Word Squares

We were interested in seeing how easy it would be to find a maximal 4 x 4 Diagonal Word Square, containing ten different four-letter words reading left-to-right, top-to-bottom, and along both diagonals left-to-right.

After a few experiments it was clear that getting beyond about 6 words was going to be impossible without a good system.

We therefore decided to find a pattern of 4 x 4 vowels and consonants that would give the widest possible range of available four-letter words to choose between.

Vowel and consonant patterns

As a starting point we created a chart showing all possible sequences of vowels and consonants, and a list of the corresponding four-letter words. For simplicity we counted "y" as a vowel.

Pattern   Words  Examples
 VVVV  0  -
 VVVC  8  aeon, eyed, eyes, year, your
 VVCV  16  aura, ease, easy, iota, oily, ooze, yoga, yoke
 VVCC  49  each, east, ions, oath, ours
 VCVV  11  aloe, area, aria, away, epee, idea, obey, oboe, urea
 VCVC  97  alas, ales, akin, atop, ones, opal, open, ores, user
 VCCV  53  ache, acne, acre, army, arty, echo, edge, else, isle, ogle, ogre, orgy, urge 
 VCCC  37  ants, arch, asks, eggs, elms, inch, owls, owns
 CVVV  9  beau, buoy, soya, quay
 CVVC  370  hued, look, real ...
 CVCV  450  cake, halo, solo, toga ...
 CVCC  925  fish, tank ...
 CCVV  44  blue, glue, shoe ...
 CCVC  289  chop, scar, ship, tsar ...
 CCCV  2  spry
 CCCC  0  -

Next we created a word square showing a layout of "c"s and "v"s that would minimise the number of hard words we'd need to find. The most successful was this:

 C  V  C  C
 V  C  ?  V
 C  ?  V  C
 C  V  C  ?

The cells marked "?" can contain either a "V" or a "C", with the exception of the two central squares which can't both contain "C"s.

The highlighted row and column show the two difficult words, which must be either VCCV or VCVV. We call these the "key words".

Key words "area" and "aria"

After a few experiments, one fruitful choice for these words was "area" and "aria":

 a  r  e  a

These led to the following solution to the original problem:

 b  a  n  d
 a  r  e  a
 l  i  a  r
 d  a  t  e

All the words are common words apart from "brae", a Scottish word for hillside.

Key words "area" and "urea"

By chance we also found a good word square with the key words "area" and "urea":

 b  u  m  p
 a  r  e  a
 r  e  a  r
 s  a  l  t

All the words are common apart from "urea", a chemical in urine.

A computer search for 4x4 word squares

I subsequently wrote a Lisp program to search for all possible 4x4 word squares; for more information see:

Lisp with Xgrid

The program found some 100 4x4 word squares; here are six of the best, avoiding obscure words:

 T  A  N  S
 A  R  E  A
 L  I  O  N
 L  A  N  D
 B  A  N  D
 A  R  E  A
 L  I  A  R
 D  A  T  E
 S  L  A  M
 T  I  L  E
 E  A  T  S
 P  R  O  S
 F  L  I  T
 L  I  N  O
 O  A  T  S
 P  R  O  S
 C  R  A  M
 R  O  L  E
 O  A  T  S
 P  R  O  S
 B  A  L  D
 A  R  E  A
 L  I  A  R
 L  A  K  E

I also searched for 5x5, 6x6, and 7x7 squares but failed to find any using common words, and including the diagonals.

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