Do sudoku answers always have a single minimal clue set?How many minimal-clue sudoku puzzles are there?How to...
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Do sudoku answers always have a single minimal clue set?
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Do sudoku answers always have a single minimal clue set?
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Suppose we have a solved sudoku, which was started from a minimal set of clues to its unique solution, i.e. 17+ numbers in specific locations, which we'll call set1. Is there a set2, (or set3, etc.), with few or no elements in common with set1?
Put another way, suppose you have Monday and Tuesday newspapers, each with an apparently different sudoku. You finish Monday, and have its 81 number solution. Tuesday seems to be a different puzzle, but when you finish, it turns out the solution is identical to that of Monday. Is that, given the mathematics of sudoku, possible?
sudoku
asked 9 hours ago
agcagc
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add a comment |
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Suppose we have a solved sudoku, which was started from a minimal set of clues to its unique solution, i.e. 17+ numbers in specific locations, which we'll call set1. Is there a set2, (or set3, etc.), with few or no elements in common with set1?
Put another way, suppose you have Monday and Tuesday newspapers, each with an apparently different sudoku. You finish Monday, and have its 81 number solution. Tuesday seems to be a different puzzle, but when you finish, it turns out the solution is identical to that of Monday. Is that, given the mathematics of sudoku, possible?
sudoku
asked 9 hours ago
agcagc
1064 bronze badges
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agc is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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$begingroup$
A few pictures would probably help.
$endgroup$
– agc
9 hours ago
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What does "and started from a minimal set of clues" mean?
$endgroup$
– Jonathan Allan
9 hours ago
$begingroup$
@JonathanAllan, It means no redundant clues, where any clue removed would make the answer insoluble or diverge to more than one solution. So33 > clues > 16, as implied by Wikipedia.
$endgroup$
– agc
5 hours ago
add a comment |
$begingroup$
Suppose we have a solved sudoku, which was started from a minimal set of clues to its unique solution, i.e. 17+ numbers in specific locations, which we'll call set1. Is there a set2, (or set3, etc.), with few or no elements in common with set1?
Put another way, suppose you have Monday and Tuesday newspapers, each with an apparently different sudoku. You finish Monday, and have its 81 number solution. Tuesday seems to be a different puzzle, but when you finish, it turns out the solution is identical to that of Monday. Is that, given the mathematics of sudoku, possible?
sudoku
asked 9 hours ago
agcagc
1064 bronze badges
New contributor
agc is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
Suppose we have a solved sudoku, which was started from a minimal set of clues to its unique solution, i.e. 17+ numbers in specific locations, which we'll call set1. Is there a set2, (or set3, etc.), with few or no elements in common with set1?
Put another way, suppose you have Monday and Tuesday newspapers, each with an apparently different sudoku. You finish Monday, and have its 81 number solution. Tuesday seems to be a different puzzle, but when you finish, it turns out the solution is identical to that of Monday. Is that, given the mathematics of sudoku, possible?
sudoku
sudoku
asked 9 hours ago
agcagc
1064 bronze badges
New contributor
agc is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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asked 9 hours ago
agcagc
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edited 5 hours ago
agc
asked 9 hours ago
agcagc
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asked 9 hours ago
agcagc
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asked 9 hours ago
agcagc
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1064 bronze badges
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$begingroup$
A few pictures would probably help.
$endgroup$
– agc
9 hours ago
$begingroup$
What does "and started from a minimal set of clues" mean?
$endgroup$
– Jonathan Allan
9 hours ago
$begingroup$
@JonathanAllan, It means no redundant clues, where any clue removed would make the answer insoluble or diverge to more than one solution. So33 > clues > 16, as implied by Wikipedia.
$endgroup$
– agc
5 hours ago
add a comment |
$begingroup$
A few pictures would probably help.
$endgroup$
– agc
9 hours ago
$begingroup$
What does "and started from a minimal set of clues" mean?
$endgroup$
– Jonathan Allan
9 hours ago
$begingroup$
@JonathanAllan, It means no redundant clues, where any clue removed would make the answer insoluble or diverge to more than one solution. So33 > clues > 16, as implied by Wikipedia.
$endgroup$
– agc
5 hours ago
$begingroup$
A few pictures would probably help.
$endgroup$
– agc
9 hours ago
$begingroup$
A few pictures would probably help.
$endgroup$
– agc
9 hours ago
$begingroup$
What does "and started from a minimal set of clues" mean?
$endgroup$
– Jonathan Allan
9 hours ago
$begingroup$
What does "and started from a minimal set of clues" mean?
