Hi everyone,

this week provided two nice extra puzzles:

[url]is.gd/290_one_more_more_str8t[/url]

and

[url]is.gd/290_one_more_more_str8t[/url]

After still not having found THE simple way for that one, this week's puzzle with its symmetric grid is a good opportunity to talk about symmetries in str8ts as mentioned some days earlier.

In general:

There's the flip symmetry in vertical and horizontal direction: If you flip both grid and solution, you will also get a valid str8ts solution. However, there's no non-trivial (white cells in row E only) solution that is symmetric to itself owing to the rules of the games. Given the hints, you can be sure that the flipped version violates. Given a symmetric grid, there needs to be a hint somewhere off-central for uniqueness.

There's a symmetry in the numbers ('global hi/low'): If you flip all numbers (1->9, 2->8,...) of a solution, you will also get a valid str8ts. Again, the hints will only allow for one of the two. Follows that there's no unique solution with just digit 5 as hints.

There's a point reflection symmetry: With symmetric grid (as here), you also get a valid solution by turning your solution 180 degree. Again, there's no all symmetric solution itself (except row E and col 5 empty).

There's also reflection symmetry in diagonal direction: Same arguments. But this time there might be an all symmetric solution.

The symmetries and the uniqueness considerations give rise to a (my oppinon new) str8ts rule: Given a symmetric grid and all symmetric hints (in the current puzzle e.g. B2=H8=2, C7=G3=1) there cannot be a unique solution (nothing to determine the asymetric part). In turn, given a symmetric grid and hints being symmetrical but one (e.g. only D6=1 or D6=1 and B2=H8=2), you can delete the unsymmetric hint digit directly from the corresponding cell as candidate (here F4<>1). Very seldomly applicable, but might be a very powerful tool for solving. If in both versions the corresponding symmetric hints were given in addition, solving becomes extremely neat whenapplying that rule (maybe a plea to Klaus...).

In the special case of this puzzle:

As mentined before, the grid is a very restrictive one, see the 4-cell hi/low interactions and many more. A lot of inner strings that reduce the number of hints needed. For example, there's a lot of conclusion even w/o any hints given. HJ3 and AB7 automatically become 23 (78) and AB4 HJ6 78(23). Furthermore, you can easily show (disproving via solver with deciding on hi/low before) that E5=3(7), setti 3,4,6,7, no 5 in A2-8 and J2-8, the well-known 7-fish, maybe also more (like no 5 in the outer ring at all).

If you see the solutions, there's a lot of symmetric fields (for both versions): A123478,B1479,D9,E5+corresponding lower half. Would guess that these are general to the geometry, not specific to the hints given. Guess also the non-symmetric part is more or less fixed, too. Would guess (??) that this puzzle has only one possible basic solution plus the variants of point reflection and global hi/low (number symmetry), so 4 in total.

For the only black cell hint versions: You will at least need two hints: One to decide on the asymmetric part and one for the high low decision. C7 or G3 determine hi/low, but nothing about (a)symmetry. Conversely, D5 or F5 decide (a)symmetry but not hi/low. BH28 don't help for neither, so only needed if there's more variants. So possibly redundant...

For the white cell hint versions: 2 hint versions are, as presented by Klaus, possible. In general, 1 number could determine both binary variants (hi/low + symmetry). Maybe there's also an 1 hint unique solutions starting from one of the non-symmetric, non-5 fields...

@Klaus:

a)In the current version, does the puzzle also solve omitting the B2=2? Even if very 'chainy' then...Does neither influence the hi/low nor the (a)symmetric question.

b)If there existed a 1 hint solution, it must be a hint answering both options. How about e.g. F7=9, D6=1 or C3=9, does your solver provide a solution for any of those cases? Would be keen to know...

@Leren: Asymmetry is induced by D5 or F5 and the range of the corresponding rows, the set of numbers for col 5 is fixed by the grid. Hope this answers your previous findings why the cols have the same range but not the rows...

CL