## Minimal Futoshiki/Sudoshiki Puzzles

For fans of Killer Sudoku, Samurai Sudoku and other variants

### Re: Minimal Futoshiki/Sudoshiki Puzzles

.
Some further observations on blue's two 17-clue Sudoshiki puzzles:

• a standard Sudoku puzzle on the first solution grid would require at least 18 givens
• a standard Sudoku puzzle on the second solution grid would require at least 19 givens!

Pretty amazing, really.

Mathimagics
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Location: Canberra

### Re: Minimal Futoshiki/Sudoshiki Puzzles

ryokousha and I have been investigating minimal thermo sudoku for a while - very much like this "Sudoshiki" but not restricted to orthogonally adjacent cells.

The best known in terms of cell count is 21 cells: Cascade
This comes out to 18 "clues" in terms of the greater-than relationships (but with some of these diagonal). ryokousha has gotten down to 2 solutions with 20 cells - the fact that 17-clue Sudoshiki exist makes me believe 20 cells is likely possible, trading the orthogonal restriction for the connectedness restriction of only three thermos.

We also have a puzzle with 18 "2CT" (2-cell thermos) here: https://f-puzzles.com/?id=yjy4qfna - that is, 18 clues (some - in fact almost all - diagonal) such that none of them are chained (one pair does share a bulb - that is, the cell is less than two other cells, but there is no direct relation between those two cells). I am less inclined to think 17 2CT is possible, based on the ridiculous geometry tricks we needed to pull of even 18, but can't rule it out either. It's worth noting that this one can be reduced to 16 2CT and 1 given:
Hidden Text: Show
the shared bulb is a 1
mith

Posts: 859
Joined: 14 July 2020

### Re: Minimal Futoshiki/Sudoshiki Puzzles

mith wrote:I am less inclined to think 17 2CT is possible, based on the ridiculous geometry tricks we needed to pull of even 18 ...

I'm inclined to agree ...

I do note that the first of blue's two 17-clue Sudoshikis has only 18 total cells in thermo terms (6 x 2CT, 2 x 3CT)

Mathimagics
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Location: Canberra

### Re: Minimal Futoshiki/Sudoshiki Puzzles

Mathimagics wrote:
mith wrote:I am less inclined to think 17 2CT is possible, based on the ridiculous geometry tricks we needed to pull of even 18 ...

I'm inclined to agree ...

I do note that the first of blue's two 17-clue Sudoshikis has only 18 total cells in thermo terms (6 x 2CT, 2 x 3CT)

I think you're forgetting the 9CT there.
mith

Posts: 859
Joined: 14 July 2020

### Re: Minimal Futoshiki/Sudoshiki Puzzles

For some reason my grid-rendering function has been infected with something (COVID?) ... it's playing "hide -the-chain" !

Mathimagics
2017 Supporter

Posts: 1901
Joined: 27 May 2015
Location: Canberra

### Re: Minimal Futoshiki/Sudoshiki Puzzles

mith wrote:ryokousha and I have been investigating minimal thermo sudoku for a while - very much like this "Sudoshiki" but not restricted to orthogonally adjacent cells.

The best known in terms of cell count is 21 cells: Cascade
This comes out to 18 "clues" in terms of the greater-than relationships (but with some of these diagonal). ryokousha has gotten down to 2 solutions with 20 cells - the fact that 17-clue Sudoshiki exist makes me believe 20 cells is likely possible, trading the orthogonal restriction for the connectedness restriction of only three thermos.

Hi mith,

I looked through the 18c puzzles that I have, and found two "nice" (and 6 total) 21-cell puzzles.
They both have 3 isolated (bent up) "thermos" with lengths of 9,8 and 4.

I don't have a very good collection -- generated from only a few good "seeds".
With that, I wouldn't be surprised if a 20-cell Sudoshiki exists.