$endgroup$
– Jonathan Allan
9 hours ago
$begingroup$
@JonathanAllan, It means no redundant clues, where any clue removed would make the answer insoluble or diverge to more than one solution. So
33 > clues > 16, as implied by Wikipedia.$endgroup$
– agc
5 hours ago
$begingroup$
@JonathanAllan, It means no redundant clues, where any clue removed would make the answer insoluble or diverge to more than one solution. So
33 > clues > 16, as implied by Wikipedia.$endgroup$
– agc
5 hours ago
add a comment |
2 Answers
2
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$begingroup$
Yes, it's completely possible that two Sudoku puzzles with disjoint clues have the same solution.
The two puzzles below:
123|456| 89 | |7
4 6|7 9|12 5 | 8 | 3
78 | 23|4 9|1 | 56
---+---+--- ---+---+---
3 |567| 2 4| |891
567|8 | | 1|234
891| |5 7 |234| 6
---+---+--- ---+---+---
34 | 8|9 5|67 | 12
6 | 1 | 4 78|9 2|3 5
2| | 91 |345|678
both have the same (unique) solution, but share no clues in the same positions.
answered 9 hours ago
Deusovi♦Deusovi
67.5k7 gold badges232 silver badges296 bronze badges
$endgroup$
1
$begingroup$
I wonder what the maximum number of disjoint clue-sets that determine the same solution is. Obviously no bigger than floor(81/17)=4, given that 17 is the minimum number of clues to make a unique solution; I bet 3 is easy but 4 might be difficult or impossible. (On the handwavy grounds that 81 is nearly 5*17, my guess is that 4 is possible.)
$endgroup$
– Gareth McCaughan♦
7 hours ago
$begingroup$
Interesting. That example is in the right direction, but the clue-sets, (at 41 and 40 clues apiece), contradict the word minimal in the question title. With 41 clues, it's hardly what anyone would think of as a sudoku at all. The answer should also note the method (not necessarily the details) of determining that each clue-set is soluble.
$endgroup$
– agc
6 hours ago
$begingroup$
@agc Each of these has a single solution - i.e. they are both what are termed "proper sudoku".
$endgroup$
– Jonathan Allan
5 hours ago
$begingroup$
@agc Making these sudokus minimal is left as an exercise to the reader.
$endgroup$
– Alex F
5 hours ago
$begingroup$
@agc "it's hardly what anyone would think of as a sudoku at all"? They both seem like perfectly fine Sudoku puzzles to me (if a bit easy). But you can also make them minimal by just removing clues until you can't remove any more without losing uniqueness -- that part isn't really a constraint on the clues. If there are nonminimal examples, then there are minimal ones.
$endgroup$
– Deusovi♦
4 hours ago
add a comment |
$begingroup$
Here are two proper, irreducible sudoku with the same solution as each other and disjoint sets of clues (24 & 25 clues, respectively).
1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9
·-------·-------·-------· ·-------·-------·-------·
A| · · · | 4 · · | 7 · · | A| 1 2 3 | 4 5 6 | 7 8 9 |
B| · · 6 | · 8 · | 1 · · | B| 4 5 6 | 7 8 9 | 1 2 3 |
C| 7 · · | · · 3 | · 5 · | C| 7 8 9 | 1 2 3 | 4 5 6 |
·-------+-------+-------· ·-------+-------+-------·
D| 8 · 7 | · 3 · | · · 4 | D| 8 9 7 | 2 3 1 | 5 6 4 |
E| · · 1 | · · · | · · · | E| 2 3 1 | 5 6 4 | 8 9 7 |
F| · 6 · | · · · | 2 · · | F| 5 6 4 | 8 9 7 | 2 3 1 |
·-------+-------+-------· ·-------+-------+-------·
G| 3 · · | · · 5 | · · 8 | G| 3 1 2 | 6 4 5 | 9 7 8 |
H| · 4 · | 9 · · | · · 2 | H| 6 4 5 | 9 7 8 | 3 1 2 |
J| · · · | · 1 · | 6 · · | J| 9 7 8 | 3 1 2 | 6 4 5 |
·-------·-------·-------· ·-------·-------·-------·
1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9
·-------·-------·-------· ·-------·-------·-------·
A| · · 3 | · · · | · · · | A| 1 2 3 | 4 5 6 | 7 8 9 |
B| 4 · · | · · · | · 2 · | B| 4 5 6 | 7 8 9 | 1 2 3 |
C| · 8 · | 1 2 · | · · 6 | C| 7 8 9 | 1 2 3 | 4 5 6 |
·-------+-------+-------· ·-------+-------+-------·
D| · · · | · · · | · · · | D| 8 9 7 | 2 3 1 | 5 6 4 |
E| 2 · · | · 6 · | · · 7 | E| 2 3 1 | 5 6 4 | 8 9 7 |
F| · · · | 8 · 7 | · 3 1 | F| 5 6 4 | 8 9 7 | 2 3 1 |
·-------+-------+-------· ·-------+-------+-------·
G| · 1 · | 6 4 · | 9 · · | G| 3 1 2 | 6 4 5 | 9 7 8 |
H| 6 · 5 | · · 8 | · · · | H| 6 4 5 | 9 7 8 | 3 1 2 |
J| 9 · 8 | 3 · · | · 4 · | J| 9 7 8 | 3 1 2 | 6 4 5 |
·-------·-------·-------· ·-------·-------·-------·
Proper: Having a single, unique solution
Irreducible: Removing any clue would make the resulting puzzle no longer proper
Disjoint: Having no elements in common
I found these by running some Python code.
