Maximal Hook Chains, OR
The Seedy Side of Scrabble

A while back on a Scrabble mailing list, an Australian player by the name of Tony Kalayzich, asked for the longest possible sequence of successive hooks (whether front or back) that can be made using the current dictionary (which at the time of writing is CSW - Collins Scrabble Words).

His example should give you the idea...

ME => MET => METH => METHO => METHOD => METHODS

I'm going to assume that we can add a hook to either the front or the back at each step, so that the sequence doesn't have to be made up solely of fronthooks, or solely of backhooks. Such mixtures, for example...

AT => BAT => BATE => ABATE => ABATES

make the task more complex, but also considerably more interesting.

I am going to answer Tony's question, but first I'll need to introduce you to some terminology...

We'll refer to a word as a hooker if it can be built up from a two-letter word by a sequence of successive hooks. The operation of further extending a hooker, by the addition of a hook, will be called a trick (or more verbosely 'doing/taking a trick').

A hooker capable of doing another trick will be referred to as a working hooker, while one which is incapable of taking another trick will be called a retired hooker. A sequence of tricks, considered in its entirety, will be called a career.

It will also be convenient to have a name for the fundamental two-letter word upon which a hooker's tricks depend. We'll call such a word a pimp. Although the rules state that a pimp must be a two-letter word, it is often interesting to consider pimps with three letters or more. To distinguish these, we will use the adjective 'illegal', and thereby speak of illegal hookers and so forth.

The aim of the present investigation can now be phrased as follows...

We would like to identify those hookers having the longest careers. Such words will be said to be madams, and it is to this charming lexical stratum that we now turn our attention.

Now I realize that you're very keen to meet some madams, but first I'd like to introduce you to their pimps. Here they are...

AI
AM
AN
AR
AS
AT
DE
EA
HI
IN
IS
LA
LI
MI
MO
OD
PA
PI
TE

What do these two letter words mean?

So, each of these pimps (and only these pimps) can lead you to at least one madam.

At this point, you might want to have a go at locating a few retired hookers belonging to these pimps, to give you a feel for the level of combinatorial complexity of this problem. Or, you might prefer to just look up the word 'combinatorial' instead.

(Note that retired hookers are much easier to identify than madams, because you don't care about the length of their careers - you only need to verify that they can't take any more tricks.)

Some Interesting Results

I'll leave the curricula vitae of all of our madams for you to browse shortly, but before I go I'll just make a handful of observations.

Some interesting working hookers and their tricks

Browsing the careers of all these madams is bound to teach you a few tricks. Here are a couple of cute working hookers along with their tricks.

You might have guessed that CANTIC was a working hooker, but did you realise just how many tricks CANTIC will perform before retirement?

CANTIC => CANTICO => CANTICOY => CANTICOYS

PIRATED comes across to many as a retired hooker. But like CANTIC, PIRATED still has back-to-back tricks waiting in the corridor...

PIRATED => SPIRATED => ASPIRATED

Illegal madams

As mentioned earlier, one can consider the illegal careers of hookers based upon pimps that violate the rules by having three letters or more.

One of the more interesting illegal madams I encountered in this investigation had the following illustrious career...

QUIN => QUINT => QUINTE => QUINTET => QUINTETT => QUINTETTE => QUINTETTES

The Full Results

Ok, without further ado, here are all of Collins' madams, organised by their pimps, along with a summary of their careers and the tricks constituting them...

---AI---
AI=>AIS=>RAIS=>RAISE=>PRAISE=>PRAISER=>PRAISERS=>UPRAISERS
AI=>AIS=>RAIS=>RAISE=>PRAISE=>PRAISER=>UPRAISER=>UPRAISERS
AI=>AIS=>RAIS=>RAISE=>PRAISE=>UPRAISE=>UPRAISER=>UPRAISERS
AI=>AIS=>RAIS=>RAISE=>RAISER=>PRAISER=>PRAISERS=>UPRAISERS
AI=>AIS=>RAIS=>RAISE=>RAISER=>PRAISER=>UPRAISER=>UPRAISERS
AI=>AIS=>RAIS=>RAISE=>RAISER=>RAISERS=>PRAISERS=>UPRAISERS
AI=>RAI=>RAIS=>RAISE=>PRAISE=>PRAISER=>PRAISERS=>UPRAISERS
AI=>RAI=>RAIS=>RAISE=>PRAISE=>PRAISER=>UPRAISER=>UPRAISERS
AI=>RAI=>RAIS=>RAISE=>PRAISE=>UPRAISE=>UPRAISER=>UPRAISERS
AI=>RAI=>RAIS=>RAISE=>RAISER=>PRAISER=>PRAISERS=>UPRAISERS
AI=>RAI=>RAIS=>RAISE=>RAISER=>PRAISER=>UPRAISER=>UPRAISERS
AI=>RAI=>RAIS=>RAISE=>RAISER=>RAISERS=>PRAISERS=>UPRAISERS