#1
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`18c, 21 cells.................................................................................----<--------<--------<-----------<<-----<------<-----------------<----------->------>------>--<------->--------<--------<--------<<<-----------+-----------+-----------+-----------+    +-----------+-----------+-----------+| .   .   . | .   . < . | .   .   . |    | 4   5   9 | 6   1 < 2 | 7   3   8 ||           |         ^ |           |    |           |         ^ |           || .   .   . | .   .   . < .   .   . |    | 6   7   1 | 5   8   3 < 4   2   9 ||           |           | ^         |    |           |           | ^         || .   .   . | .   .   . | . < .   . |    | 8   2   3 | 4   7   9 | 5 < 6   1 |+-----------+-----------+---- ^ ----+    +-----------+-----------+---- ^ ----+| .   .   . | .   .   . | .   .   . |    | 9   8   2 | 3   4   1 | 6   7   5 ||           |           |     ^     |    |           |           |     ^     || .   .   . < . < .   . | .   .   . |    | 3   1   6 < 7 < 9   5 | 2   8   4 ||         v |           |     ^     |    |         v |           |     ^     || .   . < . | .   .   . | .   .   . |    | 7   4 < 5 | 2   6   8 | 1   9   3 |+---- v ----+-----------+-----------+    +---- v ----+-----------+-----------+| . < .   . | .   .   . | .   .   . |    | 2 < 3   8 | 1   5   6 | 9   4   7 || v         |           |           |    | v         |           |           || .   .   . | .   .   . | .   .   . |    | 1   6   4 | 9   3   7 | 8   5   2 ||         ^ | v         |           |    |         ^ | v         |           || .   .   . < .   .   . | .   .   . |    | 5   9   7 < 8   2   4 | 3   1   6 |+-----------+-----------+-----------+    +-----------+-----------+-----------+459612738671583429823479561982341675316795284745268193238156947164937852597824316`

#2
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`18c, 21 cells.................................................................................---->-------->-------->---<----->---------------->-------->>------------------------>>----<--->----<----------->>-------->-------->>------------+-----------+-----------+-----------+    +-----------+-----------+-----------+| .   .   . | .   . > . | .   .   . |    | 3   5   8 | 4   9 > 7 | 2   6   1 ||           |         v |           |    |           |         v |           || .   .   . | .   .   . > .   .   . |    | 4   2   7 | 1   3   6 > 5   8   9 ||           |           | v         |    |           |           | v         || .   .   . | .   .   . | . > .   . |    | 6   9   1 | 8   5   2 | 4 > 3   7 |+-------- ^ +-----------+---- v ----+    +-------- ^ +-----------+---- v ----+| .   .   . < .   .   . | .   .   . |    | 1   4   3 < 5   6   9 | 7   2   8 ||           | ^         |     v     |    |           | ^         |     v     || . > .   . | .   .   . | .   .   . |    | 9 > 8   2 | 7   4   3 | 6   1   5 ||     v     |           |           |    |     v     |           |           || .   .   . | .   .   . | .   .   . |    | 5   7   6 | 2   8   1 | 3   9   4 |+---- v ----+-----------+-----------+    +---- v ----+-----------+-----------+| .   . > . | .   .   . | .   .   . |    | 2   6 > 5 | 9   7   8 | 1   4   3 ||         v |           |           |    |         v |           |           || .   .   . > . > .   . | .   .   . |    | 8   1   4 > 3 > 2   5 | 9   7   6 ||           |     v     |           |    |           |     v     |           || .   .   . | .   .   . | .   .   . |    | 7   3   9 | 6   1   4 | 8   5   2 |+-----------+-----------+-----------+    +-----------+-----------+-----------+358497261427136589691852437143569728982743615576281394265978143814325976739614852`

#3 (forked (>) chains)
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`18c, 21 cells.................................................................................------------<<--------<-----------<--->--<-------<---------<-----<<------------------->----->>--------->--------<--------<--------<<------------+-----------+-----------+-----------+    +-----------+-----------+-----------+| .   .   . | .   .   . | .   .   . |    | 4   7   2 | 5   6   1 | 8   3   9 ||           |         ^ |           |    |           |         ^ |           || .   .   . | .   . < . < .   .   . |    | 5   6   8 | 9   2 < 3 < 4   1   7 ||           |           | ^         |    |           |           | ^         || .   .   . | .   .   . | . < .   . |    | 3   9   1 | 4   7   8 | 5 < 6   2 |+-----------+-----------+---- ^ ----+    +-----------+-----------+---- ^ ----+| .   .   . | .   .   . | .   .   . |    | 1   8   9 | 3   5   6 | 2   7   4 ||           |           |     ^     |    |           |           |     ^     || .   .   . < .   .   . | . > .   . |    | 2   5   6 < 7   1   4 | 9 > 8   3 ||         v |           |           |    |         v |           |           || .   . < . | .   .   . | .   .   . |    | 7   3 < 4 | 2   8   9 | 6   5   1 |+-------- v +-----------+-----------+    +-------- v +-----------+-----------+| .   . < . | .   .   . | .   .   . |    | 8   2 < 3 | 1   4   5 | 7   9   6 ||     v     |           |           |    |     v     |           |           || .   .   . | . < .   . | .   .   . |    | 6   1   7 | 8 < 9   2 | 3   4   5 ||           | v         |           |    |           | v         |           || .   . < . < .   .   . | .   .   . |    | 9   4 < 5 < 6   3   7 | 1   2   8 |+-----------+-----------+-----------+    +-----------+-----------+-----------+472561839568923417391478562189356274256714983734289651823145796617892345945637128`