Note: The first is very, very difficult, but the second is extremely easy!
answered 4 hours ago
Jonathan AllanJonathan Allan
18.3k1 gold badge47 silver badges99 bronze badges
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The diagram looks very pretty. +1 :)
$endgroup$
– Mr Pie
57 mins ago
add a comment |
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2 Answers
2
active
oldest
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2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Yes, it's completely possible that two Sudoku puzzles with disjoint clues have the same solution.
The two puzzles below:
123|456| 89 | |7
4 6|7 9|12 5 | 8 | 3
78 | 23|4 9|1 | 56
---+---+--- ---+---+---
3 |567| 2 4| |891
567|8 | | 1|234
891| |5 7 |234| 6
---+---+--- ---+---+---
34 | 8|9 5|67 | 12
6 | 1 | 4 78|9 2|3 5
2| | 91 |345|678
both have the same (unique) solution, but share no clues in the same positions.
answered 9 hours ago
Deusovi♦Deusovi
67.5k7 gold badges232 silver badges296 bronze badges
$endgroup$
1
$begingroup$
I wonder what the maximum number of disjoint clue-sets that determine the same solution is. Obviously no bigger than floor(81/17)=4, given that 17 is the minimum number of clues to make a unique solution; I bet 3 is easy but 4 might be difficult or impossible. (On the handwavy grounds that 81 is nearly 5*17, my guess is that 4 is possible.)
$endgroup$
– Gareth McCaughan♦
7 hours ago
$begingroup$
Interesting. That example is in the right direction, but the clue-sets, (at 41 and 40 clues apiece), contradict the word minimal in the question title. With 41 clues, it's hardly what anyone would think of as a sudoku at all. The answer should also note the method (not necessarily the details) of determining that each clue-set is soluble.
$endgroup$
– agc
6 hours ago
$begingroup$
@agc Each of these has a single solution - i.e. they are both what are termed "proper sudoku".
$endgroup$
– Jonathan Allan
5 hours ago
$begingroup$
@agc Making these sudokus minimal is left as an exercise to the reader.
$endgroup$
– Alex F
5 hours ago
$begingroup$
@agc "it's hardly what anyone would think of as a sudoku at all"? They both seem like perfectly fine Sudoku puzzles to me (if a bit easy). But you can also make them minimal by just removing clues until you can't remove any more without losing uniqueness -- that part isn't really a constraint on the clues. If there are nonminimal examples, then there are minimal ones.
$endgroup$
– Deusovi♦
4 hours ago
add a comment |
$begingroup$
Yes, it's completely possible that two Sudoku puzzles with disjoint clues have the same solution.
The two puzzles below:
123|456| 89 | |7
4 6|7 9|12 5 | 8 | 3
78 | 23|4 9|1 | 56
---+---+--- ---+---+---
3 |567| 2 4| |891
567|8 | | 1|234
891| |5 7 |234| 6
---+---+--- ---+---+---
34 | 8|9 5|67 | 12
6 | 1 | 4 78|9 2|3 5
2| | 91 |345|678
both have the same (unique) solution, but share no clues in the same positions.
answered 9 hours ago
Deusovi♦Deusovi
67.5k7 gold badges232 silver badges296 bronze badges
$endgroup$
1
$begingroup$
I wonder what the maximum number of disjoint clue-sets that determine the same solution is. Obviously no bigger than floor(81/17)=4, given that 17 is the minimum number of clues to make a unique solution; I bet 3 is easy but 4 might be difficult or impossible. (On the handwavy grounds that 81 is nearly 5*17, my guess is that 4 is possible.)
