# A wordy digression: the game of HIPE.

In Mathematical Mind-Benders, Peter Winkler. Wellesley, MA: A K Peters, 2007. Reprinted by permission of A K Peters, Ltd (www.akpeters.com).

How every fool can play upon the word!

--Lorenzo, in Shakespeare's The Merchant of Venice

This is a free chapter--my publisher assures me that the book would be the same price without it. Think of it as an intermission, a non-mathematical break. Many mathematicians do love word games, though, and (in my experience) this one in particular.

You may have heard the following puzzle: what English word contains four consecutive letters that are consecutive letters of the alphabet? Answer: undeRSTUdy. Inspired by this and other word puzzles, I and three other high-school juniors (6) at a 1963 National Science Foundation summer program began to fire letter combinations at one another, asking for a word containing that combination.

The combination had to appear in that order and with no other letters in between. Example: WKW = aWKWard, NSW = aNSWer. Some were only two letters, e.g., HQ = eartHQuake; ZV = rendeZVous (or another borrowed word, mitZVah). Try BV yourself (hint: this one is obvious).

Double letters (WW, VV, HH) can be fun too; but the most deadly combinations we found were three or four letters, as in GNT, PTC, THAC, HEMU--answers at the end of the chapter. We named, the game after one of our favorite combinations, HIPE = arcHIPElago (despite later discovering that "worshiper" can be spelled with one "p"). Of course, HIPE has no doubt been invented and. reinvented thousands of times over history, and you are under no obligation to use our name--but it's helpful to call it something. In devising a HIPE, a natural objective is for it to be artfully concealed in the solution; and it's additionally satisfying for the solution to be a common word that's hard to think of. For example, ONIG has a solution that is among the most common words in the language: can you find it? And what fun it is to stump a friend with a HIPE like LYLY, then tell them they have to add only one more letter.

Ideally, you'd like the solution to be unique (among, say, Scrabble[TM]-eligible, non-capitalized words), but insisting on this leads to unnecessary quibbling. Still, a really good HIPE should have one standout answer. Back in 1963 we agreed on the following HIPE etiquette: when your victim produces a valid solution to your HIPE, his or her obligation is fully discharged. In particular, you are not allowed to ask for the solution you had in mind.

During the summers of 1967 and 1968 I gave a HIPE a day to all the (supposedly) gifted kids at Science and Arts Camps, in Port Ewen, NY. Here are some of their favorites: SPB, RAOR, XS, DQ, HCR, UDU, YEB, YNG, TIK, XOP, and BEK--answers at end of chapter.

Rarely do good HIPEs of more than four letters arise, since they tend to give away too much information. Two cute ones, notable for their pleasing repetitiveness, are ACHACH and TANTAN (solutions at end).

Who is good at solving HIPEs? Over the years, I have made an informal study of this question. Of course, as with mathematical puzzles, you never know until the game is played. People you think would be terrific fall flat on their faces, while others shock you by spouting answers you haven't even thought of.

There are some interesting tendencies, though. Once I found an excuse to introduce the game while teaching a class on automata about context-free languages; one student proved tO be better at it than the rest of the class put together. She was the first educated person in her family and was a voracious reader who could spell many more words than she could pronounce. Naturally, she was less likely to be misled by tricky HIPEs; she knew words by their appearance.

How, indeed, do we "know" a word? Is it primarily by the word's meaning, by the word's sound, or by the word's appearance? Would you believe none of the above? My psychologist friends tell me that the kinetic sense of how to produce a word dominates, at least in some sense, the other aspects. Now, if that's really the case, then people who can't hear well, especially those who customarily communicate in a sign language, might be expected to have an advantage in playing HIPE. Their concept of a word might be expected to be short on sound and production (with the mouth) and thus relatively long on spelling.

Resolved to test this hypothesis, I introduced HIPE to a group of hearing-impaired employees at a government agency, who sat together and conversed in ASL daily at lunch. They found the game completely trivial; as fast as I wrote HIPEs on napkins, they wrote solutions around them. To them it was a mystery why anyone would think HIPE was any kind of challenge.

Looking back on this episode, it occurs to me that the fact that some of the folks at this table were professional cryptanalysts should have made me a bit hesitant about drawing conclusions. But is the connection between hearing loss and code-breaking a coincidence? My advice is, if you need someone to (say) read a garbled telegram, try your hearing-impaired friends first.

