The New Sudoku Players' Forum

[JPF]

6x6 Word squares

Word squares (WS) are defined here.
Here are examples of 6x6 word squares:

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A F F O R D    B I P E D S    C A R D E R    D A U T E D    E D I T E D
L E A S E R    A N E M I A    A R E O L A    I N P O U R    R O M A N O
M A M M E Y    S W I P E Y    R E A R M S    P I R O G I    I N A R M S
O R I O L E    T O N I T E    D O R M I E    S C A T H E    A G G R O S
S E L L E R    O V E R E D    E L M I E R    O U T L E R    C L E A V E
T R Y E R S    S E D E R S    R A S E R S    S T E E N S    H E R S E D
            
F A S T E D    G A R R E D    H O R D E S    I D I O T S    J A M B O K
A L P I N E    A S I A G O    O L E I N E    M I N C E D    A R A A R A
S P I R A L    P I N G E D    R E D O N S    P A T E R A    M A G N O N
T I R I T I    P A G I N G    D I O T A S    A P I A R Y    B A N D I T
E N A T E S    E G E N C E    E N N A G E    R E N N I N    U R E I D E
D E L I S T    D O D G E S    D E S S E S    T R E S T S    L A S T E D
            
K E C K E D    L E A V E S    M A T A T A    N E C T A R    O S M U N D
E N H A L O    U N L E A L    A G A P E S    A C H A G E    S T U P O R
G L O M U S    S T A R R Y    R I P E N S    C H Y P R E    C A R S E Y
G E L A T E    T A M I N E    A L E X I A    R I P P E D    U P R I S E
E V I L E R    E M O T E S    R E T E A M    E N R A G E    L E A D E R
D E C A D S    D E S E R T    I R I S E S    D I E S E S    A S S E S S
            
P A S T E D    Q U O T A S    R E I R D S    S M A R T S    T I T L E S
A P P E A R    U N R E S T    A P N O E A    P I L E U P    U N W I R E
S P E N S E    O B E N T O    S H A N T I    A S P I R E    T H A N G S
S E I N E S    H O I D E N    H O R N E R    R E I T E R    S A N G A S
E A S E R S    O R D U R E    E R M I N E    R A N E E S    A L G A T E
D R E S S Y    G E E S T S    D I S E U R    E T E R N E    N E S S E S
            
U M A M I S    V E S S E L    W A S H E S    X Y S T U S    Y A W N E D
M O B I L E    E T H E N E    I M M A N E    Y S H E N T    A S H A M E
A B O L L A    R H O N E S    S M A L T I    S H A M B A    W H E L P S
M I L I U M    S A V O R S    S O L V E S    T E M P O S    N A L L A S
I L L U D E    E N E R V E    E N T E T E    U N B O N E    E M P A R E
S E A M E R    T E R S E R    S O S S E S    S T A S E S    R E S H E S
            
Z A P P E D             
A R R I B A             
P R O T O N             
P I T H O I             
E B O O K S             
D A N I S H             


The words used above are part of a set S of 22,410 six-letter words allowed in Scrabble.
This site is the Collins Scrabble checker.

Using the set S, there are 62,376,843 different word squares.
By using the main diagonal symmetry, each word square may be linked to its symmetrical:

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E L A P S E    E M P A R T
M E R L I N    L E A G U E
P A N I N O    A R N I C A
A G I N G S    P L I N K S
R U C K L E    S I N G L E
T E A S E S    E N O S E S


The two word squares may be identical if the WS has a main diagonal symmetry:

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E S P A D A
S L O K E N
P O T A S S
A K A T E A
D E S E R T
A N S A T E


There are 10,415,005 symmetrical word squares.

Here are some remarkable WS:
6 times the same word

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M A M M A L
A T A A T A
M A M M A L
M A M M A L
A T A A T A
L A L L A N


3 letters only!

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A T A A T A
T A R T A R
A R A A R A
A T A A T A
T A R T A R
A R A A R A


Puzzle:
A puzzle is a sub-word-square with only one solution.

Here are two "easy" puzzles:

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R A D A R .    C A V A S .
A N I C . T    A L A S . A
F U R . N E    L I N . A Y
F R . N N E    I N . A T E
L . S T E R    G . E K E D
. A T H L Y    . R D E R S


less easy

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F E M U . .    Z O N U . .
E L A . . T    O Z O . . S
D E . . D E    N O . . U T
O . . T O R    U . . O R E
. . C A N E    . . U R G E
. N O L E .    . S T E E .


more challenging...

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H E . . T S    G O . . I P
E N . . R E    O P . . M E
. . O C . .    . . I G . .
. . N I . .    . . G N . .
T R . . N E    I M . . W S
H E . . E R    P E . . S T


more...