---AM---
AM=>AMI=>AMIN=>AMINE=>AMINES=>TAMINES=>ETAMINES=>KETAMINES
AM=>AMI=>AMIN=>AMINE=>TAMINE=>ETAMINE=>ETAMINES=>KETAMINES
AM=>AMI=>AMIN=>AMINE=>TAMINE=>ETAMINE=>KETAMINE=>KETAMINES
AM=>AMI=>AMIN=>AMINE=>TAMINE=>TAMINES=>ETAMINES=>KETAMINES
AM=>AMI=>AMIN=>TAMIN=>ETAMIN=>ETAMINE=>ETAMINES=>KETAMINES
AM=>AMI=>AMIN=>TAMIN=>ETAMIN=>ETAMINE=>KETAMINE=>KETAMINES
AM=>AMI=>AMIN=>TAMIN=>TAMINE=>ETAMINE=>ETAMINES=>KETAMINES
AM=>AMI=>AMIN=>TAMIN=>TAMINE=>ETAMINE=>KETAMINE=>KETAMINES
AM=>AMI=>AMIN=>TAMIN=>TAMINE=>TAMINES=>ETAMINES=>KETAMINES

---AN---
AN=>AND=>LAND=>LANDE=>LANDER=>LANDERS=>SLANDERS=>ISLANDERS
AN=>AND=>LAND=>LANDE=>LANDER=>SLANDER=>ISLANDER=>ISLANDERS
AN=>AND=>LAND=>LANDE=>LANDER=>SLANDER=>SLANDERS=>ISLANDERS
AN=>ANT=>ANTI=>ANTIC=>CANTIC=>CANTICO=>CANTICOY=>CANTICOYS
AN=>EAN=>EANS=>LEANS=>CLEANS=>CLEANSE=>CLEANSER=>CLEANSERS
AN=>EAN=>LEAN=>CLEAN=>CLEANS=>CLEANSE=>CLEANSER=>CLEANSERS
AN=>EAN=>LEAN=>LEANS=>CLEANS=>CLEANSE=>CLEANSER=>CLEANSERS

---AR---
AR=>ARK=>PARK=>PARKI=>PARKIE=>PARKIES=>PARKIEST=>SPARKIEST
AR=>ARK=>PARK=>PARKI=>PARKIE=>PARKIES=>SPARKIES=>SPARKIEST
AR=>ARK=>PARK=>PARKI=>PARKIE=>SPARKIE=>SPARKIES=>SPARKIEST
AR=>PAR=>PARK=>PARKI=>PARKIE=>PARKIES=>PARKIEST=>SPARKIEST
AR=>PAR=>PARK=>PARKI=>PARKIE=>PARKIES=>SPARKIES=>SPARKIEST
AR=>PAR=>PARK=>PARKI=>PARKIE=>SPARKIE=>SPARKIES=>SPARKIEST

---AS---
AS=>ASS=>LASS=>LASSI=>LASSIE=>GLASSIE=>GLASSIES=>GLASSIEST
AS=>ASS=>LASS=>LASSI=>LASSIE=>LASSIES=>GLASSIES=>GLASSIEST
AS=>ASS=>LASS=>LASSI=>LASSIS=>CLASSIS=>CLASSISM=>CLASSISMS
AS=>ASS=>LASS=>LASSI=>LASSIS=>CLASSIS=>CLASSIST=>CLASSISTS
AS=>LAS=>LASS=>LASSI=>LASSIE=>GLASSIE=>GLASSIES=>GLASSIEST
AS=>LAS=>LASS=>LASSI=>LASSIE=>LASSIES=>GLASSIES=>GLASSIEST
AS=>LAS=>LASS=>LASSI=>LASSIS=>CLASSIS=>CLASSISM=>CLASSISMS
AS=>LAS=>LASS=>LASSI=>LASSIS=>CLASSIS=>CLASSIST=>CLASSISTS