Cheers,
Blue.
blue

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### Re: Minimal Futoshiki/Sudoshiki Puzzles

Very nice finds
mith

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Joined: 14 July 2020

### Re: Minimal Futoshiki/Sudoshiki Puzzles

mith wrote:We also have a puzzle with 18 "2CT" (2-cell thermos) here - that is, 18 clues (some - in fact almost all - diagonal) such that none of them are chained (one pair does share a bulb - that is, the cell is less than two other cells, but there is no direct relation between those two cells). I am less inclined to think 17 2CT is possible, based on the ridiculous geometry tricks we needed to pull of even 18, but can't rule it out either. It's worth noting that this one can be reduced to 16 2CT and 1 given ...

Perhaps that puzzle should be called a "16 x 2C + 3C", rather than "18 x 2C" puzzle. A "transitive" 3-cell set ("sharing a bulb") might not be as strong a constraint as a 3-cell chain, but it is stronger than a 2-cell chain.

So, before we look at whether 17 x 2C is possible or not, we perhaps need to ask whether in fact there is a (true) "18 x 2C"?

By the way, I note that the "16 x 2C" without the given, has 1,140,537,139 solutions (according to sigh's ISS solver). I don't think that I have ever seen such a huge reduction in a solution space with 1 given.

Cheers
MM

Mathimagics
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### Re: Minimal Futoshiki/Sudoshiki Puzzles

I agree it's not ideal, though I don't know that we can say definitively that a shared-bulb is always stronger than *two* 2CTs.

This particular puzzle was the result of a search on all morphs of the setup of bulbs and tips - b134679 have two orientations (whether the thermos follow positive or negative diagonals), plus whether the bulbs in b28 align or not - followed by adding one more 2CT in all possible ways. This was the only "18 2CT" found. Of course, that doesn't mean there's no way to do it without sharing a bulb... but the setup is *very* restrictive, and we really only came up with one other setup that even came close (yielding a 19 2CT - and it's not *that* different, it has the same structure in b134679 along with c45, just something different to force things with c6).

What I may try at some point is an exhaustive search on the 15 2CT "setup" shared by the two 19 2CT found + three more thermos in all possible locations, maybe we're missing some slightly different way to handle c6 that this will catch. Otherwise, we need something substantially different.
mith

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Joined: 14 July 2020

### Re: Minimal Futoshiki/Sudoshiki Puzzles

blue wrote:Hi mith,

I looked through the 18c puzzles that I have, and found two "nice" (and 6 total) 21-cell puzzles.
They both have 3 isolated (bent up) "thermos" with lengths of 9,8 and 4.

I don't have a very good collection -- generated from only a few good "seeds".
With that, I wouldn't be surprised if a 20-cell Sudoshiki exists.

I've found a few 20-celll puzzle, now.
Here are two of the nicer ones.

#1 - 18c, one clean, and one forked '>' chain:
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`.................................................................................---------->>----->>------------->-------------->------>------>--------------------<<-----<-------->>------------------>-------<------<-----<<---+-----------+-----------+-----------+    +-----------+-----------+-----------+| .   .   . | .   .   . | .   .   . |    | 1   3   8 | 9   5   7 | 4   2   6 ||           |           |           |    |           |           |           || .   .   . > . > .   . | .   .   . |    | 7   9   4 > 2 > 1   6 | 3   8   5 ||         ^ |           |           |    |         ^ |           |           || .   . > . > .   .   . | .   .   . |    | 2   6 > 5 > 4   8   3 | 1   9   7 |+---- ^ ----+ v --------+-----------+    +---- ^ ----+ v --------+-----------+| .   .   . | .   .   . | .   .   . |    | 4   7   2 | 3   9   5 | 8   6   1 ||     ^     | v         |         ^ |    |     ^     | v         |         ^ || . > .   . | .   .   . | .   .   . |    | 9 > 8   3 | 1   6   4 | 5   7   2 ||           |           |         ^ |    |           |           |         ^ || .   .   . | .   .   . | .   . > . |    | 5   1   6 | 8   7   2 | 9   4 > 3 |+-----------+-----------+---- ^ ----+    +-----------+-----------+---- ^ ----+| .   .   . | .   .   . | . > .   . |    | 3   4   1 | 7   2   9 | 6 > 5   8 ||           |         v | ^         |    |           |         v | ^         || .   .   . | .   .   . > .   .   . |    | 6   2   9 | 5   3   8 > 7   1   4 ||           |           |           |    |           |           |           || .   .   . | .   .   . | .   .   . |    | 8   5   7 | 6   4   1 | 2   3   9 |+-----------+-----------+-----------+    +-----------+-----------+-----------+138957426794216385265483197472395861983164572516872943341729658629538714857641239`