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L E . . T H    M A . . M E    N A . . R E
A R . . I A    E M . . E D    A T . . E S
. . . . . .    . . . . . .    . . . . . .
. . . . . .    . . . . . .    . . . . . .
M O . . E S    L I . . E R    L I . . E R
S W . . R T    E N . . R S    S A . . D S


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T . . . . .    S . . . . S
. A . . . .    . N . . N .
. . R . . .    . . O O . .
. . . G . .    . . O R . .
. . . . E .    . P . . T .
. . . . . T    S . . . . S


insane

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B . . . . C    G . . . . R   E . . . . Z
. . . . . .    . . . . . .   . . . . . .
. . . . . .    . . . . . .   . . . . . .
. . . . . .    . . . . . .   . . . . . .
. . . . . .    . . . . . .   . . . . . .
O . . . . E    S . . . . P   S . . . . Y


and finally...

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X . . . . .    M . . . . .    N . . . . .    U . . . . .    U . . . . .    V . . . . .
. . . . . .    . . . . . .    . . . . . .    . . . . . .    . . . . . .    . . . . . .
. . . . . .    . . . . . .    . . . . . .    . . . . . .    . . . . . .    . . . . . .
. . . . . .    . . . . . .    . . . . . .    . . . . . .    . . . . . .    . . . . . .
. . . . . .    . . . . . .    . . . . . .    . . . . . .    . . . . . .    . . . . . .
. . . . . X    . . . . . V    . . . . . J    . . . . . J    . . . . . V    . . . . . J


There is no minimal puzzle with only one clue.

JPF

[Leren]

Easy, less easy and more challenging:

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R A D A R S   C A V A S S   F E M U R S   Z O N U R E   H E A R T S   G O S S I P
A N I C U T   A L A S K A   E L A N E T   O Z O N E S   E N T I R E   O P T I M E
F U R A N E   L I N H A Y   D E C I D E   N O B B U T   A T O C I A   S T I G M A
F R E N N E   I N N A T E   O V A T O R   U N B O R E   L I N I E R   S I G N E R
L I S T E R   G E E K E D   R E C A N E   R E U R G E   T R I E N E   I M M E W S
E A T H L Y   O R D E R S   A N O L E S   E S T E E M   H E A R E R   P E A R S T

more ...

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L E N G T H   M A D A M E   N A T U R E   T E S T E D   S U S H I S
A R A L I A   E M U L E D   A T O N E S   E A T A G E   A N N O N A
R E P O L L   A I R B A G   S A L L E T   S T R I G S   D R O O G S
U N P I L E   S D A I N E   A X L I K E   T A I G L E   H O O R A H
M O I R E S   L I N T E R   L I E V E R   E G G L E R   U P D A T E
S W E E R T   E N T E R S   S A Y E D S   D E S E R T   S E S H E S

insane

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B R O M I C   G A F F E R   E R S A T Z
R E T I R E   U L L A G E   G O O R O O
A T T O R N   E T U D E S   E S L O I N
S E A T E D   S H E E S H   S C I L L A
C A V I A R   T E N U T I   T O O L E R
O R A C L E   S A T R A P   S E N A R Y

and finally ...

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X Y S T U S   M A L L A M   N O S E R S   U N G A G S   U N S E A M   V R A I C S
Y B L E N T   A R I O S O   O U T L A W   N A R R O W   N I E L L O   R E G R O W
S L A N G Y   L I V O R S   S T R U M A   G R A M P A   S E L V A S   A G G A D A
T E N O U R   L O O F A H   E L U T O R   A R M O U R   E L V I S H   I R A T E R
U N G U L A   A S R A M A   R A M O N A   G O P U R A   A L A S K A   C O D E I A
S T Y R A X   M O S H A V   S W A R A J   S W A R A J   M O S H A V   S W A R A J

Leren

[JPF]

Excellent, Congratulations!

Scrabble score:
Let's give to each letter its Scrabble number of points (A=1, B=3, C=3,..., Z=10)

What is the scrabble-like word square with the highest score?

Please note that there are only 9 A, 2 B, ..., 1 Z - see here.

Example :

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A B A C A S
B I V I U M
A V E N G E
C I N Q U E
A U G U S T
S M E E T H


score* : 136 = 2 x 68

*edited, thanks to Leren

JPF

[Leren]

Hi JPF. I'm not quite sure how you get a Scrabble score of 130 for your example. The Scrabble total letter score is 68. Presumably you count the words horizontally and vertically, effectively doubling the letter count.

On that basis I make the score for your example 136. My first attempt is the following.

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Q U A Z Z Y
U P G O E S
A G U N A H
Z O N A T E
Z E A T I N
Y S H E N D

The score for this is 196. QUAZZY is the highest scoring six letter word with a score of 36 but that doesn't mean that the word square is necessarily the highest scoring. I'll continue searching unless you can confirm that it is the highest.

Leren

<Edit> Just beat my previous effort with the following:

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B E Z I L S
E P I Z O A
Z I Z Z E D
I Z Z A R D
L O E R I E
S A D D E D

Score 216. It's getting late, the search will continue tomorrow ...

