## Can You Solve This Without Trial and Error?

For fans of Kakuro

### Re: Can You Solve This Without Trial and Error?

saul wrote:Have a look at ATK's M84472. It seems to be another example of a ridiculously difficult medium puzzle. I've worked at it for close to two hours and still have cells with 6, 7, or even 8 candidates!

Have you found the naked quads in columns c4 and c6 ?
denis_berthier
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### Re: Can You Solve This Without Trial and Error?

Just to be sure, I'm assuming by c4 and c6, you mean segments with sums of 30 and 31, right?

No, I haven't found any quads, naked or clothed. Joking aside, "naked" quad means that there are 4 cells, all of whose candidates are elements of some 4-set, is that right? I don't see any of these yet. However, when I looked back at the puzzle, I saw some hidden singles (that is, required values that occurred only once as a candidate, if that's the right term) and made quite a bit of progress. I still haven't solved it, but at least I don't have any cell with more than 6 candidates now.

Maybe this puzzle isn't as hard as I thought. I seem to be having a bad day.
saul

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### Re: Can You Solve This Without Trial and Error?

saul wrote:Just to be sure, I'm assuming by c4 and c6, you mean segments with sums of 30 and 31, right?

saul wrote:"naked" quad means that there are 4 cells, all of whose candidates are elements of some 4-set, is that right?

4 cells in the same sector...
denis_berthier
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### Re: Can You Solve This Without Trial and Error?

I found one more puzzle that you should like, with lots of surface sums: H83722.
denis_berthier
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### Re: Can You Solve This Without Trial and Error?

denis_berthier wrote:I found one more puzzle that you should like, with lots of surface sums: H83722.

Yes, zillions of sum doubularities, one of which leads to an almost immediate singularity. Not to mention a global difference doubularity at r6c6, r7c7.

This H puzzle was a lot easier than that other M puzzle you guys pointed out.

Bill Smythe
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### Re: Can You Solve This Without Trial and Error?

Smythe Dakota wrote:This H puzzle was a lot easier than that other M puzzle you guys pointed out.

Yes. And it is decomposable, in the sense I've already discussed: each of the w, e, nw, ne and se sub-puzzles can easily be solved (by whips[2]) completely independently of the rest (this is not true of the sw sub-puzzle). Taking these partial results into account, the rest can easily be solved by g-whips[3].
However, if one ignores surface sums, it is in W10, which is hard.

It is thus an interesting example of how using surface sums (in the relatively rare cases when they are available) can drastically simplify a puzzle.

If you liked it, try also H83732 (similar pattern).
It seems that the atk generator creates a pattern of black cells and tries to fill it with different combinations of clues (sometimes after rotation or up/down symmetry). So there may be similar other cases between H83722 and H83732 (I didn't check).
denis_berthier
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### Re: Can You Solve This Without Trial and Error?

Take a look at M73901. It's another example of a puzzle with so few internal black cells, and such wide white paths, that one worries the solution may not be unique. It is, though.

The puzzle contains ten 8-digit words, no 7-digit words, only four 6-digit words, and only four 4-digit words.

Bill Smythe
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### Re: Can You Solve This Without Trial and Error?

Smythe Dakota wrote:Take a look at M73901. It's another example of a puzzle with so few internal black cells, and such wide white paths, that one worries the solution may not be unique. It is, though.
The puzzle contains ten 8-digit words, no 7-digit words, only four 6-digit words, and only four 4-digit words.

Interesting puzzle. It shows how difficult it is to rate a puzzle on its appearance. M73901 is in W2+BC3, i.e. it can be solved using only whips[2] plus a few bivalue-chains[3].
(Whips[2] are always easy patterns in Kakuro, each being based on a contradiction local to a single sector - except for the special case of X-Wings, not present here.)

I had a similar example (same pattern, rotated) with M73961.
denis_berthier
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### Re: Can You Solve This Without Trial and Error?

I think the essential reason this puzzle is so easy is the large number of uniques in the short segments. For example, 7 out of the 8 columns crossing the top row in the NW are uniques. And of course, an 8-cell segment is always unique, so that cuts out even more possibilities.
saul

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### Re: Can You Solve This Without Trial and Error?

saul wrote:I think the essential reason this puzzle is so easy is the large number of uniques in the short segments. For example, 7 out of the 8 columns crossing the top row in the NW are uniques. And of course, an 8-cell segment is always unique, so that cuts out even more possibilities.

It'd be interesting to see if all the puzzles with this pattern of black cells (some of those between M73901 and M73961 ???) are as easy. But I'm already busy with too many things at the same time (and making errors in my analyses of Slitherlink puzzles).
denis_berthier
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### Re: Can You Solve This Without Trial and Error?

I know what you mean. I've got to learn the Go programming language in the next few days.
saul

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### Re: Can You Solve This Without Trial and Error?

denis_berthier wrote:It'd be interesting to see if all the puzzles with this pattern of black cells (some of those between M73901 and M73961 ???) are as easy.

I've finally looked for other similar puzzles in the atk collection. Between two puzzle numbers, there can be no puzzle. I found the following 6 with the same pattern:
M73901, M73902, M73961, M73962, M73971, M73972

But they are essentially the same puzzle modulo rotation and/or symmetry and/or the following transformation:
- for each sector of size s, replace the sector sum S by 10*s - S (this obviously replaces every solution number n by 10-n).
denis_berthier
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### Re: Can You Solve This Without Trial and Error?

Hmm, dirty trick. But who's going to notice?

Bill Smythe
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### Re: Can You Solve This Without Trial and Error?

Smythe Dakota wrote:Hmm, dirty trick. But who's going to notice?

We.
After checking the atk collection I've manually copied and solved with CSP-Rules (several hundred puzzles), such types of variants appear often.

On the other hand, if you find (essentially) different puzzles with the same patterns of black cells, I'm curious about it.
denis_berthier
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### Re: Can You Solve This Without Trial and Error?

denis_berthier wrote:
On the other hand, if you find (essentially) different puzzles with the same patterns of black cells, I'm curious about it.

I find it interesting that you apparently haven't found such examples. I would build a collection like this by using black-cell patterns from manually-constructed puzzles. Then I would randomly assign numbers to the cells and test whether my solver could solve it using whatever inference rules I had programmed. I would rate the puzzles according to the difficulty I perceived in applying those rules. (I might allow some limited trial-and-error too, but in that case, one has to do exhaustive search to ensure there is a unique solution.)

I have noticed that there seem to be more patterns than this would suggest, but if you've been systematically checking and have only found the same pattern of black cells for equivalent puzzles, then they must be using an entirely different approach. Of course, you didn't say that you have been systematically checking, did you? .
saul

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