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 2021-02-06, 02:47 #1 Bobby Jacobs     May 2018 D516 Posts 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?
2021-02-06, 11:39   #2
xilman
Bamboozled!

"πΊππ·π·π­"
May 2003
Down not across

2×72×109 Posts

Quote:
 Originally Posted by Bobby Jacobs 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?
Blocks or bars?

2021-02-06, 11:44   #3
retina
Undefined

"The unspeakable one"
Jun 2006
My evil lair

22×5×307 Posts

Quote:
 Originally Posted by Bobby Jacobs 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?
Does the answer have to follow those usual rules? Or are those three usual rules just for informational purposes?

 2021-02-06, 13:04 #4 Dr Sardonicus     Feb 2017 Nowhere 13·349 Posts 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.
2021-02-06, 16:48   #5
Uncwilly
6809 > 6502

"""""""""""""""""""
Aug 2003
101Γ103 Posts

2×4,783 Posts

Quote:
 Originally Posted by xilman Blocks or bars?
Absent word from the OP, assume blocks. That is the norm on-line and much of the word. It adds to the challenge.
Quote:
 Originally Posted by retina Does the answer have to follow those usual rules? Or are those three usual rules just for informational purposes?
Those 3 are the recognized rules for a typical CW. A Sunday one might have a larger grid. One done for your local HΕ«sker DΕ«? club with members names likely won't fulfill the symmetric rule, nor filling the square.

2021-02-06, 17:56   #6
xilman
Bamboozled!

"πΊππ·π·π­"
May 2003
Down not across

2·72·109 Posts

Quote:
 Originally Posted by Uncwilly Absent word from the OP, assume blocks. That is the norm on-line and much of the word. It adds to the challenge.
This challenge, perhaps. I have not tried to solve it.

The most challenging regularly published crossword is generally held to be The Listener 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.

2021-02-07, 00:37   #7
slandrum

Jan 2021
California

22·17 Posts

Quote:
 Originally Posted by Dr Sardonicus 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.
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.

Last fiddled with by slandrum on 2021-02-07 at 00:39

 2021-02-07, 00:53 #8 retina Undefined     "The unspeakable one" Jun 2006 My evil lair 17FC16 Posts 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 3600 Last fiddled with by retina on 2021-02-07 at 00:53
 2021-02-08, 04:24 #9 0scar   Jan 2020 111112 Posts 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? 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. "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? 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 Last fiddled with by 0scar on 2021-02-08 at 04:37
 2021-02-11, 23:54 #10 Bobby Jacobs     May 2018 110101012 Posts 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.
 2021-02-12, 02:18 #11 SmartMersenne   Sep 2017 6416 Posts I think the problem is not very clear to everyone. You may help by providing some small cases to clarify all your points.