Crosswords
 

Crosswords usually obey the following 3 rules.

1. The grid is symmetrical, the same upside-down.
2. The grid is connected.
3. Every word has at least 3 letters.

What is the most words that can fit in a 15*15 crossword puzzle?

[QUOTE=Bobby Jacobs;570965]Crosswords usually obey the following 3 rules.

1. The grid is symmetrical, the same upside-down.
2. The grid is connected.
3. Every word has at least 3 letters.

What is the most words that can fit in a 15*15 crossword puzzle?[/QUOTE]Blocks or bars?

[QUOTE=Bobby Jacobs;570965]Crosswords usually obey the following 3 rules.

1. The grid is symmetrical, the same upside-down.
2. The grid is connected.
3. Every word has at least 3 letters.

What is the most words that can fit in a 15*15 crossword puzzle?[/QUOTE]Does the answer have to follow those usual rules? Or are those three usual rules just for informational purposes?

Assuming each word is within a single row or column, and is at least three letters, there can't be more than 5 words in any row or column. So there can't be more than 75 "across" words or 75 "down" words. No more than 150 words all told.

Assuming further that the beginning and end of each word is at either an edge or a blacked-in square (a general crossword rule which I cheerfully ignored in the above) will knock down the number of possible words. Trivially the number of "across" and "down" words drops to at most 60 each, and the total to at most 120.

The number of "across" and "down" clues in most daily (15x15) crosswords I've worked have each generally been in the mid to upper thirties IIRC (it's been a while) so in practice there are probably between 70 and 80 words in a 15x15 crossword puzzle.

For printed puzzles, the number of words is also limited by the amount of space available.

[QUOTE=xilman;570972]Blocks or bars?[/QUOTE]
Absent word from the OP, assume blocks. That is the norm on-line and much of the word. It adds to the challenge.
[QUOTE=retina;570974]Does the answer have to follow those usual rules? Or are those three usual rules just for informational purposes?[/QUOTE]
Those 3 are the recognized rules for a typical CW. A Sunday one might have a larger grid. One done for your local Hūsker Dū? club with members names likely won't fulfill the symmetric rule, nor filling the square.

[QUOTE=Uncwilly;571002]Absent word from the OP, assume blocks. That is the norm on-line and much of the word. It adds to the challenge.[/QUOTE]This challenge, perhaps. I have not tried to solve it.

The most challenging regularly published crossword is generally held to be [I][URL="https://listenercrossword.com/"]The Listener[/URL][/I] which almost always uses bars. I have solved one of their puzzles fewer than ten times in about 30 years of trying (though I haven't tried every one of them) and won precisely once, about 25 years ago.

[QUOTE=Dr Sardonicus;570978]Assuming each word is within a single row or column, and is at least three letters, there can't be more than 5 words in any row or column. So there can't be more than 75 "across" words or 75 "down" words. No more than 150 words all told.

Assuming further that the beginning and end of each word is at either an edge or a blacked-in square (a general crossword rule which I cheerfully ignored in the above) will knock down the number of possible words. Trivially the number of "across" and "down" words drops to at most 60 each, and the total to at most 120.[/QUOTE]

If there are 60 across words, then there are blocks completely filling 3 columns and no other blocks on the grid, so the number of down words would be 12 words (of 15 letters apiece) which would lead to a total of only 72 words. If the grid is sub-sectioned into 3x3 grids then you have 48 across and 48 down for a total of 96 words. This though violates another rule - the grid is not connected.

If you completely fill the grid with 15 letter words, then counting all embedded words will give the maximum possible.

15 + 15 = 30 x 15-letters
15x2 + 15x2 = 60 x 14-letters
...
15x14 + 15x14 = 420 x 2-letters
15x15 + 15x15 = 450 x 1-letter
[code]~ echo 30*{1..15}+ 0|bc
[b]3600[/b][/code]:showoff:

nice puzzle; not sure about minimality of my candidates.

"bars" version, fixing 150-word near-solution by Dr Sardonicus.
5 words and 4 bars per row/column;
150 words, 120 bars placed along 8 lines, 25 disconnected 3x3 squares;
symmetry holds, restore grid connection by removing 24 bars;
any bar removal merges two words, so 150 - 24 = 126 words?