$endgroup$
– Gareth McCaughan♦
7 hours ago
$begingroup$
Interesting. That example is in the right direction, but the clue-sets, (at 41 and 40 clues apiece), contradict the word minimal in the question title. With 41 clues, it's hardly what anyone would think of as a sudoku at all. The answer should also note the method (not necessarily the details) of determining that each clue-set is soluble.
$endgroup$
– agc
6 hours ago
$begingroup$
@agc Each of these has a single solution - i.e. they are both what are termed "proper sudoku".
$endgroup$
– Jonathan Allan
5 hours ago
$begingroup$
@agc Making these sudokus minimal is left as an exercise to the reader.
$endgroup$
– Alex F
5 hours ago
$begingroup$
@agc "it's hardly what anyone would think of as a sudoku at all"? They both seem like perfectly fine Sudoku puzzles to me (if a bit easy). But you can also make them minimal by just removing clues until you can't remove any more without losing uniqueness -- that part isn't really a constraint on the clues. If there are nonminimal examples, then there are minimal ones.
$endgroup$
– Deusovi♦
4 hours ago
add a comment |
$begingroup$
Yes, it's completely possible that two Sudoku puzzles with disjoint clues have the same solution.
The two puzzles below:
123|456| 89 | |7
4 6|7 9|12 5 | 8 | 3
78 | 23|4 9|1 | 56
---+---+--- ---+---+---
3 |567| 2 4| |891
567|8 | | 1|234
891| |5 7 |234| 6
---+---+--- ---+---+---
34 | 8|9 5|67 | 12
6 | 1 | 4 78|9 2|3 5
2| | 91 |345|678
both have the same (unique) solution, but share no clues in the same positions.
answered 9 hours ago
Deusovi♦Deusovi
67.5k7 gold badges232 silver badges296 bronze badges
$endgroup$
Yes, it's completely possible that two Sudoku puzzles with disjoint clues have the same solution.
The two puzzles below:
123|456| 89 | |7
4 6|7 9|12 5 | 8 | 3
78 | 23|4 9|1 | 56
---+---+--- ---+---+---
3 |567| 2 4| |891
567|8 | | 1|234
891| |5 7 |234| 6
---+---+--- ---+---+---
34 | 8|9 5|67 | 12
6 | 1 | 4 78|9 2|3 5
2| | 91 |345|678
both have the same (unique) solution, but share no clues in the same positions.
answered 9 hours ago
Deusovi♦Deusovi
67.5k7 gold badges232 silver badges296 bronze badges
answered 9 hours ago
Deusovi♦Deusovi
67.5k7 gold badges232 silver badges296 bronze badges
answered 9 hours ago
Deusovi♦Deusovi
67.5k7 gold badges232 silver badges296 bronze badges
answered 9 hours ago
Deusovi♦Deusovi
67.5k7 gold badges232 silver badges296 bronze badges
67.5k7 gold badges232 silver badges296 bronze badges
1
$begingroup$
I wonder what the maximum number of disjoint clue-sets that determine the same solution is. Obviously no bigger than floor(81/17)=4, given that 17 is the minimum number of clues to make a unique solution; I bet 3 is easy but 4 might be difficult or impossible. (On the handwavy grounds that 81 is nearly 5*17, my guess is that 4 is possible.)
$endgroup$
– Gareth McCaughan♦
7 hours ago
$begingroup$
Interesting. That example is in the right direction, but the clue-sets, (at 41 and 40 clues apiece), contradict the word minimal in the question title. With 41 clues, it's hardly what anyone would think of as a sudoku at all. The answer should also note the method (not necessarily the details) of determining that each clue-set is soluble.
$endgroup$
– agc
6 hours ago
$begingroup$
@agc Each of these has a single solution - i.e. they are both what are termed "proper sudoku".
$endgroup$
– Jonathan Allan
5 hours ago
$begingroup$
@agc Making these sudokus minimal is left as an exercise to the reader.
$endgroup$
– Alex F
5 hours ago
$begingroup$
@agc "it's hardly what anyone would think of as a sudoku at all"? They both seem like perfectly fine Sudoku puzzles to me (if a bit easy). But you can also make them minimal by just removing clues until you can't remove any more without losing uniqueness -- that part isn't really a constraint on the clues. If there are nonminimal examples, then there are minimal ones.
$endgroup$
– Deusovi♦
4 hours ago
add a comment |
1
$begingroup$
I wonder what the maximum number of disjoint clue-sets that determine the same solution is. Obviously no bigger than floor(81/17)=4, given that 17 is the minimum number of clues to make a unique solution; I bet 3 is easy but 4 might be difficult or impossible. (On the handwavy grounds that 81 is nearly 5*17, my guess is that 4 is possible.)