---AT---
AT=>ATE=>LATE=>ELATE=>ELATES=>RELATES=>PRELATES=>PRELATESS
AT=>ATE=>LATE=>ELATE=>RELATE=>PRELATE=>PRELATES=>PRELATESS
AT=>ATE=>LATE=>ELATE=>RELATE=>RELATES=>PRELATES=>PRELATESS
AT=>ATE=>RATE=>IRATE=>PIRATE=>PIRATED=>SPIRATED=>ASPIRATED
AT=>EAT=>EATH=>HEATH=>SHEATH=>SHEATHE=>SHEATHER=>SHEATHERS
AT=>EAT=>HEAT=>HEATH=>SHEATH=>SHEATHE=>SHEATHER=>SHEATHERS
AT=>LAT=>LATE=>ELATE=>ELATES=>RELATES=>PRELATES=>PRELATESS
AT=>LAT=>LATE=>ELATE=>RELATE=>PRELATE=>PRELATES=>PRELATESS
AT=>LAT=>LATE=>ELATE=>RELATE=>RELATES=>PRELATES=>PRELATESS
AT=>RAT=>RATE=>IRATE=>PIRATE=>PIRATED=>SPIRATED=>ASPIRATED

---DE---
DE=>ODE=>MODE=>MODER=>MODERN=>MODERNE=>MODERNES=>MODERNEST

---EA---
EA=>EAN=>EANS=>LEANS=>CLEANS=>CLEANSE=>CLEANSER=>CLEANSERS
EA=>EAN=>LEAN=>CLEAN=>CLEANS=>CLEANSE=>CLEANSER=>CLEANSERS
EA=>EAN=>LEAN=>LEANS=>CLEANS=>CLEANSE=>CLEANSER=>CLEANSERS
EA=>EAT=>EATH=>HEATH=>SHEATH=>SHEATHE=>SHEATHER=>SHEATHERS
EA=>EAT=>HEAT=>HEATH=>SHEATH=>SHEATHE=>SHEATHER=>SHEATHERS
EA=>LEA=>LEAN=>CLEAN=>CLEANS=>CLEANSE=>CLEANSER=>CLEANSERS
EA=>LEA=>LEAN=>LEANS=>CLEANS=>CLEANSE=>CLEANSER=>CLEANSERS

---HI---
HI=>HIE=>HIES=>SHIES=>SHIEST=>ASHIEST=>WASHIEST=>SWASHIEST
HI=>HIN=>SHIN=>SHINE=>SHINES=>SHINESS=>ASHINESS=>WASHINESS

---IN---
IN=>HIN=>SHIN=>SHINE=>SHINES=>SHINESS=>ASHINESS=>WASHINESS
IN=>LIN=>BLIN=>BLING=>ABLING=>ABLINGS=>TABLINGS=>STABLINGS
IN=>LIN=>BLIN=>BLING=>ABLING=>TABLING=>STABLING=>STABLINGS
IN=>LIN=>BLIN=>BLING=>ABLING=>TABLING=>TABLINGS=>STABLINGS
IN=>LIN=>BLIN=>BLING=>BLINGS=>ABLINGS=>TABLINGS=>STABLINGS
IN=>LIN=>LING=>BLING=>ABLING=>ABLINGS=>TABLINGS=>STABLINGS
IN=>LIN=>LING=>BLING=>ABLING=>TABLING=>STABLING=>STABLINGS
IN=>LIN=>LING=>BLING=>ABLING=>TABLING=>TABLINGS=>STABLINGS
IN=>LIN=>LING=>BLING=>BLINGS=>ABLINGS=>TABLINGS=>STABLINGS
IN=>LIN=>LING=>LINGS=>BLINGS=>ABLINGS=>TABLINGS=>STABLINGS
IN=>PIN=>PING=>APING=>RAPING=>CRAPING=>SCRAPING=>SCRAPINGS
IN=>PIN=>PING=>OPING=>OOPING=>COOPING=>SCOOPING=>SCOOPINGS
IN=>PIN=>PING=>OPING=>OOPING=>HOOPING=>WHOOPING=>WHOOPINGS
IN=>WIN=>WING=>OWING=>ROWING=>TROWING=>STROWING=>STROWINGS

---IS---
IS=>AIS=>RAIS=>RAISE=>PRAISE=>PRAISER=>PRAISERS=>UPRAISERS
IS=>AIS=>RAIS=>RAISE=>PRAISE=>PRAISER=>UPRAISER=>UPRAISERS
IS=>AIS=>RAIS=>RAISE=>PRAISE=>UPRAISE=>UPRAISER=>UPRAISERS
IS=>AIS=>RAIS=>RAISE=>RAISER=>PRAISER=>PRAISERS=>UPRAISERS
IS=>AIS=>RAIS=>RAISE=>RAISER=>PRAISER=>UPRAISER=>UPRAISERS
IS=>AIS=>RAIS=>RAISE=>RAISER=>RAISERS=>PRAISERS=>UPRAISERS