Here's a longer list of HIPEs, with no solutions provided. Of course, you can find solutions for any of them easily on your computer, by downloading a word list from the web and searching for a given HIPE combination by your favorite method. But I suggest trying out your brain first.
```BG   CM   FC   FW   GC   GJ
KC   GD   GZ   HK   IJ   KG
KJ   LJ   LQ   MD   MQ   PJ
TJ   TK   TV   UH   UQ   XF
XG   XN   XQ   ZB   ZK   ZM
ZP   ZW

AFY   AIE   AIQ   AKN   AKT
AMT   ANW   AOH   AOT   APK
ATG   AWG   BFR   BOF   BOJ
BPL   BPO   BTF   CAQ   CEC
CEK   CHG   CTY   CYH   CYO
CZE   DDB   DDM   DEK   DEQ
DHP   DSC   DSK   DTE   DWR
DYB   DYG   DYM   EGW   EKD
EOI   ETD   EWG   EZA   FEG
FEK   FSA   FSI   FTB   FTT
GEC   GEF   GGN   GGP   GHG
GNP   GOC   GSK   HDI   HDU
HLR   HMM   HPL   HPR   HSH
HSK   HTC   HTG   HTM   HYN
IAI   IAU   IDP   IEI   IFA
IJI   IMF   IOA   IOE   IOI
IOV   IPK   IUT   LYF   IZU
KAC   KBL   KCL   KEB   KSG
KSK   KSU   KTR   KUS   KYC
KYR   LAL   LCL   LDB   LDT
LEK   LEQ   LPL   LII   LKH
LML   LNO   LPF   LPL   LSC
LTC   LTP   LYD   LYF   LYV

MBB   MBN   MBP   MCH   MEU
MKH   MND   MNL   MPF   MPG
MPM   MPP   MSK   MSM   MSP
MSU   MVE   MWI   NDD   NDJ
NDK   NGN   NKG   NKM   NNK
NYH   NYP   OAB   OAU   OAV
OEQ   OEU   OHO   OHY   OIE
OKC   OUA   OUQ   OWU   OYH
OYO   OYR   PEV   PIP   PIU
PFR   PPH   PSF   PSM   PSP
PYC   PYW   RDP   RDV   RFD
RJU   RLH   RMC   RMP   RNH
RPM   RSB   RTG   RYD   SDR
SHH   SIQ   SKR   SSK   SUO
SYW   TBL   THC   THT   TIW
TMU   TOZ   TYD   UAH   UDB
UEO   UFA   UMC   UXU   VEF
VEP   VEW   VYH   VYS   WAW
WPE   WNC   WNM   WNP   WNU
XTB   XYE   XYM   XYS   YAG
YFR   YHA   YOE   YRD   YSC
YSL   YSY   YUN   YZY   ZEF

CEBE    DROB    ECIB    ELEL
ERYO    EWHA    FELE    FRAR
GHAG    GUAG    FODI    HA00
LSEL    IKEA    ITCA    MECA
MEON    MELT    MYRO    NIKE
OELA    OFOL    OFTO    OOMM
PIOC    PLEL    PONR    RESK
ROOR    SPES    TYPU    UBBU
UGUG    ZIPA
```

Promised solutions to earlier HIPEs:

* gloWWorm, poWWow

* saVVy, reVVing

* hitcHHike, batHHouse

* sovereiGNTy

* bankruPTCy

* tooTHAChe

* chrysantHEMUm

* raSPBerry

* extRAORdinary

* coXSwain

* witcHCRaft

* fraUDUlent

* eYEBrow, eYEBall

* larYNGitis

* swasTIKa, baTIK

* saXOPhone

* unBEKnown

* stomACHACHe

* insTANTANeous

I hope you've enjoyed HIPE (7) and will use the game to drive your friends crazy. Now back to mathematical puzzles.

(6) Richard Thurston, Robert Webber, and Robert Winternitz--yes, roommates were assigned alphabetically.

(7) For those who think games like this are good for nothing, picture your author as a high-school senior trying to get into Harvard. He is asked to write about himself. He is from Fair Lawn, NJ, and is good at math. He is competing with kids who were raised by monks in Mandalay, played solo bassoon for the Queen of England, and cloned a moose at age 12. What can he say? Hoping to sucker the admissions officers into playing HIPE, he writes a light essay called "The HIPE Story" in which he describes how he and his friends started a local craze. Four years later, he is a senior at Harvard and overhears a tutor who had served on the admissions committee torturing a colleague with HYPEs. What's more, the tutor is calling them HIPEs. So, I figure HIPEs got me into Harvard, and I owe them this chapter.

PETER WINKLER