The New Sudoku Players' Forum

[enxio27]

This is really the first greater-than-less-than puzzle I've worked. I get the basic idea (that the arrows point to the lower numbers), but there must be some techniques that are unique to GTLT that I haven't discovered yet.

I don't know of any software that can handle these puzzles, either, so no way of getting any hints. (Unfortunately, even though I can import the candidates into Richard Piarina's SudokuSolver and print/save the puzzle, I can't put the arrows in there, much less use the program to solve it.)

This is supposed to be an easy puzzle, but I've made very little headway on it. Here's the puzzle and what I've done so far (along with LCs on 1r2c2/r2c3, 9r7c1/r9c1 and 1r8c8/r9c8). Any ideas on how to proceed?

[Marty R.]

enxio27,

I know I encouraged you to post your problem puzzles and here you've received no responses. Most Sudoku players I know are quick to help others. Obviously speaking only for myself, I have no idea what this puzzle is or what to do with it. It has a 9x9 grid but otherwise doesn't look like the Sudoku puzzles that I'm familiar with.

Wish I could help.

[blue]

Hello enxio27,

It looks like you're on the right track, just missing some of the eliminations.
I get this, after doing the initial "GTLT" eliminations

gtlt1.jpg

Then this, after doing normal sudoku things with the 1's, and updating the GTLT eliminations.
Edit: After eleven's post, I see I forgot to handle the details in in box 2
... eliminating 1's and updateing GTLT stuff.

gtlt2.jpg

A general rule, is that when you have this: (Edit: reworded)

A < B ... A,B are cells
'm' is the smallest candidate in A
'n' is the largest candidate in B.

You can eliminate:

all candidates 'm' or smaller in cell B
all candidates 'n' or larger in cell A.

[David P Bird]

There is one trick missing from Blue's initial eliminations.

Looking at r8c5, it must be less than 4 cells, r7c5, r8c4, r9c5 & r9c6 so it can't hold a digit higher than 5
Similarly r4c2 must be greater than 4 cells, r456c1 & r5c2, so can't hold a digit lower than 5

This simple counting exercise is only valid at the start.
Should the GTLT markers extend across box boundaries in harder puzzles, then the counts should be confined to individual houses.

(I'm not sure if this is the right thread for posting puzzles for variants.)
Topic moved to Sudoku Variants area. JasonLion-Admin

[enxio27]

After I posted, I reprinted my puzzle progress (helps to make some of the pencil marks less obscure). Then I started noticing chains of greater-than (> > > >) and less-than (< < < <), and found quite a few eliminations that way. I haven't yet had time since then to complete the puzzle, but I think I'm on my way.

[eleven]

There is one trick missing from Blue's initial eliminations.

Also simply note that having the 1 in r3c5 the 2 cannot be in r1c4 (which gives you a number).

[blue]

There is one trick missing from Blue's initial eliminations.

Good point.

I think you can use it (carefully), at points beyond the initial eliminations too.
As you say ... only when the <> relations are between cells of the same house.

A < B, A < C ... A,B,C unfilled cells in the same house.
n1,n2 -- the largest unplaced digits in the house.
==> A < min(B,C) <= min(n1,n2)
Eliminate anything >= the smaller of n1,n2, in cell A.

That isn't the end of it though.

Normally in cell A, you can eliminate anything >= the largest candidate in B, and anything >= the largest candidate in C.
When the largest candidates are different, I think that takes you as far as you can go (?).
When they're the same value ... 'n' ... you can eliminate anything >= (n-1) in cell A.

Added: That still isn't the end of things: the (n-1) above, can be replaced by the larger of the "2nd largest candidates" in the two cells. Adding more cells like B and C, would make things even more complicated.

[David P Bird]

Added: That still isn't the end of things: the (n-1) above, can be replaced by the larger of the "2nd largest candidates" in the two cells. Adding more cells like B and C, would make things even more complicated.

Yes, I could see that there would be scope for combining the different inequality chains in the way you describe, but without actually trying to solve the puzzle, guessed it probably wouldn't be needed – hence my "simple" counting qualifier.

These puzzles will tend to be solved progressively from the extreme digits (1) and (9) towards the middle, with (5) probably being the last one to solve. The early regular Sudoku based eliminations will therefore primarily be single digit ones – naked and hidden tuples, box-line eliminations, simple colouring & fish. This gives me the impression that solving these puzzles will largely be mechanical, and therefore less interesting.

David

[denis_berthier]

Hi all,

This puzzle is a combination of Futoshiki and Sudoku. You can therefore apply the rules of both.
For Futoshiki, you can find a general detailed formulation of the rules for ascending-chains, hills and valleys (classical rules, no standard names) in my last book (http://arxiv.org/abs/1304.1628, chapter 14).
Note that there's no reason for not having ascending-chains across block boundaries and there's no reason for limiting their application to the start of the puzzle.