Leren

[JPF]

Hi JPF. I'm not quite sure how you get a Scrabble score of 130 for your example. The Scrabble total letter score is 68. Presumably you count the words horizontally and vertically, effectively doubling the letter count.
On that basis I make the score for your example 136.

Thanks, it's a mistake I made when I read my computer output.
I took the position to double count the letters because words can be read horizontally and vertically...

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Q U A Z Z Y
U P G O E S
A G U N A H
Z O N A T E
Z E A T I N
Y S H E N D

The score for this is 196. QUAZZY is the highest scoring six letter word with a score of 36 but that doesn't mean that the word square is necessarily the highest scoring. I'll continue searching unless you can confirm that it is the highest.

Well it's not bad, but you forgot a rule I mentioned about the number of letters existing in a scrabble game:
there is only one z for instance.
More generally there are 9A, 2B, etc..

In any case, without that constraint, 196 is not the maximum. Here is an example:

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H E M P E N
E Q U I N E
M U Z Z L E
P I Z Z A Z
E N L A C E
N E E Z E D


score = 226

JPF

[Leren]

I suggest that we just stick to the simple rule of counting the score for each letter (twice) irrespective of how many Scrabble tiles of each letter there are.

The reason is that, if you want to be picky, there are also 2 blank tiles in Scrabble so, for example, words with 3 Z's are legal but 2 of the Z's have to be provided by blank tiles so their scores don't count. There is also the issue of premium squares. You could argue that the word square should be Scrabble-like and have triple word scores for words covering the 4 corner cells, double scores for words covering the cells diagonally in from the four corners etc etc. All these different possible scoring algorithms could produce different results for the "best" word square. How about we just keep it simple at this stage ?

BTW I just put PIZZAZ in the same position as your example and came up with the following:

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T E M P E H
E Q U I N E
M U Z Z L E
P I Z Z A Z
E N L A C E
H E E Z E D

Score 232 - a new record for me. The search goes on ...

Leren

[JPF]

I suggest that we just stick to the simple rule of counting the score for each letter (twice) irrespective of how many Scrabble tiles of each letter there are.

OK.

Here is my next proposal:

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M A H Z O R
A G U I Z E
H U Z Z A S
Z I Z Z L E
O Z A L I D
R E S E D A


score = 238

JPF

[Leren]

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L E Z Z A S
E Q U I P E
Z U Z Z I M
Z I Z Z E S
A P I E C E
S E M S E M

Score 258. How long will this new record stand ?

Leren

[JPF]

Score 258. How long will this new record stand ?

Congratulations! That's the highest score -and the same word square- I found after a comprehensive search.

Unfortunately for me, you missed this one:

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L E Z Z A S
E Q U I N E
Z U Z Z I M
Z I Z Z L E
A N I L I N
S E M E N S


score = 242

JPF

[Leren]

OK, I'm now willing to try word squares with Scrabble letter limits, but with no "blank" tiles or premium cells (I think this was your original intention).

BTW were 238, 242 and 248 the top 3 scores and what was the top non-symmetric score ?

Leren

[JPF]

BTW were 238, 242 and 248 the top 3 scores and what was the top non-symmetric score ?

yep! 238, 242 and 258

There are 4 non-symmetric WS with a top score equal to 224

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J A Z Z E D   J A Z Z E S
A S I A G O   A S I A G O
Z I Z Z E S   Z I Z Z E S
Z A Z E N S   Z A Z E N S
E G E N C E   E G E N C E
S O S S E D   D O S S E D
   
S P R I T S   S P R I T S
P I A Z Z A   P I A Z Z E
R A Z Z E S   R A Z Z E S
I Z Z A T S   I Z Z A T S
T Z E T Z E   T Z E T Z E
S E S S E S   S A S S E S


JPF

[Leren]

Here's a suggestion for the highest scoring symmetric Scrabble Like word square - no blank tiles or premium squares.

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B E H E S T
E X U V I A
H U M E F Y
E V E J A R
S I F A K A
T A Y R A S

Score 164

Leren

[JPF]

Here's a suggestion for the highest scoring symmetric Scrabble Like word square - no blank tiles or premium squares.

Code: Select all

B E H E S T
E X U V I A
H U M E F Y
E V E J A R
S I F A K A
T A Y R A S

Score 164

Correct, that's the highest score!

JPF

[Leren]

Does that include non-symmetric squares as well ? Symmetric squares are much faster to find so I'm sticking to those at the moment. If it is the highest what is the highest scoring non-symmetric square ?

I've begun to suspect that there might be a theorem that proves that the highest scoring square will always be symmetric. Of course that might depend on what restriction you put on the squares (eg some quirky effect of the
Scrabble letter limit rules).

Leren

[JPF]

Does that include non-symmetric squares as well ?

the highest score for non-symmetric squares is 160.
I'll let you find the (two) corresponding word squares.

JPF