Squares, Hexagons , an d Triangle s
1.1 Th e squar e grid . Imagin e a n infinitel y larg e gri d o f points .
We ca n onl y dra w a smal l par t o f th e picture , s o yo u hav e t o imagin e i t
goes o n forever . Thin k o f a gian t wal l wit h thi s a s th e wallpape r patter n o n it .
Now imagin e tha t w e too k th e whol e gri d an d move d everythin g on e uni t
up. I t ma y hel p t o clos e you r eye s a s yo u imagin e this . Af t er shiftin g th e gri d
up, i t look s exactl y th e same . Whereve r ther e was a dot, ther e stil l is a dot , an d
no do t ha s appeare d wher e ther e wasn' t on e previously . I n th e diagra m above ,
it appear s tha t movin g everythin g u p on e spac e will leav e a n empt y ro w a t th e
bottom. Bu t remember , th e gri d keep s goin g o n forever , an d ther e i s anothe r
row below that on e which will move up t o tak e it s place, an d anothe r ro w belo w
that, an d s o on . I t i s key tha t th e gri d goe s o n forever ; ther e i s alway s anothe r
row t o tak e th e plac e o f on e tha t jus t move d up .
Likewise, if we move th e entir e grid left , right , up , o r dow n a whole numbe r
of units , the n th e gri d will en d u p lookin g exactl y th e same . Wit h thi s i n min d
we set th e following :
Today's Lega l Moves :
1) Move the entire grid a whole number of units up or down.
2) Move the entire grid a whole number of units left or right.
3) Any combination of 1) and 2).
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http://dx.doi.org/10.1090/mawrld/005/01
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