Comments on: Unscrabbled http://bit-player.org/2008/unscrabbled An amateur's outlook on computation and mathematics. Mon, 01 Dec 2008 20:20:50 +0000 http://wordpress.org/?v=2.6.3 By: Mark Gritter http://bit-player.org/2008/unscrabbled#comment-1723 Mark Gritter Wed, 02 Jul 2008 23:08:01 +0000 http://bit-player.org/?p=152#comment-1723 An anonymous commenter on my blog offers this legal path to another one of the 4x4 blocks, so we do have a good bound of 16 tiles. This requires the use of one of the blank tiles for 'J'. GJUS GJUS *I** *V** *Y** GJUS *I** *V** SYED GJUS *I*Y *V*E SYED GJUS *I*Y UVAE SYED GJUS JIVY UVAE SYED An anonymous commenter on my blog offers this legal path to another one of the 4×4 blocks, so we do have a good bound of 16 tiles. This requires the use of one of the blank tiles for ‘J’.

GJUS

GJUS
*I**
*V**
*Y**

GJUS
*I**
*V**
SYED

GJUS
*I*Y
*V*E
SYED

GJUS
*I*Y
UVAE
SYED

GJUS
JIVY
UVAE
SYED

]]> By: Barry Cipra http://bit-player.org/2008/unscrabbled#comment-1722 Barry Cipra Wed, 02 Jul 2008 21:43:10 +0000 http://bit-player.org/?p=152#comment-1722 I'll try that again, in case it's so obscure even Brian can't figure out what I meant: *HEY **** **** **** *HEY **Y* **R* **Y* *HEY HIYA **R* **Y* THEY HIYA E*R* Y*Y* THEY HIYA E*R* YAYS THEY HIYA EYRY YAYS (I was trying to use -'s for empty squares and run all the steps across one "line" in the first attempt, but the wordprocessing here seems to ignore repeated space bar entries, and interprets multiple -'s in some weird way.) I’ll try that again, in case it’s so obscure even Brian can’t figure out what I meant:

*HEY
****
****
****

*HEY
**Y*
**R*
**Y*

*HEY
HIYA
**R*
**Y*

THEY
HIYA
E*R*
Y*Y*

THEY
HIYA
E*R*
YAYS

THEY
HIYA
EYRY
YAYS

(I was trying to use -’s for empty squares and run all the steps across one “line” in the first attempt, but the wordprocessing here seems to ignore repeated space bar entries, and interprets multiple -’s in some weird way.)

]]> By: Barry Cipra http://bit-player.org/2008/unscrabbled#comment-1721 Barry Cipra Wed, 02 Jul 2008 21:29:30 +0000 http://bit-player.org/?p=152#comment-1721 Mark Gritter writes "I think the middle one is actually reachable via legal play." Here's one way to do it, making use of the SOWPODS intermediates HEY, HI, and YA: <pre> -HEY -HEY -HEY THEY THEY THEY ---- --Y- HIYA HIYA HIYA HIYA ---- --R- --R- E-R- E-R- EYRY ---- --Y- --Y- Y-Y- YAYS YAYS </pre> My apologies if the columns don't line up well (I hope Brian will fix it so they do). Mark Gritter writes “I think the middle one is actually reachable via legal play.” Here’s one way to do it, making use of the SOWPODS intermediates HEY, HI, and YA:

-HEY        -HEY       -HEY       THEY        THEY       THEY
----        --Y-       HIYA       HIYA        HIYA       HIYA
----        --R-       --R-       E-R-        E-R-       EYRY
----        --Y-       --Y-       Y-Y-        YAYS       YAYS

My apologies if the columns don’t line up well (I hope Brian will fix it so they do).

]]> By: Craig Kaplan http://bit-player.org/2008/unscrabbled#comment-1719 Craig Kaplan Wed, 02 Jul 2008 21:14:00 +0000 http://bit-player.org/?p=152#comment-1719 Another note -- the rules don't say anything explicit about what to do when continued play is impossible. But players always have the option of passing. I found <a href="http://www.farscapegames.co.uk/shopscr743.html" rel="nofollow">one site</a> that says that the game ends if everyone passes twice in a row. Hasbro (and my physical Scrabble game) <a href="http://www.hasbro.com/games/adult-games/scrabble/home.cfm?page=About/gameplay" rel="nofollow">says</a> "The game ends when all letters have been drawn and one player uses his or her last letter; or when all possible plays have been made," which could be taken to include stymied positions. Another note — the rules don’t say anything explicit about what to do when continued play is impossible. But players always have the option of passing. I found one site that says that the game ends if everyone passes twice in a row. Hasbro (and my physical Scrabble game) says “The game ends when all letters have been drawn and one player uses his or her last letter; or when all possible plays have been made,” which could be taken to include stymied positions.