$endgroup$
– Gareth McCaughan♦
7 hours ago
$begingroup$
Interesting. That example is in the right direction, but the clue-sets, (at 41 and 40 clues apiece), contradict the word minimal in the question title. With 41 clues, it's hardly what anyone would think of as a sudoku at all. The answer should also note the method (not necessarily the details) of determining that each clue-set is soluble.
$endgroup$
– agc
6 hours ago
$begingroup$
@agc Each of these has a single solution - i.e. they are both what are termed "proper sudoku".
$endgroup$
– Jonathan Allan
5 hours ago
$begingroup$
@agc Making these sudokus minimal is left as an exercise to the reader.
$endgroup$
– Alex F
5 hours ago
$begingroup$
@agc "it's hardly what anyone would think of as a sudoku at all"? They both seem like perfectly fine Sudoku puzzles to me (if a bit easy). But you can also make them minimal by just removing clues until you can't remove any more without losing uniqueness -- that part isn't really a constraint on the clues. If there are nonminimal examples, then there are minimal ones.
$endgroup$
– Deusovi♦
4 hours ago
1
1
$begingroup$
I wonder what the maximum number of disjoint clue-sets that determine the same solution is. Obviously no bigger than floor(81/17)=4, given that 17 is the minimum number of clues to make a unique solution; I bet 3 is easy but 4 might be difficult or impossible. (On the handwavy grounds that 81 is nearly 5*17, my guess is that 4 is possible.)
$endgroup$
– Gareth McCaughan♦
7 hours ago
$begingroup$
I wonder what the maximum number of disjoint clue-sets that determine the same solution is. Obviously no bigger than floor(81/17)=4, given that 17 is the minimum number of clues to make a unique solution; I bet 3 is easy but 4 might be difficult or impossible. (On the handwavy grounds that 81 is nearly 5*17, my guess is that 4 is possible.)
$endgroup$
– Gareth McCaughan♦
7 hours ago
$begingroup$
Interesting. That example is in the right direction, but the clue-sets, (at 41 and 40 clues apiece), contradict the word minimal in the question title. With 41 clues, it's hardly what anyone would think of as a sudoku at all. The answer should also note the method (not necessarily the details) of determining that each clue-set is soluble.
$endgroup$
– agc
6 hours ago
$begingroup$
Interesting. That example is in the right direction, but the clue-sets, (at 41 and 40 clues apiece), contradict the word minimal in the question title. With 41 clues, it's hardly what anyone would think of as a sudoku at all. The answer should also note the method (not necessarily the details) of determining that each clue-set is soluble.
$endgroup$
– agc
6 hours ago
$begingroup$
@agc Each of these has a single solution - i.e. they are both what are termed "proper sudoku".
$endgroup$
– Jonathan Allan
5 hours ago
$begingroup$
@agc Each of these has a single solution - i.e. they are both what are termed "proper sudoku".
$endgroup$
– Jonathan Allan
5 hours ago
$begingroup$
@agc Making these sudokus minimal is left as an exercise to the reader.
$endgroup$
– Alex F
5 hours ago
$begingroup$
@agc Making these sudokus minimal is left as an exercise to the reader.
$endgroup$
– Alex F
5 hours ago
$begingroup$
@agc "it's hardly what anyone would think of as a sudoku at all"? They both seem like perfectly fine Sudoku puzzles to me (if a bit easy). But you can also make them minimal by just removing clues until you can't remove any more without losing uniqueness -- that part isn't really a constraint on the clues. If there are nonminimal examples, then there are minimal ones.
$endgroup$
– Deusovi♦
4 hours ago
$begingroup$
@agc "it's hardly what anyone would think of as a sudoku at all"? They both seem like perfectly fine Sudoku puzzles to me (if a bit easy). But you can also make them minimal by just removing clues until you can't remove any more without losing uniqueness -- that part isn't really a constraint on the clues. If there are nonminimal examples, then there are minimal ones.
$endgroup$
– Deusovi♦
4 hours ago
add a comment |
$begingroup$
Here are two proper, irreducible sudoku with the same solution as each other and disjoint sets of clues (24 & 25 clues, respectively).