If you're interested in pure Futoshiki puzzles (with no block constraint), the best place I've found is http://www.atksolutions.com/flashgames.html

[eleven]

... but without actually trying to solve the puzzle, guessed it probably wouldn't be needed – hence my "simple" counting qualifier.

These puzzles will tend to be solved progressively from the extreme digits (1) and (9) towards the middle, with (5) probably being the last one to solve. The early regular Sudoku based eliminations will therefore primarily be single digit ones – naked and hidden tuples, box-line eliminations, simple colouring & fish. This gives me the impression that solving these puzzles will largely be mechanical, and therefore less interesting.

Well, i tried it on paper last night, but was stuck rather early having the 1's and some 2-4s. Probably i have overlooked something, i did not count all the cells, just filled in some cells with up to 2 or 3 candidates left.
But i saw that there can be also more tricky eliminations than blue had described. Something like "3 of the 5 cells have to be 234, but if this one is 5, only 2 are left".

It's too cumbersome for me to bring my grid into a post-able form, but maybe enxio27 can post an interesting situation.

[enxio27]

A < B, A < C ... A,B,C unfilled cells in the same house.
n1,n2 -- the largest unplaced digits in the house.
==> A < min(B,C) <= min(n1,n2)
Eliminate anything >= the smaller of n1,n2, in cell A.

That isn't the end of it though.

Normally in cell A, you can eliminate anything >= the largest candidate in B, and anything >= the largest candidate in C.
When the largest candidates are different, I think that takes you as far as you can go (?).
When they're the same value ... 'n' ... you can eliminate anything >= (n-1) in cell A.

That's a good description of what I was referring to as GT chains and LT chains.

This gives me the impression that solving these puzzles will largely be mechanical, and therefore less interesting.

I don't see it as mechanical or uninteresting--just requiring some different ways of solving that aren't found in vanilla puzzles (just as jigsaw and killer have some solving techniques that are unique to them). For me, the simple fact that it isn't a vanilla puzzle and does require unusual solving strategies adds enough interest.

[blue]

Well, i tried it on paper last night, but was stuck rather early having the 1's and some 2-4s. Probably i have overlooked something, i did not count all the cells, just filled in some cells with up to 2 or 3 candidates left.
But i saw that there can be also more tricky eliminations than blue had described. Something like "3 of the 5 cells have to be 234, but if this one is 5, only 2 are left".

I was able to solve it using the simple rule from my first post, along with locked candidates and naked pairs.
I thought it was a "fun" puzzle ... nothing too complex, and not too easy either.

[eleven]

Ah, thats fine. When i wanted to finish it now (i.e. 3 hours ago!), i only made little progress. So i did the work to copy it to an editor.
I found a triple, 2 x-wings and a skyscraper, but am stuck again. So it seems, that i am blind for something. Can you give me a hint here, please.
(hope it does not spoil enxio's fun, so lets hide question and answer)

Hidden Text: Show

Code: Select all

+-------------------------------+-------------------------------+-------------------------------+
|    4   <   56789  >    2      |   3    <    56    <    678    |    1         678  <     89    |
     v                   v          v          ^          ^                     v         v
|   356      356789      1      |   2     <   78    <    89     |    4   <     56   <    678    |
     ^                   ^                     v                     v
|   567   <   678   <   789     |   4          1         56789  |    3          2       56789   |
+-------------------------------+-------------------------------+-------------------------------+
|  35678  <   56789   3456789   |   789   >    2    <    45678  |   5678  <   6789   >    1     |
     v          v                    v         ^          v                    v          ^
|    2    <   4567      5678    |   67    <   789   >     1     |  56789      4567   >    3     |
     v          ^                    v         v          ^                    ^          ^
|    1    <   5678  <   6789    |   56    >   45    >     3     |    2        6789   >   4678   |
+-------------------------------+-------------------------------+-------------------------------+
|  5678        2        4567    |    1        789       45678   |   789   >    3    <    5678   |
                         ^                     v                     v                    v
|  5678   >    1    <   5678    |   6789   >   3    >     2     |   678      456789      4567   |
     ^         ^                      v        ^          ^          v                    v 
|    9    >  345678    35678    |   5678   >  456   <    5678   |   567   >    1    <     2     |
+-------------------------------+-------------------------------+-------------------------------+

[blue]

Hi eleven,

This should get you going again ...

Hidden Text: Show

From r1c1 > r2c1 and r1c1=4, you can eliminate 56r2c1, and get a single.
This is a special case of that "simple rule", I guess ... since r2c1 is a filled cell.
(It "fits", if you consider it as a cell whose only candidate is '4').

[eleven]

Oh many thanks, so simple, i just did not look there any more. I am happy, that i asked instead of plagueing my eyes 2 more hours.