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Letter trees.

On March 13 2005, Will Shortz presented an interesting word puzzle on National Public Radio: rearrange the letters of K + ANALOGIES into a tree, where every path leading downward is a four-letter word. The tree at the left yields saga, sage, sane, sank, sine, sink, silk and silo.


There are thousands of ten-letter groups that form eight different words, so it is necessary to restrict the conditions of the problem. First, the ten letters must form a word; second, all ten letters must be different; third, the solution must be unique (only one tree arrangement is possible). If one restricts the eight words to the Official Scrabble Players Dictionary (1995), there are still at least 28 solutions:

adsorptive d-io-vtr-aesp

beansprout t-ao-bpr-usen

complaints s-cl-aoi-ntpm

dispatcher p-ai-rct-desh

emulations m-ai-uls-nteo

impersonal l-ei-anm-rsop

mailperson l-ei-anm-rsop

outdrawing g-ar-uio-dntw

panegyrics r-ai-cpn-yesg

resampling g-ai-mer-psnl

shadflower h-ao-rew-ldsf

taperingly p-ai-rln-teyg

unchastely l-au-scn-heyt

wardenship w-ai-rns-pdeh

amphiboles s-hl-oia-epmb

blandisher b-ai-snr-hedl

diplomates l-ai-mtd-peso

dogwatches s-ch-aoe-gtwd

headstrong h-ao-ren-tdsg

inculpates p-au-cnl-etsi

neutralism m-ai-ulr-ntes

outlandish s-ah-iuo-nltd

polyanthus s-hl-uoa-ntpy

restacking r-ai-tcn-sekg

stewarding d-ai-wrn-steg

tradesfolk f-ao-trl-sedk

underplays p-au-rln-desy

restamping g-ai-mer-psnt

The obvious generalization is to arrange 15 letters in a tree to generate 16 different words. Here it seems advisable to relax the three conditions. Again drawing words from the OSPD, one can construct several families of solutions. For example, the first solution places B E in place of the + + along the left edge of the tree, and A N in place of the--along the right edge of the tree.


Editor's Note: Many more solutions are possible if one enlarges the word stock. For example, using Webster's Second the letter tree s-ht-eao-lnrw-fdeks (shelf, sheld, shend, shene, shand, shane, share, shark, stand, stane, stare, stark, store, stork, stowk, stows) has at least 168 relatives in its extended family. It may be time to issue a challenge to the reader to find a letter tree consisting of 32 different six-letter words. Even a letter tree consisting of 64 seven-letter words may be possible if a sufficiently large set of words is used (Rex Gooch?).


Eagan, Minnesota
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Author:Iverson, Eric
Publication:Word Ways
Date:May 1, 2005
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