1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9
·-------·-------·-------· ·-------·-------·-------·
A| · · · | 4 · · | 7 · · | A| 1 2 3 | 4 5 6 | 7 8 9 |
B| · · 6 | · 8 · | 1 · · | B| 4 5 6 | 7 8 9 | 1 2 3 |
C| 7 · · | · · 3 | · 5 · | C| 7 8 9 | 1 2 3 | 4 5 6 |
·-------+-------+-------· ·-------+-------+-------·
D| 8 · 7 | · 3 · | · · 4 | D| 8 9 7 | 2 3 1 | 5 6 4 |
E| · · 1 | · · · | · · · | E| 2 3 1 | 5 6 4 | 8 9 7 |
F| · 6 · | · · · | 2 · · | F| 5 6 4 | 8 9 7 | 2 3 1 |
·-------+-------+-------· ·-------+-------+-------·
G| 3 · · | · · 5 | · · 8 | G| 3 1 2 | 6 4 5 | 9 7 8 |
H| · 4 · | 9 · · | · · 2 | H| 6 4 5 | 9 7 8 | 3 1 2 |
J| · · · | · 1 · | 6 · · | J| 9 7 8 | 3 1 2 | 6 4 5 |
·-------·-------·-------· ·-------·-------·-------·
1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9
·-------·-------·-------· ·-------·-------·-------·
A| · · 3 | · · · | · · · | A| 1 2 3 | 4 5 6 | 7 8 9 |
B| 4 · · | · · · | · 2 · | B| 4 5 6 | 7 8 9 | 1 2 3 |
C| · 8 · | 1 2 · | · · 6 | C| 7 8 9 | 1 2 3 | 4 5 6 |
·-------+-------+-------· ·-------+-------+-------·
D| · · · | · · · | · · · | D| 8 9 7 | 2 3 1 | 5 6 4 |
E| 2 · · | · 6 · | · · 7 | E| 2 3 1 | 5 6 4 | 8 9 7 |
F| · · · | 8 · 7 | · 3 1 | F| 5 6 4 | 8 9 7 | 2 3 1 |
·-------+-------+-------· ·-------+-------+-------·
G| · 1 · | 6 4 · | 9 · · | G| 3 1 2 | 6 4 5 | 9 7 8 |
H| 6 · 5 | · · 8 | · · · | H| 6 4 5 | 9 7 8 | 3 1 2 |
J| 9 · 8 | 3 · · | · 4 · | J| 9 7 8 | 3 1 2 | 6 4 5 |
·-------·-------·-------· ·-------·-------·-------·
Proper: Having a single, unique solution
Irreducible: Removing any clue would make the resulting puzzle no longer proper
Disjoint: Having no elements in common
I found these by running some Python code.
Note: The first is very, very difficult, but the second is extremely easy!
answered 4 hours ago
Jonathan AllanJonathan Allan
18.3k1 gold badge47 silver badges99 bronze badges
$endgroup$
$begingroup$
The diagram looks very pretty. +1 :)
$endgroup$
– Mr Pie
57 mins ago
add a comment |
$begingroup$
Here are two proper, irreducible sudoku with the same solution as each other and disjoint sets of clues (24 & 25 clues, respectively).
1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9
·-------·-------·-------· ·-------·-------·-------·
A| · · · | 4 · · | 7 · · | A| 1 2 3 | 4 5 6 | 7 8 9 |
B| · · 6 | · 8 · | 1 · · | B| 4 5 6 | 7 8 9 | 1 2 3 |
C| 7 · · | · · 3 | · 5 · | C| 7 8 9 | 1 2 3 | 4 5 6 |
·-------+-------+-------· ·-------+-------+-------·
D| 8 · 7 | · 3 · | · · 4 | D| 8 9 7 | 2 3 1 | 5 6 4 |
E| · · 1 | · · · | · · · | E| 2 3 1 | 5 6 4 | 8 9 7 |
F| · 6 · | · · · | 2 · · | F| 5 6 4 | 8 9 7 | 2 3 1 |
·-------+-------+-------· ·-------+-------+-------·
G| 3 · · | · · 5 | · · 8 | G| 3 1 2 | 6 4 5 | 9 7 8 |
H| · 4 · | 9 · · | · · 2 | H| 6 4 5 | 9 7 8 | 3 1 2 |
J| · · · | · 1 · | 6 · · | J| 9 7 8 | 3 1 2 | 6 4 5 |
·-------·-------·-------· ·-------·-------·-------·
1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9
·-------·-------·-------· ·-------·-------·-------·
A| · · 3 | · · · | · · · | A| 1 2 3 | 4 5 6 | 7 8 9 |
B| 4 · · | · · · | · 2 · | B| 4 5 6 | 7 8 9 | 1 2 3 |
C| · 8 · | 1 2 · | · · 6 | C| 7 8 9 | 1 2 3 | 4 5 6 |
·-------+-------+-------· ·-------+-------+-------·
D| · · · | · · · | · · · | D| 8 9 7 | 2 3 1 | 5 6 4 |
E| 2 · · | · 6 · | · · 7 | E| 2 3 1 | 5 6 4 | 8 9 7 |
F| · · · | 8 · 7 | · 3 1 | F| 5 6 4 | 8 9 7 | 2 3 1 |
·-------+-------+-------· ·-------+-------+-------·
G| · 1 · | 6 4 · | 9 · · | G| 3 1 2 | 6 4 5 | 9 7 8 |
H| 6 · 5 | · · 8 | · · · | H| 6 4 5 | 9 7 8 | 3 1 2 |
J| 9 · 8 | 3 · · | · 4 · | J| 9 7 8 | 3 1 2 | 6 4 5 |
·-------·-------·-------· ·-------·-------·-------·
Proper: Having a single, unique solution
Irreducible: Removing any clue would make the resulting puzzle no longer proper
Disjoint: Having no elements in common
I found these by running some Python code.
