## Three nearly identical puzzles

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

### Three nearly identical puzzles

These three puzzles are identical except for the two givens in the center box, yet they each have a unique solution and have wildly different difficulty.

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

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

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

What is the maximum number of valid Sudokus that can be made in this mold -- identical placement of givens, all but two givens identical in all puzzles?
tso

Posts: 798
Joined: 22 June 2005

tso wrote:What is the maximum number of valid Sudokus that can be made in this mold -- identical placement of givens, all but two givens identical in all puzzles?

What is the maximum number (p) of valid Sudokus that can be made with all but one given identical in all puzzles ?

By "randomly" generating a small sample of 1000 minimal puzzles, I got the following distribution for p :
Code: Select all
`p   4     5.5 %3    33.32    57.9            1     3.3     ----    100.0 %`

p=1 means that it's impossible to change the digit in any cell of the puzzle.
It seems to be the less probable case.

Here's an example of 4 valid puzzles, different by only the digit in a given cell.

Code: Select all
`[1,2,6,9 are valid digits in R9C3] . 1 8 | 6 . . | 7 . . 4 . . | . . . | 6 . . . . 3 | . . 1 | 8 . .-------+-------+------- . . . | . 8 2 | . . . . . 4 | . . . | 3 . 1 2 6 . | . . . | . 9 .-------+-------+------- . . . | 3 . . | . . 7 . . . | . 9 4 | . . . . 4 1 | . . 5 | . . .`

and some more (p=4) :
Code: Select all
`020600010000800003000730500015000230403000608600070000007210000000000000081000050020600010000800003000730500015000230403000608600070000007240000000000000081000050020600010000800003000730500015000230403000608600070000007250000000000000081000050020600010000800003000730500015000230403000608600070000007290000000000000081000050200000000040070020006054007020006005400009800000020000073010500900300100100500060200000000040070020006054007020006005400009800000020000073010500900300100100500070200000000040070020006054007020006005400009800000020000073010500900300100100500080200000000040070020006054007020006005400009800000020000073010500900300100100500090200000450805000700007010008000900000406000100090800006500090070000063000080007020200000450805000700007020008000900000406000100090800006500090070000063000080007020200000450805000700007040008000900000406000100090800006500090070000063000080007020200000450805000700007050008000900000406000100090800006500090070000063000080007020040090023005130090060000080000010000102083009080200000000500007706020000010060000040090023005130090060000080000010000102083009080600000000500007706020000010060000040090023005130090060000080000010000102083009080700000000500007706020000010060000040090023005130090060000080000010000102083009080900000000500007706020000010060000000001200047089000230500090002046000000000010070000004020000300090000170000803650000001200047089000230500090003046000000000010070000004020000300090000170000803650000001200047089000230500090005046000000000010070000004020000300090000170000803650000001200047089000230500090009046000000000010070000004020000300090000170000803650038020000001000800620050090080060007006900400090004008000080500050007100000300020038020000004000800620050090080060007006900400090004008000080500050007100000300020038020000005000800620050090080060007006900400090004008000080500050007100000300020038020000009000800620050090080060007006900400090004008000080500050007100000300020005000400042000008000100053708000090014030000030700600000080940000020000061040800005000400043000008000100053708000090014030000030700600000080940000020000061040800005000400046000008000100053708000090014030000030700600000080940000020000061040800005000400047000008000100053708000090014030000030700600000080940000020000061040800803070500000000900270085000004300209000020000600007000008000010030090450060400300803070500000000900270085000004300209000020000600007000008000020030090450060400300803070500000000900270085000004300209000020000600007000008000060030090450060400300803070500000000900270085000004300209000020000600007000008000070030090450060400300`

and some "untouchables" (p=1)
Code: Select all
`010400800206000000800050009000003210020010000000007600000340060000080000000670031002800004300000000000600700000004806009010020001082005000500908508291000000000000501790002207040000080030000040003690010020500000800007100980200000002070000000006030000204028000000400000700370002006201070000006500070000806045600920000000000000000000000008000060005026740060305000000000002591062000900007300700850000000600800000000901700028060000005000600004090004000006083006007210000000059000410040000002040006010073004600008000000004200903700000400036000000000000009209000000010035200`
JPF
JPF
2017 Supporter

Posts: 3781
Joined: 06 December 2005
Location: Paris, France

Cool. They're especially interesting when changing a single clue leads to a solution grid with the greatest divergence and/or the puzzle's subjective difficulty changes the most.

I suppose that's another challenge question -- what is the greatest number of digits that can differ in the solution grids of two sudokus that differ only by the value of a single clue? (65 would be the upper limit.)

This is similar to a question I posed some time ago here.
tso

Posts: 798
Joined: 22 June 2005

Here are 5 valid puzzles, only different by the digit in a given cell.
Code: Select all
`[1,2,3,7,9 are valid digits in r5c6] . 1 . | . . 6 | . . . . 5 7 | . 9 . | 2 . . 6 . . | . . 8 | 4 . 5-------+-------+------- . 7 . | 4 8 . | . . . . . . | . . 7 | . . . . . . | . . . | 7 . 4-------+-------+------- 2 4 5 | . . . | . 9 . . . 9 | . 6 . | . . . . . . | . 3 . | . 8 .`

Unfortunately, all of them are very easy (7 to 9-steppers).
only r5c6 = 7 or 9 make the puzzle minimal.