#2 - 17c, 3 clean chains
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`.................................................................................---------->>----->------>------------------>--->------>------>------------>-------<------<-----------<--<------------->-------<------<------<---+-----------+-----------+-----------+    +-----------+-----------+-----------+| .   .   . | .   .   . | .   .   . |    | 2   3   7 | 9   1   5 | 4   8   6 ||           |     ^     |           |    |           |     ^     |           || .   .   . > . > .   . | .   .   . |    | 1   8   4 > 3 > 2   6 | 5   9   7 ||         ^ |           |           |    |         ^ |           |           || .   . > . | .   .   . | .   .   . |    | 9   6 > 5 | 8   7   4 | 2   1   3 |+ v - ^ ----+-----------+-----------+    + v - ^ ----+-----------+-----------+| . > .   . | .   .   . | .   .   . |    | 8 > 7   2 | 4   9   1 | 3   6   5 ||           |           |           |    |           |           |           || .   .   . | .   .   . | .   .   . |    | 5   4   6 | 2   8   3 | 9   7   1 ||           |           |         ^ |    |           |           |         ^ || .   .   . | . > .   . | .   . > . |    | 3   1   9 | 6 > 5   7 | 8   4 > 2 |+-----------+ ^ --------+---- ^ ----+    +-----------+ ^ --------+---- ^ ----+| .   .   . | .   .   . | . > .   . |    | 4   2   1 | 7   3   9 | 6 > 5   8 ||           |         v | ^         |    |           |         v | ^         || .   .   . | .   .   . > .   .   . |    | 6   5   3 | 1   4   8 > 7   2   9 ||           |           |           |    |           |           |           || .   .   . | .   .   . | .   .   . |    | 7   9   8 | 5   6   2 | 1   3   4 |+-----------+-----------+-----------+    +-----------+-----------+-----------+237915486184326597965874213872491365546283971319657842421739658653148729798562134`
blue

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### Re: Minimal Futoshiki/Sudoshiki Puzzles

Confirmed both of these examples ...

Great work, blue!

And all this with purely orthogonal chains (ie without resorting to diagonal adjacency) ...

Mathimagics
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### Re: Minimal Futoshiki/Sudoshiki Puzzles

Beautiful, nicely done blue

The 17c has a cool solve path, as well.
mith

Posts: 859
Joined: 14 July 2020

### Re: Minimal Futoshiki/Sudoshiki Puzzles

That 17c is pretty good the way it propagates the solution
eg
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`+---+---+---+|...|.1.|...||..4|32.|...||965|...|...|+---+---+---+|87.|...|...||...|...|..1||...|...|.32|+---+---+---+|...|..8|54.||...|..7|6..||...|...|...|+---+---+---+  has 0 solutions - a 1@r5c9 is forbidden somehow`

maybe this one is a good seed ?
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`+---+---+---+|...|...|.87||3..|...|.9.||5..|4..|...|+---+---+---+|..6|5..|...||.87|...|...||.9.|...|...|+---+---+---+|...|.98|...||...|.7.|32.||...|..1|4..|+---+---+---+  17C with 11 clues in the diagonal B348 -  with [optional] 2 at r8c8 `
coloin

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Location: Devon

### Re: Minimal Futoshiki/Sudoshiki Puzzles

ryokousha is using the 17c as a seed, and has 19 thermo cells down to 10 solutions already with diagonals allowed. We'll keep you posted
mith

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Joined: 14 July 2020

### Re: Minimal Futoshiki/Sudoshiki Puzzles

I got it down to 3 solutions, but not making any good progress on modifying it from here.

We do now have several examples with a pair of 9CTs plus one 2CT. For example: https://f-puzzles.com/?id=258rmyq5 (remove the r8c5 bulb for the 3 solution 19 cell)
mith

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Joined: 14 July 2020

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