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 on forever . Thin k o f a giant wal l with thi s a s the wallpape r patter n o n it . Now imagin e tha t w e took 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 exactly th e same . Whereve r ther e was a dot, ther e still 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 leave 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 to take its place, and anothe r ro w below that, an d s o on . I t i s key tha t th e gri d goe s on forever ther e i s always anothe r row to take th e plac e o f one that jus t move d up . Likewise, if we move the entire grid left , right , up , or down a whole numbe r of units, then th e gri d will end u p lookin g exactly 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
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