Note: The first is very, very difficult, but the second is extremely easy!
answered 4 hours ago
Jonathan AllanJonathan Allan
18.3k1 gold badge47 silver badges99 bronze badges
$endgroup$
$begingroup$
The diagram looks very pretty. +1 :)
$endgroup$
– Mr Pie
57 mins ago
add a comment |
$begingroup$
Here are two proper, irreducible sudoku with the same solution as each other and disjoint sets of clues (24 & 25 clues, respectively).
1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9
·-------·-------·-------· ·-------·-------·-------·
A| · · · | 4 · · | 7 · · | A| 1 2 3 | 4 5 6 | 7 8 9 |
B| · · 6 | · 8 · | 1 · · | B| 4 5 6 | 7 8 9 | 1 2 3 |
C| 7 · · | · · 3 | · 5 · | C| 7 8 9 | 1 2 3 | 4 5 6 |
·-------+-------+-------· ·-------+-------+-------·
D| 8 · 7 | · 3 · | · · 4 | D| 8 9 7 | 2 3 1 | 5 6 4 |
E| · · 1 | · · · | · · · | E| 2 3 1 | 5 6 4 | 8 9 7 |
F| · 6 · | · · · | 2 · · | F| 5 6 4 | 8 9 7 | 2 3 1 |
·-------+-------+-------· ·-------+-------+-------·
G| 3 · · | · · 5 | · · 8 | G| 3 1 2 | 6 4 5 | 9 7 8 |
H| · 4 · | 9 · · | · · 2 | H| 6 4 5 | 9 7 8 | 3 1 2 |
J| · · · | · 1 · | 6 · · | J| 9 7 8 | 3 1 2 | 6 4 5 |
·-------·-------·-------· ·-------·-------·-------·
1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9
·-------·-------·-------· ·-------·-------·-------·
A| · · 3 | · · · | · · · | A| 1 2 3 | 4 5 6 | 7 8 9 |
B| 4 · · | · · · | · 2 · | B| 4 5 6 | 7 8 9 | 1 2 3 |
C| · 8 · | 1 2 · | · · 6 | C| 7 8 9 | 1 2 3 | 4 5 6 |
·-------+-------+-------· ·-------+-------+-------·
D| · · · | · · · | · · · | D| 8 9 7 | 2 3 1 | 5 6 4 |
E| 2 · · | · 6 · | · · 7 | E| 2 3 1 | 5 6 4 | 8 9 7 |
F| · · · | 8 · 7 | · 3 1 | F| 5 6 4 | 8 9 7 | 2 3 1 |
·-------+-------+-------· ·-------+-------+-------·
G| · 1 · | 6 4 · | 9 · · | G| 3 1 2 | 6 4 5 | 9 7 8 |
H| 6 · 5 | · · 8 | · · · | H| 6 4 5 | 9 7 8 | 3 1 2 |
J| 9 · 8 | 3 · · | · 4 · | J| 9 7 8 | 3 1 2 | 6 4 5 |
·-------·-------·-------· ·-------·-------·-------·
Proper: Having a single, unique solution
Irreducible: Removing any clue would make the resulting puzzle no longer proper
Disjoint: Having no elements in common
I found these by running some Python code.
Note: The first is very, very difficult, but the second is extremely easy!
answered 4 hours ago
Jonathan AllanJonathan Allan
18.3k1 gold badge47 silver badges99 bronze badges
$endgroup$
Here are two proper, irreducible sudoku with the same solution as each other and disjoint sets of clues (24 & 25 clues, respectively).
