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1. SQUARES , HEXAGONS, AND TRIANGLE S
The importan t fac t fo r no w is : Today' s Lega l Move s leav e th e gri d lookin g
exactly th e same . W e sa y "Today' s Lega l Moves " becaus e o n othe r day s w e
might choos e a differen t collectio n o f Lega l Moves . Fo r example , Today' s Lega l
Moves don' t permi t u s t o tur n o r rotat e th e grid , bu t i n th e futur e w e ma y
permit thos e move s also .
Now, sinc e th e Lega l Move s don' t d o anythin g t o th e grid , w e hav e t o ad d
something t o mak e i t interes t ing. Suppos e w e colo r on e square :
Let's se e wha t happen s whe n w e appl y ou r Lega l Moves . Som e exampl e
moves are :
a) Left 2
b) Down 1
c) Up 1 then Right 3.
Think o f th e squar e a s connecte d t o th e grid : whe n w e mov e th e grid , th e
square moves along with it. A s we do each of the moves a, b , c, the squar e move s
to th e place s marke d below :
For aj , we moved the original square two spaces left. The n fo r bj , we moved
the original square on e spac e down . Tha t is , w e star t fres h eac h time , an d al l
Legal Moves refer t o th e origina l beginnin g position . I f you think o f the origina l
square a s a rubber stam p covere d wit h ink , colorin g the squar e i t land s on , the n
a Lega l Mov e jus t teli s yo u whic h squar e t o color . Eac h Lega l Mov e color s a
square, an d thi s lead s t o th e followin g question :
Question: Wha t happen s afte r w e appl y al l possibl e Lega l Move s t o th e
original square ?
There ar e infinitel y man y Lega l Moves , bu t w e ca n stil l imagin e the y hav e
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