# QUEEN'S MOVE GRAPHING

by Ross Eckler
Word Ways, 1996

In chess, the queen is the most versatile piece, moving in any of eight directions. Unlike the king, which is constrained to move only one square at a time, she can move in a straight line until another piece is encountered. Translated to logology, this means that any word whose separate letters can be placed on chessboard squares so that it can be spelled out by a king, can also be spelled out by a queen. Furthermore, there will exist some words that can only be spelled out by queen's moves.

Ascertaining necessary and sufficient conditions for queen's move graphability is an extraordinarily difficult problem. An obvious necessary condition for a queen graphable word is that no letter in it participates in bigrams with more than eight different other letters. For example, CHOLEDOCHODUODENOSTOMIES cannot be queen graphed because it contains nine different letters sharing a bigram with O: HO,OL,DO,OC,UO,NO,OS,TO,OM. However, this is not a particularly useful condition; few if any words fail queen graphability solely because of this. The generalization of this requirement to two letters, each of which participates in bigrams with a common set of other letters, is even less useful.

No other rules are known. However, a useful technique for determining queen graphability is to identify all bigram triads such as EN,EC,CN or AT,ET,EA that exist in a word. Bigram triads must be assembled in right-angled structures that have very little freedom to be adjusted, particularly when two or more bigram triads share a pair of letters. For example, it is easy to convince onself that INSCIENCES is not queen graphable, for it contains the four triads

```	 S	 N	 N	 C			 NC     NS
CE	SE	IE	IN			IES    IEC```

which cannot be simultaneously realized. The two diagrams at the right show how the first three triads can be assembled, but the fourth cannot be incorporated since C,N and I cannot be placed in a right-angled structure with full accessibiity (in the second diagram, E blocks I from C).

Usually, it takes a bit more analysis to prove that a word is not queen graphable. To illustrate, TETRAIODOPHENOLPHTHALEIN contains the triads OP,OL,LP as well as IO,NO,IN and EN,EI,IN. These can be assembled in several different ways:

```	  I			  IE			 EI
PONE			LON			LON
L			 P			P```

One should next look for possible short connections between letters such as P or L on the left, and E on the right. There are in fact two: PHE and LE. To achieve these, P and L must be extended.

```
H
+++
++++T
++I++
P+ONE
+++
L```

T can be added on the right to accommodate the bigrams TE and TH, but A, which participates in HA,AI and AL, cannot be located anywhere in the grid. Experimentation with the other arrays also leads to dead ends.

There are probably only a few dozen Websterian words that cannot be queen graphed, but no systematic survey of these has been made. One place to start is with words that cannot be king graphed. Of nineteen 24-letter words so identified by Leonard Gordon, five failed: the two already discussed, and PACREATICOGASTROSTOMIES, HEMATOSPECTROPHOTOMETERS and ELECTROENCEPHALOGRAPHERS.

On page 4 of his February 1995 article "Introduction to Word Graphs," Leonard Gordon lists five 15-letter words that cannot be king graphed. Of these, three cannot be queen graphed either: PROSCRIPTIONIST, OVERCONCENTRATE and UNCONSCIENTIOUS. The other two words, AURICULOCRANIAL and DIPLOSPONDYLISM, are queen graphed below.

```	 C++O			  YD			  LH
IRU+			  +I			 ETY
NAL			 +LPSM			DMNRP
N++O			 IAO
```

Two other words in Gordon's article fail queen-graphing, PNEUMORETROPERITONEUM and VENTRICULOCISTERNOSTOMIES (after assembling the four triads RN,NT,TR, ST,OS,TO, EN,NT,TE and NT,TRRN, there is no way that R, S and E, all of which must join I, can be placed in a stright line).

A word with five different letters is potentially queen graphable if it contains no more than eight of the ten possible bigrams of ten different letters. INTRATRINITARIAN, INTENSITIES and INSCIENCES, all having nine, are not. Similarly, a word with six different letters is potentially queen graphable if it contains no more than eleven of the fifteen possible bigrams of two different letters. No words that fail this requirement are known.

The shortest word that is not queen graphable is INSCIENCES, and the longest queen graphable word is DIAMINOPROPYLTETRAMETHYLENEDIAMINE, found in the Random House Unabridged. This word is actually king graphable, as shown by Leonard Gordon in the February 1995 Word Ways; it is diagrammed above.