JPF
JPF
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Posts: 3781
Joined: 06 December 2005
Location: Paris, France

Interesting puzzle nonetheless. Actually, if r5c6 is simply left empty, the puzzle has exactly 5 solutions, one for each of the 5 possibilities left for that cell. All but 17 cells can be solved without filling this cell -- which partially explains why they all have similar difficulty. A puzzle with only 16 unsolved cells has a much lower limit for complexity of the solution.
tso

Posts: 798
Joined: 22 June 2005

tso wrote:Interesting puzzle nonetheless. Actually, if r5c6 is simply left empty, the puzzle has exactly 5 solutions, one for each of the 5 possibilities left for that cell. All but 17 cells can be solved without filling this cell –

actually, only 15 cells are unknown :
Code: Select all
` 8 1 2 | 5 4 6 | 3 7 9 4 5 7 | . 9 . | 2 6 8 6 9 3 | 7 2 8 | 4 1 5-------+-------+------- 5 7 6 | 4 8 . | 9 . 1 . . 4 | . . . | 8 5 6 . . 8 | 6 5 . | 7 . 4-------+-------+------- 2 4 5 | 8 . . | 6 9 3 3 8 9 | 2 6 5 | 1 4 7 7 6 1 | 9 3 4 | 5 8 2`
in all the cases, r5c8=r6c5=5

Anyway, I found a better example :

Code: Select all
`[3,4,5,8,9 are valid digits in A = r4c9] . . . | 2 . 8 | 1 . . . 4 3 | . . . | . 8 . . 9 . | . . . | . 5 .-------+-------+------- 1 . . | . . 6 | 7 . 9 . . . | 5 . . | 2 6 . 6 . . | . 2 . | . . .-------+-------+------- . 7 . | . . . | . . . . . . | . 6 . | 9 . . . . 8 | 9 . 5 | . . 1`

A = 8 or 9 : singles (9 & 10 steppers)
A = 4 or 5 : need XY-Wing
A = 3 seems a bit more difficult.

If A is left empty, 5 solutions and 34 cells are unknown.

JPF
JPF
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Posts: 3781
Joined: 06 December 2005
Location: Paris, France

### all-but-one-given identical

JPF (2006.Jun.16) wrote:What is the maximum number of valid SuDokus that can be made with all but one given identical in all puzzles ?

5 — two examples posted above, two in another Topic
i should mention that some additional examples have been shown for a 5-mutable clue, but they lack one other property which i desire — when the cell is left empty (one less clue), the puzzle has more than 5 answers.

6
Ocean (2006.Nov.18) wrote:
Code: Select all
` . . 1 | . . 2 | . . .  . 3 . | . 4 . | . . .  5 . . | 6 . . | 7 . . -------+-------+------ . . 7 | . . . | . . 1  . 8 . | . . . | . _ .  2 . . | . . . | 6 . . -------+-------+------ . . 9 | . . 3 | . . 5  . 4 . | . 2 . | . 8 .  6 . . | 7 . . | 2 . . `

gsf (2006.Dec.20) wrote:
Code: Select all
` . . . | . . 8 | 5 . 2  . 2 3 | 5 9 1 | . . 8  . . 1 | . 7 4 | 9 . . -------+-------+------ . . . | . . . | 6 . 1  9 . 8 | . _ . | 7 . .  2 . . | . . 7 | . 3 . -------+-------+------ . 6 7 | . . 3 | . . .  8 . . | 4 . . | . . 7  . 9 . | . 8 2 | . 4 . `

7

Red Ed (2006.Nov.28) wrote:
Code: Select all
` . . . | . 8 5 | . 9 .  7 . . | . 9 3 | 4 . .  . 6 9 | 7 2 4 | 1 . . -------+-------+------ . . 3 | 4 6 8 | . . 9  . 8 . | 5 1 . | 3 . 6  5 . 6 | . . . | . . . -------+-------+------ 6 . 1 | . . . | 9 . .  . . . | . . . | . _ .  . 2 . | 9 . . | 8 . . `

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

gsf (2006.Dec.6) wrote:
Code: Select all
` . 2 . | 4 . . | . . .  4 . . | . . . | . 3 7  . 8 . | . . . | 1 . 5 -------+-------+------ . . 1 | . . 5 | 3 . 8  . . _ | 9 . . | . . .  . . . | 7 . 1 | . 6 4 -------+-------+------ . . . | 5 . 6 | 8 . .  5 . . | . 9 . | . . .  . 7 . | 2 . 3 | . . . `

8

Red Ed (2006.Nov.30) wrote:
Code: Select all
` 1 2 . | 3 4 . | 5 6 .  5 3 . | 6 . . | . . .  4 . . | . 5 . | 7 2 3 -------+-------+------ 2 . 1 | . 3 . | 8 . .  7 4 . | 2 . . | . . .  . 8 . | 7 . . | 6 . 2 -------+-------+------ . . 2 | 9 . . | . 8 .  . 7 . | . . . | 2 1 5  . . . | . . _ | 9 . . `

~ Pat

Pat

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