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A fairly common Sudoku variant is so called "Thermometer" Sudoku. In this variant arrows with a rounded end (which look like a thermometer) are added to the grid, with the rule that the numbers on the thermometer must be strictly decreasing in the direction the arrow points. Thermometers can only move between squares that are directly adjacent (no diagonal moves or jumps).

Here is an example board:

Example board

The neat thing is that with thermometer's added to the mix it is no longer required for the puzzle to start with any numbers on the board.

The question here is with a single thermometer and no additional clues what is the largest portion of the puzzle that can be filled out. Your score is thus a number out of 81.

The board should have at least 1 solution with a perfect score having exactly one solution (fills all 81 squares). (Although it should be a fairly simple corollary of this paper that no such thermometer exists)

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    $\begingroup$ I've not seen this way of showing inequality chains before. It is much more compact than the separated cells normally used in futoshiki grids. I can see it would become a bit confusing if the thermometers cross or meet, but presumably the fact that it has the 3x3 sudoku constraints reduces the need for such crossings, unlike Fukoshiki. $\endgroup$ – Jaap Scherphuis Dec 31 '19 at 13:01
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My best solution so far is

12 squares filled

This is done with the following thermometer

enter image description here Since this thermometer is 9 squares long it must contain the numbers 1 through 9 in order enter image description here Now there is only one square in the central big square that can contain 9 enter image description here With that filled in there is only one location in the central big square that can contain a 1 enter image description here And the last square in the central big square must be an 8 enter image description here

Here are three other thermometers with the same score:

enter image description here enter image description here enter image description here

If you find any more thermometers that have this score and are not symmetric to any of the three in this post I would be happy to see them.

Symmetry here means any combination of:

  • Translation

  • Rotation

  • Mirroring

  • Reversing the direction of the thermometer.


Now for an upper bound

The first thing we will notice is that if there are is a pair of columns in the same super column (columns made up by the $3\times 3$ boxes), where neither intersects the thermometer, then for any solution of the board we can swap these two columns to get another solution.
This means we cannot deduce any numbers in such columns.
Note that the same can be said for rows as well.
Now let us consider the best case scenario where we can deduce all squares other than in such columns. For a given thermometer we can draw a "deduction block" which shows all the squares that cannot be ruled out given this method or the row version of this method. For example here is the deduction block (in light green) compared with the bounding box (in dark green) for the above solution:
enter image description here
From this we can tell that no more than 36 squares can be deduced from that thermometer.
Now I will point out that the deduction block is just a function of the bounding box, so we can just consider bounding boxes. And with a limit on 9 on thermometer length there are just 5 possible bounding boxes. So here are all of them with their best possible deduction blocks
9x1 8x2 7x3 6x4 5x5
Of these the $6 \times 4$ does the best with a deduction block of size 48. Meaning in the absolute best case scenario no more than 48 squares can be filled. It does seem quite high for a upper bound I would love to see someone lower this.

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