[SPOILER]Example with both upside-down and left-right symmetry:
label rows and columns from 1 to 15;
remove all bars from row 8 and from columns 2,5,8,11,14.[/SPOILER]

"block" version, fixing 96-word near-solution by slandrum.
3 rows and 3 columns containing blocks only;
4 words and 3 blocks per remaining row/column;
96 words, 81 blocks placed along 6 lines, 16 disconnected 3x3 squares;
symmetry holds, restore grid connection by removing 15 blocks;
so 96-15 = 81 words?

[SPOILER]We can connect 4 squares by removing 3 contiguous blocks:
two pairs of 3-letter words are merged into two 7-letter words, a new 3-letter word is built.
So 96-5 = 91 words?
Example with both upside-down and left-right symmetry (0=letter, 1=block):
000100010001000
000100010001000
000100010001000
110001111100011
000100010001000
000100010001000
000100010001000
111111000111111
000100010001000
000100010001000
000100010001000
110001111100011
000100010001000
000100010001000
000100010001000[/SPOILER]

I am talking about blocked crosswords. By the way, your blocked crossword contains some 1-letter words (unchecked letters). Every letter must be part of a word going across and a word going down.

I think the problem is not very clear to everyone. You may help by providing some small cases to clarify all your points.

[QUOTE=Bobby Jacobs;571366] Every letter must be part of a word going across and a word going down.[/QUOTE]
Not really, you may have long words (unambiguous) whose non-terminal letters are not part of any other word, this is not forbidden in any kind of crosswords. For example, you have a 6 letters horizontal word starting in R1C1, then R2C4 can be black, if the letter in R1C4 is not ambiguous. Your requirement is very restrictive, it would restrict a lot the places where the black dots can go, for example, you would not have any black dot in row 2, nor any in column 2, nor any in row N-1, not any in column n-1, etc. (you can go on from here), because any such dot will insulate a letter in either row1, or col1, etc.

[QUOTE=Bobby Jacobs;571366]I am talking about blocked crosswords. By the way, your blocked crossword contains some 1-letter words (unchecked letters). Every letter must be part of a word going across and a word going down.[/QUOTE]So my solution #8 is valid, right? We just need to deduct 450 for your new rule of no 1-letter words.

Therefore: 3600 - 450 = 3150 :showoff:

[size=1]I hope we don't keep seeing more and more ever restrictive rules being added at random intervals.[/size]:sad:

[QUOTE=retina;571389]So my solution #8 is valid, right? We just need to deduct 450 for your new rule of no 1-letter words.[/QUOTE]
OP
[QUOTE=Bobby Jacobs;570965]Crosswords usually obey the following 3 rules.

1. The grid is symmetrical, the same upside-down.
2. The grid is connected.
3. Every word has at least 3 letters.[/QUOTE]

[QUOTE=Uncwilly;571392]OP[/QUOTE]See #3. It went unanswered. We don't know what rules the OP has.

[QUOTE=LaurV;571388]... for example, you would not have any black dot in row 2, nor any in column 2, nor any in row N-1, not any in column n-1, etc. (you can go on from here), because any such dot will insulate a letter in either row1, or col1, etc.[/QUOTE]You can have black spaces in any column or row. I didn't see any rule saying there can't be two together.

[QUOTE=Bobby Jacobs;571366]I am talking about blocked crosswords. By the way, your blocked crossword contains some 1-letter words (unchecked letters). Every letter must be part of a word going across and a word going down.[/QUOTE]

Block-version is the most interesting one.
I slightly modifìed my candidate to fit your additional constraint.
Less words, but more symmetries.

[SPOILER]86 words:
000100010001000
000100010001000
000000010000000
110001111100011
000000010000000
000100010001000
000100000001000
111111000111111
000100000001000
000100010001000
000000010000000
110001111100011
000000010000000
000100010001000
000100010001000
every letter belongs to at least one 3x3 square
[/SPOILER]

[QUOTE=retina;571398]You can have black spaces in any column or row. I didn't see any rule saying there can't be two together.[/QUOTE]
That's what I said.

If you ask me what are "my" rules for a good crossword, then those should be only one: no ambiguity of the solution. This means that once you show me the solution, I would open my mouth large and make big eyes of awe and say "yes!". I wouldn't care about walls, corners, and stuff.

