9x9 Sudoku puzzles with 8 givens

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9x9 Sudoku puzzles with 8 givens

Postby udosuk » Wed Jul 26, 2006 4:55 pm

It's widely believed that "vanilla" 9x9 sudoku must have at least 17 initial givens, while if we add the "long diagonal" constraints ("Sudoku X") then the minimum number known is 13 (from here).

When all symbols are treated equally and independently (e.g. no order, no "consecutive or not"), we need at least 8 initial givens to give the puzzle a unique solution... (If we have 7 or less, then at least 2 of the symbols aren't appearing among the givens. So we could "swap" those 2 symbols in one solution to get another different solution...)

So, if we add more constraints (e.g. symbols cannot repeat in some short diagonals), we can create some 9x9 sudoku variants with 8 initial givens... Here I'll show a couple (one very hard, one easier), which are based on studies by Condor in these forums a year ago...


The first variant has all 34 diagonals (long or short) involved, but relaxes on 2 of the 9 candidate digits... Only 7 of the 9 digits must not repeat diagonally (you have to determine which 7) while the remaining 2 can repeat diagonally any number of times, including on the main diagonals (so they're not necessary Sudoku X)...

Code: Select all
....4...............3.9.8...............................2...7...........1...6....


Image


And then on the opposite scale, there is this variant that requires digits to repeat diagonally the maximum of times, i.e. five:

Code: Select all
..............................123........6......789.......4......................


Image


These puzzles both have unique solutions... You can try to solve them by hand or with a program to prove it... I wonder if someone could create other 8-clue puzzles based on different additional conditions...:?: But bear in mind the requirement is that the 9 symbols used must be independent to each other, e.g. you can't compare them to see which one is higher... In other words, you can replace the digits by 9 random letters/logos, or even 9 different fruits as some have done before... You can't say an apple is less than an orange, or a grape is consecutive to a watermelon...:)
udosuk
 
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Postby HATMAN » Thu Mar 15, 2007 5:04 pm

If you have not been watching DJApes site.

For a diagonal, disjoint group, non-consequtive puzzle what is the minimum number of givens?

Note it is below 8 as the switching property Udosuk mentions does not apply in non-consequtive sudoku.

HATMAN
HATMAN
 
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