]]> By: Craig Kaplan http://bit-player.org/2008/unscrabbled#comment-1718 Craig Kaplan Wed, 02 Jul 2008 20:01:45 +0000 http://bit-player.org/?p=152#comment-1718 Come to think of it, it might be easier to generate a stymied position at the <em>end</em> of a game rather than the beginning. Let's just imagine playing out 98 tiles, leaving each player with a single V and nowhere to place it. I expect it would be pretty easy to construct such a board, though I'd be the first to admit that it's much more boring as a stymied position than a minimal grid on an otherwise empty board. Come to think of it, it might be easier to generate a stymied position at the end of a game rather than the beginning. Let’s just imagine playing out 98 tiles, leaving each player with a single V and nowhere to place it. I expect it would be pretty easy to construct such a board, though I’d be the first to admit that it’s much more boring as a stymied position than a minimal grid on an otherwise empty board.

]]> By: Craig Kaplan http://bit-player.org/2008/unscrabbled#comment-1717 Craig Kaplan Wed, 02 Jul 2008 19:56:02 +0000 http://bit-player.org/?p=152#comment-1717 Mark, yes, I found the same three 4x4 grids and came to the same conclusions. Alas, I don't think any of them can arise in legal play. I'll hunt for simple move sequences that generate stymied 5x5 grids. Your comment about words that cannot be extended because the necessary letters have already been used is very interesting. It makes the search significantly more complex, even for 4x4 grids, since in theory any word could be involved. It's unlikely, but it might even produce a stymied 3x3 grid! Mark, yes, I found the same three 4×4 grids and came to the same conclusions. Alas, I don’t think any of them can arise in legal play. I’ll hunt for simple move sequences that generate stymied 5×5 grids.

Your comment about words that cannot be extended because the necessary letters have already been used is very interesting. It makes the search significantly more complex, even for 4×4 grids, since in theory any word could be involved. It’s unlikely, but it might even produce a stymied 3×3 grid!

]]> By: brian http://bit-player.org/2008/unscrabbled#comment-1715 brian Wed, 02 Jul 2008 13:03:09 +0000 http://bit-player.org/?p=152#comment-1715 Thanks, Mark! Very impressive. Thanks, Mark! Very impressive.

]]> By: Mark Gritter http://bit-player.org/2008/unscrabbled#comment-1714 Mark Gritter Wed, 02 Jul 2008 06:35:56 +0000 http://bit-player.org/?p=152#comment-1714 This approach doesn't take letter frequencies into account. There might be a position which has words that can be extended, but only via letters already used up. (But for small positions the blank would probably allow play.) Here are the 4x4 blocks that are not extensible. Note that they are all symmetric. GJUS JIVY UVAE SYED THEY HIYA EYRY YAYS TECS ELHI CHEZ SIZY I think the middle one is actually reachable via legal play, excepct that it has 6 occurences of 'Y'. The other two look hopeless as well. This approach doesn’t take letter frequencies into account. There might be a position which has words that can be extended, but only via letters already used up. (But for small positions the blank would probably allow play.)

Here are the 4×4 blocks that are not extensible. Note that they are all symmetric.

GJUS
JIVY
UVAE
SYED

THEY
HIYA
EYRY
YAYS

TECS
ELHI
CHEZ
SIZY

I think the middle one is actually reachable via legal play, excepct that it has 6 occurences of ‘Y’. The other two look hopeless as well.

]]> By: Mark Gritter http://bit-player.org/2008/unscrabbled#comment-1713 Mark Gritter Wed, 02 Jul 2008 06:08:31 +0000 http://bit-player.org/?p=152#comment-1713 That figure should be oxxo xxxx xxxx oxxo That figure should be

oxxo
xxxx
xxxx
oxxo

]]> By: Mark Gritter http://bit-player.org/2008/unscrabbled#comment-1712 Mark Gritter Wed, 02 Jul 2008 06:08:03 +0000 http://bit-player.org/?p=152#comment-1712 No 3x3 block can be a terminal position. I took the SOWPODS list and ran it through a perl script looking for words which cannot be extended to longer legal words. The only way to get a 3x3 block which cannot be played upon is if none of the 9 words can be extended to 4 or more letters. Alas, there are only 5 words of length 3 that are not stems: SWY FAP TWP VLY QIS Obviously these cannot be assembled into a 3x3 block. (There are no nonextendable words of length two.) Any 4x3 block would have to contain four of these words along the short dimension (not necessarily unique) which doesn't look possible. There are 250 such words of length 4, leading to a search space of about 4 billion possibilities. Might be feasible. Could something smaller than a 4x4 block be terminal? Perhaps something like this xx xxxx xxxx xx would work if the 2-letter runs did not have a common letter in their extensions. No 3×3 block can be a terminal position.

I took the SOWPODS list and ran it through a perl script looking for words which cannot be extended to longer legal words. The only way to get a 3×3 block which cannot be played upon is if none of the 9 words can be extended to 4 or more letters.

Alas, there are only 5 words of length 3 that are not stems:
SWY
FAP
TWP
VLY
QIS

Obviously these cannot be assembled into a 3×3 block. (There are no nonextendable words of length two.)

Any 4×3 block would have to contain four of these words along the short dimension (not necessarily unique) which doesn’t look possible.

There are 250 such words of length 4, leading to a search space of about 4 billion possibilities. Might be feasible.

Could something smaller than a 4×4 block be terminal? Perhaps something like this

xx
xxxx
xxxx
xx

would work if the 2-letter runs did not have a common letter in their extensions.

]]>