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September 21

Moving an edge from one side to the opposite side in Sudoku

Are there any complete (9-by-9 as usual) Sudoku grids that remain valid Sudoku grids if one moves the topmost row, bottommost row, leftmost column, or rightmost column to the other side?

For example, suppose that one moves the topmost row to the bottom. Then, in the original Sudoku grid, the first three entries in the topmost row must be a derangement of the first three entries in the seventh row, and likewise for the middle three entries and the last three entries in the respective rows, in order for the bottom three rows in the new grid to meet the Sudoku condition. The same condition must also hold for the first (topmost) and fourth rows, as well as the fourth and seventh rows.

A derangement is a permutation without any fixed points. For example, 231 is a derangement of 123, since all three digits have changed their positions. In contrast, 132 is not a derangement of either 123 or 231, since the digit 1 remains in the same position as in 123, while the digit 3 remains in the same position as in 231. GeoffreyT2000 (talk) 02:17, 21 September 2020 (UTC)[reply]

The question is the disjunction of four options: T, B, L or R. If the answer to any of the three is yes, then the other three are also possible, since a completed grid remains valid under rotation. --Lambiam 07:38, 21 September 2020 (UTC)[reply]

Here is a solution for a circular left shift by one position (leftmost column becomes rightmost column):

            before                    after

     6 9 3 | 1 4 2 | 8 5 7    9 3 1 | 4 2 8 | 5 7 6
     1 5 7 | 8 9 3 | 6 2 4    5 7 8 | 9 3 6 | 2 4 1
     8 2 4 | 6 5 7 | 1 9 3    2 4 6 | 5 7 1 | 9 3 8
     ------+-------+------    ------+-------+------
     3 6 9 | 2 7 1 | 4 8 5    6 9 2 | 7 1 4 | 8 5 3
     2 7 1 | 4 8 5 | 3 6 9    7 1 4 | 8 5 3 | 6 9 2
     4 8 5 | 3 6 9 | 2 7 1    8 5 3 | 6 9 2 | 7 1 4
     ------+-------+------    ------+-------+------
     9 3 6 | 7 2 4 | 5 1 8    3 6 7 | 2 4 5 | 1 8 9
     7 1 2 | 5 3 8 | 9 4 6    1 2 5 | 3 8 9 | 4 6 7
     5 4 8 | 9 1 6 | 7 3 2    4 8 9 | 1 6 7 | 3 2 5

The derangement theorem can be seen to hold for this solution; derangements of fewer than four elements are also circular shifts. --Lambiam 10:14, 21 September 2020 (UTC)[reply]

What's the name of a set of lines with a common point?

In an Euclidean space take a point P and define a set S of all lines through P. What is S called?

Wikipedia says a similarly defined set of all planes through a chosen line is called a Sheaf of planes but I failed fo find a 'sheaf of lines'... There exists an article about a Sheaf (mathematics) itself, but that seems too general notion.

I found a sheaf of lines somewhere in the Internet, but that was a set of parallel lines, as opposed to set of lines all intersecting (in one point).

OTOH the Wolfram's Mathworld knows Sheaf of Lines, but instead of defining the term it redirects to Star, ‘sometimes called a sheaf’. That one, however, is a set of line segments with a common midpoint, and I'm interested in a set of lines, not line segments. --CiaPan (talk) 09:22, 21 September 2020 (UTC)[reply]

@CiaPan: Real projective space?--Jasper Deng (talk) 09:54, 21 September 2020 (UTC)[reply]

@Jasper Deng: Thank you, that's close - but it is too much. I need a set of lines alone, without studying its internal structure, possible metrics or topology. --CiaPan (talk) 11:15, 21 September 2020 (UTC)[reply]

Looks like a pencil of lines – Pencil (mathematics)#Pencil of lines – is what I was looking for... --CiaPan (talk) 09:58, 21 September 2020 (UTC)[reply]

Apparently (according to the Pencil article) called thus in two-space, while the term for three-space is said to be bundle of lines. The article Bundle (mathematics) seems solely to describe another term for function, though. --Lambiam 13:37, 21 September 2020 (UTC)[reply]

Note that the Pencil articles uses "Pencil of planes" for what is being called "Sheaf of planes" above, so the terminology varies. I gather this terminology is a bit dated now anyway; like saying locus instead of variety. --RDBury (talk) 16:22, 21 September 2020 (UTC)[reply]

Lines through a common point are also called "concurrent lines". --174.89.48.182 (talk) 21:54, 21 September 2020 (UTC)[reply]

That term does not have the sense, though, of denoting the totality of lines through a common point. --Lambiam 22:19, 21 September 2020 (UTC)[reply]

I suppose you're aware that such a set could never be enumerated, having infinitely many members. -- Jack of Oz ^{[pleasantries]} 20:47, 22 September 2020 (UTC)[reply]

Yes. The set of real numbers

x

such that

0 \leq x \leq 1

can also not be enumerated, by Cantor's infamous diagonal argument – which, by the way, is not acceptable to constructivists. Yet we have a name for it: "the unit interval". The set of natural numbers has infinitely many members, last time I counted them, but is nevertheless considered enumerable. --Lambiam 21:09, 22 September 2020 (UTC)[reply]

Not so about constructivists. The argument is intuitionistically valid. They differ in its interpretation. --Trovatore (talk) 18:58, 23 September 2020 (UTC)[reply]

@JackofOz: Does enumerability have anything to do with the existence of a set? A line itself is infinite, too, and not enumerable as a set of points, nevertheless some mathematicians agree lines exist. And they even have a special name: a line... --CiaPan (talk) 23:52, 22 September 2020 (UTC)[reply]

The things one learns on the WP Ref Desk ... -- Jack of Oz ^{[pleasantries]} 00:38, 23 September 2020 (UTC)[reply]

Yes, that's why I love to come to WP:RD

--CiaPan (talk) 18:04, 23 September 2020 (UTC)[reply]

Possibly worth pointing out that enumerable is not the usual word for the concept under discussion. The usual word is countable; see countable set. There is a slight ambiguity; a few workers use "countable" to mean "countably infinite", excluding finite sets. There is also an older word, denumerable, which isn't used a lot anymore, but is possibly more likely to exclude finite sets, though so few people actually use it that I can't say I'm sure of that. The word "enumerable" is more typically used as part of the formulation computably enumerable, which implies countability but not vice versa. --Trovatore (talk) 18:37, 23 September 2020 (UTC)[reply]