1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9
·-------·-------·-------· ·-------·-------·-------·
A| · · · | 4 · · | 7 · · | A| 1 2 3 | 4 5 6 | 7 8 9 |
B| · · 6 | · 8 · | 1 · · | B| 4 5 6 | 7 8 9 | 1 2 3 |
C| 7 · · | · · 3 | · 5 · | C| 7 8 9 | 1 2 3 | 4 5 6 |
·-------+-------+-------· ·-------+-------+-------·
D| 8 · 7 | · 3 · | · · 4 | D| 8 9 7 | 2 3 1 | 5 6 4 |
E| · · 1 | · · · | · · · | E| 2 3 1 | 5 6 4 | 8 9 7 |
F| · 6 · | · · · | 2 · · | F| 5 6 4 | 8 9 7 | 2 3 1 |
·-------+-------+-------· ·-------+-------+-------·
G| 3 · · | · · 5 | · · 8 | G| 3 1 2 | 6 4 5 | 9 7 8 |
H| · 4 · | 9 · · | · · 2 | H| 6 4 5 | 9 7 8 | 3 1 2 |
J| · · · | · 1 · | 6 · · | J| 9 7 8 | 3 1 2 | 6 4 5 |
·-------·-------·-------· ·-------·-------·-------·
1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9
·-------·-------·-------· ·-------·-------·-------·
A| · · 3 | · · · | · · · | A| 1 2 3 | 4 5 6 | 7 8 9 |
B| 4 · · | · · · | · 2 · | B| 4 5 6 | 7 8 9 | 1 2 3 |
C| · 8 · | 1 2 · | · · 6 | C| 7 8 9 | 1 2 3 | 4 5 6 |
·-------+-------+-------· ·-------+-------+-------·
D| · · · | · · · | · · · | D| 8 9 7 | 2 3 1 | 5 6 4 |
E| 2 · · | · 6 · | · · 7 | E| 2 3 1 | 5 6 4 | 8 9 7 |
F| · · · | 8 · 7 | · 3 1 | F| 5 6 4 | 8 9 7 | 2 3 1 |
·-------+-------+-------· ·-------+-------+-------·
G| · 1 · | 6 4 · | 9 · · | G| 3 1 2 | 6 4 5 | 9 7 8 |
H| 6 · 5 | · · 8 | · · · | H| 6 4 5 | 9 7 8 | 3 1 2 |
J| 9 · 8 | 3 · · | · 4 · | J| 9 7 8 | 3 1 2 | 6 4 5 |
·-------·-------·-------· ·-------·-------·-------·
Proper: Having a single, unique solution
Irreducible: Removing any clue would make the resulting puzzle no longer proper
Disjoint: Having no elements in common
I found these by running some Python code.
Note: The first is very, very difficult, but the second is extremely easy!
answered 4 hours ago
Jonathan AllanJonathan Allan
18.3k1 gold badge47 silver badges99 bronze badges
edited 4 hours ago
answered 4 hours ago
Jonathan AllanJonathan Allan
18.3k1 gold badge47 silver badges99 bronze badges
answered 4 hours ago
Jonathan AllanJonathan Allan
18.3k1 gold badge47 silver badges99 bronze badges
answered 4 hours ago
Jonathan AllanJonathan Allan
18.3k1 gold badge47 silver badges99 bronze badges
18.3k1 gold badge47 silver badges99 bronze badges
$begingroup$
The diagram looks very pretty. +1 :)
$endgroup$
– Mr Pie
57 mins ago
add a comment |
$begingroup$
The diagram looks very pretty. +1 :)
$endgroup$
– Mr Pie
57 mins ago
$begingroup$
The diagram looks very pretty. +1 :)
$endgroup$
– Mr Pie
57 mins ago
$begingroup$
The diagram looks very pretty. +1 :)
$endgroup$
– Mr Pie
57 mins ago
add a comment |
agc is a new contributor. Be nice, and check out our Code of Conduct.
agc is a new contributor. Be nice, and check out our Code of Conduct.
agc is a new contributor. Be nice, and check out our Code of Conduct.
agc is a new contributor. Be nice, and check out our Code of Conduct.
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$begingroup$
A few pictures would probably help.
$endgroup$
– agc
9 hours ago
$begingroup$
What does "and started from a minimal set of clues" mean?
$endgroup$
– Jonathan Allan
9 hours ago
$begingroup$
@JonathanAllan, It means no redundant clues, where any clue removed would make the answer insoluble or diverge to more than one solution. So
33 > clues > 16, as implied by Wikipedia.$endgroup$
– agc
5 hours ago