---LA---
LA=>LAP=>LAPS=>LAPSE=>ELAPSE=>RELAPSE=>RELAPSER=>RELAPSERS
LA=>LAS=>LASS=>LASSI=>LASSIE=>GLASSIE=>GLASSIES=>GLASSIEST
LA=>LAS=>LASS=>LASSI=>LASSIE=>LASSIES=>GLASSIES=>GLASSIEST
LA=>LAS=>LASS=>LASSI=>LASSIS=>CLASSIS=>CLASSISM=>CLASSISMS
LA=>LAS=>LASS=>LASSI=>LASSIS=>CLASSIS=>CLASSIST=>CLASSISTS
LA=>LAT=>LATE=>ELATE=>ELATES=>RELATES=>PRELATES=>PRELATESS
LA=>LAT=>LATE=>ELATE=>RELATE=>PRELATE=>PRELATES=>PRELATESS
LA=>LAT=>LATE=>ELATE=>RELATE=>RELATES=>PRELATES=>PRELATESS

---LI---
LI=>LIN=>BLIN=>BLING=>ABLING=>ABLINGS=>TABLINGS=>STABLINGS
LI=>LIN=>BLIN=>BLING=>ABLING=>TABLING=>STABLING=>STABLINGS
LI=>LIN=>BLIN=>BLING=>ABLING=>TABLING=>TABLINGS=>STABLINGS
LI=>LIN=>BLIN=>BLING=>BLINGS=>ABLINGS=>TABLINGS=>STABLINGS
LI=>LIN=>LING=>BLING=>ABLING=>ABLINGS=>TABLINGS=>STABLINGS
LI=>LIN=>LING=>BLING=>ABLING=>TABLING=>STABLING=>STABLINGS
LI=>LIN=>LING=>BLING=>ABLING=>TABLING=>TABLINGS=>STABLINGS
LI=>LIN=>LING=>BLING=>BLINGS=>ABLINGS=>TABLINGS=>STABLINGS
LI=>LIN=>LING=>LINGS=>BLINGS=>ABLINGS=>TABLINGS=>STABLINGS

---MI---
MI=>AMI=>AMIN=>AMINE=>AMINES=>TAMINES=>ETAMINES=>KETAMINES
MI=>AMI=>AMIN=>AMINE=>TAMINE=>ETAMINE=>ETAMINES=>KETAMINES
MI=>AMI=>AMIN=>AMINE=>TAMINE=>ETAMINE=>KETAMINE=>KETAMINES
MI=>AMI=>AMIN=>AMINE=>TAMINE=>TAMINES=>ETAMINES=>KETAMINES
MI=>AMI=>AMIN=>TAMIN=>ETAMIN=>ETAMINE=>ETAMINES=>KETAMINES
MI=>AMI=>AMIN=>TAMIN=>ETAMIN=>ETAMINE=>KETAMINE=>KETAMINES
MI=>AMI=>AMIN=>TAMIN=>TAMINE=>ETAMINE=>ETAMINES=>KETAMINES
MI=>AMI=>AMIN=>TAMIN=>TAMINE=>ETAMINE=>KETAMINE=>KETAMINES
MI=>AMI=>AMIN=>TAMIN=>TAMINE=>TAMINES=>ETAMINES=>KETAMINES

---MO---
MO=>MOD=>MODE=>MODER=>MODERN=>MODERNE=>MODERNES=>MODERNEST

---OD---
OD=>MOD=>MODE=>MODER=>MODERN=>MODERNE=>MODERNES=>MODERNEST
OD=>ODE=>MODE=>MODER=>MODERN=>MODERNE=>MODERNES=>MODERNEST

---PA---
PA=>PAR=>PARK=>PARKI=>PARKIE=>PARKIES=>PARKIEST=>SPARKIEST
PA=>PAR=>PARK=>PARKI=>PARKIE=>PARKIES=>SPARKIES=>SPARKIEST
PA=>PAR=>PARK=>PARKI=>PARKIE=>SPARKIE=>SPARKIES=>SPARKIEST

---PI---
PI=>PIN=>PING=>APING=>RAPING=>CRAPING=>SCRAPING=>SCRAPINGS
PI=>PIN=>PING=>OPING=>OOPING=>COOPING=>SCOOPING=>SCOOPINGS
PI=>PIN=>PING=>OPING=>OOPING=>HOOPING=>WHOOPING=>WHOOPINGS

---TE---
TE=>ATE=>LATE=>ELATE=>ELATES=>RELATES=>PRELATES=>PRELATESS
TE=>ATE=>LATE=>ELATE=>RELATE=>PRELATE=>PRELATES=>PRELATESS
TE=>ATE=>LATE=>ELATE=>RELATE=>RELATES=>PRELATES=>PRELATESS
TE=>ATE=>RATE=>IRATE=>PIRATE=>PIRATED=>SPIRATED=>ASPIRATED

NOTE — You might have noticed that some, indeed most, madams have more than one career under the one pimp. It is not my intention, either here or elsewhere, to provide plausible interpretations of this practice.
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