But the crosswords I used to solve a hundred years ago, when the crosswords were "something", i.e. before a lot of easy crap flooded the markets, had very well defined rules, like "no closed corners" (which means you could not have black squares in, say, R1C4 and R2C3, because in this case the R1C3 is a "closed" cell, and upper-left corner becomes a "closed corner", you would have a lot of possibilities to fill it, especially if you speak a "flexed" language, as Romanian; however, closed cells are accepted if not a lot, and not ambiguous by definition), or "no walls" (i.e. no two black squares can share an edge), etc. But what Bobby asked, every letter be part of two words, that would be a VERY restrictive condition, which I didn't see in almost any rebus I solved, except accidentally (in very good, awarded, rebuses). (P.S. In Romanian, rebus is just an advanced form of crossword - Ro="cuvinte încrucișate" - and in many contexts, they mean the same stuff and are interchangeable, but I just realized now, looking on wikipedia, that, in English, rebus is a different kind of logic game, so forgive me if I use the term to mean "crosswords").

So, what is [URL="https://en.wikipedia.org/wiki/Crossword#Types"]shown on wikipedia[/URL] as "american" or "australian/british", or "japanese" -style grids, would all be forbidden in competitions I used to participate in my youth (they all have closed corners, walls, and ambiguities), and they won't be published in "good" magazines. Also, the symmetry was not a requirement, but it was highly appreciated by solvers. Example: (random from web, Romanian Rebus Magazine, sometime in the '80s):

[ATTACH]24316[/ATTACH]
Please note that Bobby's rules will forbid all 3 grids in the example, as they all have at least 4 letters which are part of a single word.

[QUOTE=LaurV;571492]That's what I said.[/QUOTE]You said the opposite. You stated you can't, I stated you can. :confused:

[QUOTE=retina;571496]You said the opposite. You stated you can't, I stated you can. :confused:[/QUOTE]
Read again.

[QUOTE=LaurV;571498]Read again.[/QUOTE]Nope, I don't see it?[QUOTE=LaurV;571388]... for example, you would [b]not have any black dot in row 2[/b], nor any in column 2, nor any in row N-1, not any in column n-1, etc. (you can go on from here), because any such dot will insulate a letter in either row1, or col1, etc.[/QUOTE][QUOTE=retina;571398]You [b]can have black spaces in any column or row[/b].[/QUOTE]:confused:

Which part of "if" (your condition applies) and "would" don't you get?

I thought you are native speaker.

If "any letter must be a part of 2 words", then there [U]would[/U] be no dark square allowed in the second ring.

Or... oh? do you mean that you [U]can[/U] have black square in the second ring, in case the correspondent "singled" square in the first ring is also black? Like a R2C2 black square will force all R1C1, R1C2 and R2C1 to be black?

Grrr, yeah, this way you can, but I somehow interdicted "walls" in my concept... :bow:

[QUOTE=LaurV;571505]Or... oh? do you mean that you [U]can[/U] have black square in the second ring, in case the correspondent "singled" square in the first ring is also black? Like a R2C2 black square will force all R1C1, R1C2 and R2C1 to be black?[/QUOTE]Yup. Exactly. The example posted by 0scar also shows how all columns/rows can have all options.

You can do better than 86 words.

Is anybody going to try to solve this puzzle and find the 15*15 crossword with the most words?

[QUOTE=Bobby Jacobs;574201]Is anybody going to try to solve this puzzle and find the 15*15 crossword with the most words?[/QUOTE]If you answered when asked questions about clarifying the rules then people might be more interested. Specifically, you didn't answer any of my queries.

[QUOTE=Bobby Jacobs;574201]Is anybody going to try to solve this puzzle and find the 15*15 crossword with the most words?[/QUOTE]

After such an explicit challenge...

[SPOILER]96 words:
000100010001000
000100010001000
000100010001000
110001000100011
000010001000000
000000100010000
000100010001000
111100010001111
000100010001000
000010001000000
000000100010000
110001000100011
000100010001000
000100010001000
000100010001000
[/SPOILER]

That is correct! The most words possible in a 15*15 crossword is 96, if the grid is symmetrical and connected, and every word has at least 3 letters.

000010000100000
000010000100000
000010000100000
111000111000111
000001000010000
000100000001000
000010000100000
111000111000111
000001000010000
000100000001000
000010000100000
111000111000111
000001000010000
000001000010000
000001000010000