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).

1

http://dx.doi.org/10.1090/mawrld/005/01