots are empty spots

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` 6 7 4 | 8 9 2 | 5 3 1  5 3 9 | 6 1 7 | 4 8 2  1 2 8 | 4 5 3 | 7 6 9  - - - + - - - + - - -  4 6 2 | 9 3 8 | 1 5 7  7 9 5 | 2 6 1 | 8 4 3  8 1 3 | 7 4 5 | 9 2 6  - - - + - - - + - - -  2 4 1 | 3 8 9 | 6 7 5  3 5 6 | 1 7 4 | 2 9 8  9 8 7 | 5 2 6 | 3 1 4  `

That was the solutions I came up with. Its sound with 1 thru 9 in the columns, rows, and squares.

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` 6 3 4 | 7 9 2 | 5 8 1  5 7 9 | 8 1 3 | 4 6 2  1 2 8 | 6 5 4 | 7 3 9  - - - + - - - + - - -  4 6 2 | 9 3 8 | 1 5 7  7 9 5 | 2 6 1 | 8 4 3  8 1 3 | 4 7 5 | 9 2 6  - - - + - - - + - - -  2 4 1 | 3 8 9 | 6 7 5  3 5 6 | 1 4 7 | 2 9 8  9 8 7 | 5 2 6 | 3 1 4 `

That is the solution posted on the back of the card. It doesn't match mine, but its also correct. Am I missing something? Is there something wrong with the puzzles?

Also, take this example. Im only posting the first set of rows.

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` 5 2 1 | 4 9 8 | 3 7 6  7 8 9 | 3 5 6 | 1 4 2  4 6 3 | 7 1 2 | 5 9 8  - - - + - - - + - - -   ^         ^       ^ `

The three columns marked all have the same numbers. You can move the numbers around in those three columns, and not effect the rest of the puzzle below because the numbers are in the same column at all times. Therefore, there would be like 9 different solutions. Am I wrong? Please help. This site says there should only be one answer!
kinome79

Posts: 2
Joined: 08 January 2006

a poorly designed puzzle can have multiple solutions. Puzzles from Pappocom Sudoku (available from this website) will only ever have one solution.
Shazbot

Posts: 220
Joined: 24 September 2005

There is actually 3 different solutions for that 1st puzzle
Last edited by tarek on Mon Jan 09, 2006 8:26 am, edited 2 times in total.

tarek

Posts: 2650
Joined: 05 January 2006

The puzzles that are created are of a really poor quality. (Especially since they say that it has one unique solution while that is NOT true)
Animator

Posts: 469
Joined: 08 April 2005

Yeah, the bicycle instructions don't claim there is only 1 solution, but I really don't see a point in a puzzle with more than one solution. A puzzle that requires random guessing isn't too much fun, I like my logic. Oh well, I'll keep doing them. Thanks for the input, for a while I thought I was just doing them wrong.
kinome79

Posts: 2
Joined: 08 January 2006

A valid Sudoku is *assumed* to have exactly one solution -- unless explicitly stated otherwise. That's just the default. It's perfectly acceptable to create a Sudoku that has, for instance, two solutions -- as long as this information is included with the puzzle. For example, this one by dukuso:

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

... from this thread, has exactly two solutions. 44 of the empty cells are different in one solution compared with the other.

If a Sudoku were created that had a large number of solutions -- even if this fact were withheld, the puzzle could be valid, if it were presented, not as a Sudoku, but instead, this way: "Fill the cells in this 9x9 grid with the numbers from 1 to 9 so that no two rows, columns or boxes have duplicates. What number is in the central cell?" As long as all possible solutions have the same number in the central cell, this is a fair challenge -- and not an uncommon type of challenge in the logic puzzle world.

In each case, some tactics are useless, new tactics might help.
tso

Posts: 798
Joined: 22 June 2005

kinome79 wrote:Yeah, the bicycle instructions don't claim there is only 1 solution...

tso wrote:A valid Sudoku is *assumed* to have exactly one solution -- unless explicitly stated otherwise.

Especially when a "solution" is printed on the back of the card, it is assumed that there is only one solution. IMHO, these cards are defective.
snoeyink

Posts: 3
Joined: 30 December 2005