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Heterogrammatic eight-squares.

Eight-squares exist in overwhelming numbers (a few million million by my guess, and 27 million million by Chris Long's formula), so there are many 'perfect' examples. A 1902 effort by Charles B Stewart had a two-word place name and a word with an initial capital: decisive egomania computes impotent sauterne interned Viennese East Eden. In November 1926 (in Enigma) Mrs Burkholder came close to perfection with words from a single dictionary: Agaricus (genus) generant anaconda recanter ironwort cantonal underage (Webster2 is needed to avoid hyphen) startled. Finally, a 'perfect' square by Darryl Francis was reported in W88-80: rosetter overrule sequined eruptive tritical tunicate elevates redeless, all in Web3, except for redeless in Web2.

The major challenge is therefore to find eight-squares with some additional attributes. I had failed to find a nine-square made from heterogrammatic words by a margin of one word (Nine-Square Roundup in August 2003), so it was natural to hunt for heterogrammatic eight-squares. This turned out to be a worthwhile challenge, as I found many (18000), but none quite perfect. Given that I also found a very few diagonal nine-squares with a single diagonal, the supposition is that there is a legion of such eight-squares, but what about heterogrammatic eights with diagonal words?

Throughout the article, the squares in each section are presented in approximately descending order of quality, marks being deducted for phrases, hyphens, apostrophes, non-headwords, initial caps, foreign words etc, multiple deductions applying to a single word. All letters in the squares are normally lower-case, except where boldface indicates a capital letter. For some squares, the notes give alternative words after a semicolon.

Heterogrammatic Eight-Squares

Despite being restricted to just 15% of the 8-letter vocabulary of around 750,000, the heterogrammatic squares below are almost perfect, in that virtually all words are solid lower-case dictionary headwords (or regular derivations). They are thus comparable in quality to the best ten-squares taken from a full vocabulary. In fact, I have failed to find any previous heterogrammatic word squares in Word Ways larger than 5x5 (eg agami gamin amine miner inert in W69-226).

Here is a small selection of the better examples from the 18,000 or so squares found. BISCEOPA heads three of the best squares.


Heterogrammatic Eight-Squares with SW-NE diagonal

Of the 18,000 heterogrammatic squares, about 300 (1.7%) have a word as the SW-NE diagonal. Some of the better squares showing each of the ten different SW-NE diagonal words are given below. No squares with either of the NW or SE diagonals were found. The SW-NE diagonal is perforce palindromic, so cannot be a heterogram. All except one of the diagonals are to be found in Palindromicon II. Disregarding the merits of the diagonal word itself, the diagonals SIISSII'S, HRRRRRRH, AAAAAAAA, and EEEEEEEE yield the best squares, in which half the words are 'perfect'.

1. Suussuu's: 3 found


2. siisii's: 8 found and given


3. Saassaas (places): 1 found


4. hr-r-r-r-r-rh (EDD): 7 found


5. tuuttuut: 2 found


6. AAAAAAAA (tutoring service): 175 found.


7. Dddddddd (OED signature 6a 1864q "Detached Sheet: 4to; signature, Dddddddd"): 3 found


8. EEEEEEEE (James Joyce, Ulysses): 86 found


9. MMMMMMMM (Washington Post, 22 Dec 1978, c4/1 "Mmmmmmmm" at Travolta's hunky nearness"): 5 found.


10. OOOOOOOO (Bluebottle, in The Goon Show): 11 found.


Popular Top Row Words in Heterogrammatic squares

A frequent everyday top row word (0.16% of squares) is DESPATCH. More frequent are the following, with the TOMBRACH the most popular, occurring at the top of 0.43% of all squares:


Popular Words anywhere in Heterogrammatic Squares

A frequent everyday word is ISLANDER, at 0.07% of all 150,000 words in the squares; KINGSLEY (Water Babies) and TUCKINGS are a little more frequent; but the most frequent, occurring up to 0.225% of the time, are listed here in ascending frequency:


Sources: Dorland = Dorland's Illustrated Medical Dictionary; ITIS = Integrated Taxonomic Information System of the USDA; ITISa = ITIS animal; ITISp = ITIS plant; NIMA = National Imagery and Mapping Agency (USA, default source for place names); OSPD = Official Scrabble Players Dictionary.


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Author:Gooch, Rex
Publication:Word Ways
Date:May 1, 2005
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