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Heterogrammic Word Squares

Yesterday's LOGOLOG entry dealt with the notion of pangrams, arrangements of letters which contains every letter of the alphabet at least once. I also hinted at the complementary concept: the "heterogram". An heterogram is a collection of letters wherein none of the letters is repeated.

The smallest possible pangram would be 26 letters, while the largest possible heterogram would be 26 letters. With exactly one instance of each letter, pangrammacy and heterogrammacy become equivalent.

This past week I decided to tackle a pretty tough puzzle which popped into my head (though, as I learned later, once again it has been tackled by others in the past). I wanted to see if I could construct a 4x4 Word Square which was heterogrammic. In otherwords, I needed to use 16 different leters of the alphabet to form a 4x4 word grid such that each row and column forms a valid word.

Creating a 3x3 example is not that hard at all:

EFS
LIT
DRY

EFS, more than one "F"
LIT, past tense of "LIGHT"
DRY, not wet
ELD, archaic word: "old person" (old word: "archaic person"?)
FIR, type of evergreen tree
STY, a pigpen

Those are all fairly common words, and I bet the reader can find a 3x3 example with even more familiar terms.

A 4x4 is a different monster altogether. Suddenly, the vowels are at a premium and familiar consonant digraphs are important. 16 of 26 letters wouldn't seem to be too difficult until you understand just how well they have to fit together. Each word affects the other, and finding valid combinations is no trivial task.

In fact, after a couple of hours, I was beginning to think it might be impossible.

I had come pretty close a couple of times. For example, I put these 15 letters down just to be foiled by that first cell:

?PID
ULNA
ROCK
ETHS

I also found a configuration that formed a solid 4x4 but required a 17th letter (the "I"!) to be tacked onto the bottom:

MOCH
UGLY
FRAP
TENS
I   

Oh the frustration!

Determination prevailed, however, and by toying with that last configuration a bit, I finally hit upon my solution. As usual, I was looking for words from a single word source (the MW3 in this case). No hyphens, phrases, capitalizations, or abbreviations:

YOCK
UGLI
FRAP
TENS

YOCK, to laugh
UGLI, a hybrid fruit (tangelo)
FRAP, to bind tightly with ropes
TENS, more than one 10
YUFT, Russia leather
OGRE, a giant
CLAN, a small tribe
KIPS, sleeps

The "N" may be replaced with the "D" or "W" to get slightly different solutions.

I was very pleased when I created this configuration. Of course, had I consulted my various logology references beforehand I would have seen that a 4x4 heterogrammic word square is not a new thing. The book Making the Alphabet Dance mentions a few (the real challenge of logology is to find a word puzzle that has not already been tackled ad nauseum...)

Still, my 4x4 word square was different than those mentioned before, and I was proud enough of it to create a little plaque for myself:

While a 5x5 square would only require 25 of the 26 letters of the alphabet, I can say with near certainty that an heterogrammic square of that size is impossible. A pair of 3x3s (using 18 letters total) might be doable, however...

-- Eric

Update: Web friend Wei-Hwa Huang tackled this in a much more reasonable way than my shuffling of Scrabble tiles. He wrote a program to generate all possible 4x4 heterogrammic word squares (he used the MW2 as a word source). These are his results.

Another Update: I was inspired to write my own heterogrammic word square crunching program to look for pairs of 3x3s. Using the OSPD3 wordlist I found 15 such pairs.

[3 January 2006]
   
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